Yu. S. Kivshar- Bright and dark spatial solitons in non-Kerr media

I N V I T E D P A P E R

Bright and dark spatial solitonsin non-Kerr media

Y U . S . K I V S H A R

Australian Photonics Cooperative Research Centre, Optical Sciences Centre,Research School of Physical Sciences and Engineering, The AustralianNational University, Canberra ACT 2000, Australia

An overview of the theory of self-guided optical beams, spatial optical solitons supportedby non-Kerr non-linearities, is presented. This includes bright and dark solitons in opticalmedia with intensity-dependent non-linear response as well as two-component solitarywaves supported by parametric wave mixing in quadratic or cubic media. The propertiesof non-linear spatially localized waves are discussed for qualitatively different types ofsoliton bearing non-integrable non-linear models, including the scalar model describedby a generalized non-linear SchroÈdinger equation and the models of the second- andthird-harmonic generation. Special attention is paid to the recent advances of the theoryof soliton stability and soliton internal modes.

1. IntroductionRecent years have shown increased interest from di�erent experimental and theoreticalgroups in the study of self-guided optical beams that propagate in slab waveguides or bulknon-linear media without supporting waveguide structures. Such beams are commonlyreferred to by physicists as spatial optical solitons even though they do not possess allproperties of the formal mathematical de®nition of solitons, spatially localized non-linearwaves of integrable non-linear models. The property of integrability is valid only for theso-called Kerr solitons, the �1� 1�-dimensional beams of cubic non-linearity described inthe paraxial approximation.

Simple physics explains the existence of spatial optical solitons in a generalized self-focusing non-linear medium. First, we recall the physics of optical waveguides (see, forexample [1] and references therein). Optical beams have an innate tendency to spread(di�ract) as they propagate in a homogeneous medium. However, the beam's di�ractioncan be compensated for by beam refraction if the refractive index is increased in the regionof the beam. An optical waveguide is an important means to provide a balance betweendiffraction and refraction if the medium is uniform in the direction of propagation. Thecorresponding propagation of the light is con®ned in the transverse direction of thewaveguide, and it is described by the so-called linear guided modes, spatially localizedeigenmodes of the electric ®eld in the waveguide.

Optical and Quantum Electronics 30 (1998) 571±614

0306±8919 Ó 1998 Chapman & Hall 571

As was discovered a long time ago [2], a similar e�ect, i.e. suppression of di�ractionthrough a local change of the refractive index, can be produced solely by non-linearity. Ashas been well established in many experiments (for the most recent overview of experi-mental observations of di�erent types of spatial optical solitons see [3]), some materialscan display considerable optical non-linearities when their properties are modi®ed by thelight propagation. In particular, if the non-linearity leads to a change of the refractiveindex of the medium in such a way that it generates an e�ective positive lens to the beam,the beam can become self-trapped and propagates unchanged without any externalwaveguiding structure [2]. Such stationary self-guided beams are called these days spatialoptical solitons, they exist with pro®les of a certain form allowing a local compensation ofthe beam diffraction by the non-linearity-induced change in the material refractive index.

Until recently, the theory of spatial optical solitons has been based on the non-linearSchroÈ dinger (NLS) equation with a cubic non-linearity, which is exactly integrable bymeans of the inverse scattering (IST) technique [4]. Generally speaking, the integrabilitymeans that any localized input beam will be decomposed into stable solitary waves (orsolitons) and radiation, and also that interaction of solitons is elastic. From the physicalpoint of view, the integrable NLS equation describes the �1� 1�-dimensional (i.e. onetransverse and one longitudinal dimensions) beams in a Kerr (cubic) non-linear medium inthe framework of the so-called paraxial approximation. The cubic NLS equation is knownto be a good model for temporal optical solitons propagating enormous distances alongexisting waveguides, optical ®bres. In application to spatial optical solitons, the cubic NLSequation becomes an inappropriate model. First, for spatial optical solitons much higherinput powers are required to compensate for diffraction, meaning that the refractive indexexperiences large deviations from a linear (Kerr) dependence. Second, as was recognized along time ago (see, for example [5]), radially symmetric stationary localized solutions ofthe �2� 1�-dimensional NLS equation are unstable and may display collapse (for acomprehensive overview of the wave collapse phenomenon see [6]). To deal with realisticoptical models, saturation had been suggested as a way to stabilize such a catastrophicself-focusing and produce stable solitary waves of higher dimensions (see, for example [7,8]), the effect also discussed in some other applications [9, 10]. Accounting for this effectimmediately leads to non-integrable models of generalized non-linearities, not possessing theproperties of integrability and elastic soliton collisions. Another mechanism of non-Kerrnon-linearities and enhanced non-linear properties of optical materials is a resonant,phase-matched interaction between the modes of different frequencies. In this latter case,multi-component solitary waves are created, and the mutual beam coupling can modifydrastically the properties of single beams, as it occurs in the case of the so-called quadraticsolitons of cascaded non-linearities (see, for example, the recent excellent review paper oncascaded effects in v�2� materials [11]).In spite of the fact that, generally speaking, there exist no universal analytical tools for

analysing solitary waves and their interactions in non-integrable models, recent advancesof the theory suggest that many of the properties of optical solitons in non-Kerr media aresimilar, and they can be approached with the help of rather general physical concepts. Oneof the recent advances in this ®eld was the developments of the unifying conceptualapproach (see [12] and references therein) based on the notion that a self-guided beaminduces a waveguide. This allows one to employ the theory of linear guided waves sup-ported by prescribed waveguiding structures to understand, at least qualitatively, theproperties and physics of self-guided waves existing without supporting waveguides.

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The observation that a beam creates a waveguide and guides itself in it is, as a matter offact, not new, this concept is known from the very ®rst days of non-linear optics [13], andit was also clearly stated in some earlier papers on the light self-focusing [2, 14]. The sameconcept, in a di�erent context, has been rediscovered often since that time (see, forexample [15] as a typical example), and it was also realized recently that this concept alonecould be useful to make novel predictions, extending our knowledge of self-guided beamsin non-Kerr media [12].

Some of the recent advances in the theory of self-guided beams based on the physics ofthe induced waveguides suggest that the behaviour of beams in non-Kerr materials arequalitatively similar [16]. This has led to a number of predictions about spatial solitonsand their dynamical evolution which do not follow from the analysis based solely on the�1� 1�-dimensional integrable cubic NLS equation. From this perspective we understandthat there is no simple mapping between temporal and spatial optical solitons. Spatialsolitons are a much richer and more complicated phenomenon, and this has already beendemonstrated by a number of elegant experiments in this ®eld.

In particular, it has recently been demonstrated experimentally, that self-guided beamscan be observed in materials with strong photorefractive and photovoltaic e�ect [17±20],in vapours with a strong saturation of the intensity-dependent refractive index [21, 22],and also as a result of the mutual self-focusing due to the phase-matched three-wavemixing in quadratic (or v�2�� non-linear crystals [23]. In all these cases, propagation of self-guided waves is observed in non-Kerr materials which are described by the models moregeneral than the cubic NLS equation.

The main purpose of this paper is to discuss the most recent advances in the theory ofself-guided beams in non-integrable models of non-Kerr non-linearities. Because a com-prehensive review of Kerr spatial solitons is available in the same issue [24], here weconcentrate mostly on the general properties of solitary waves of non-Kerr (e.g. saturableor v�2�� non-linearities described by non-integrable non-linear models. We also discuss thedifferent physical mechanisms leading to such non-Kerr solitary waves, including theeffect of parametric interaction between phase-matched beams of different frequencies.

2. Models of non-Kerr spatial solitons2.1. Why temporal and spatial solitons are differentBecause the phenomenon of the long-distance propagation of temporal optical solitons inoptical ®bres is known to a much broader community of researchers in optics and non-linear physics, ®rst we emphasize the di�erence between spatial and temporal solitons.Indeed, stationary beam propagation in planar waveguides has been considered somewhatsimilar to the pulse propagation in ®bres. This is a direct manifestation of the so-calledspatio-temporal analogy in wave propagation [25, 26], meaning that the propagation co-ordinate z is treated as the evolution variable and the spatial beam pro®le along thetransverse direction, for the case of waveguides, is similar to the temporal pulse pro®le, forthe case of ®bres. This analogy has been employed for many years, and it is based on asimple notion that both beam evolution and pulse propagation can be described by thecubic NLS equation (see, for example, [27, 28]). However, contrary to this widely acceptedopinion, we point out a crucial difference between these two phenomena. Indeed, in thecase of the non-stationary pulse propagation in ®bres, the operation wavelength is usuallyselected near the zero of the group-velocity dispersion. This means that the absolute valueof the ®bre dispersion is small enough to be compensated by a week non-linearity such as

Bright and dark spatial solitons

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that produced by the (very weak) Kerr effect in optical ®bres which leads to a relative non-linearity-induced change in the refractive index very small. Therefore, non-linearity in suchsystems is always weak and it should be well modelled by the same form of the cubic NLSequation, which is known to be integrable be means of the IST technique [4, 29, 30].However, for very short (fs) pulses the cubic NLS equation describing the long-distancepropagation of pulses should be corrected to include some additional (but still small)effects such as higher-order dispersion, Raman scattering, etc. (for example, [31, 32]). Allthese corrections can be taken into account by a perturbation theory (for an overview ofthe perturbation theory of solitons see [33]). Thus, in ®bres non-linear effects are veryweak and they become important only when dispersion is small (near the zero-dispersionpoint) affecting the pulse propagation over large distances (of order of hundreds of metresor even kilometres).

In contrary to pulse propagation in optical ®bres, the physics underlying stationarybeam propagation in planar waveguides and bulk media is di�erent. In this case the non-linear change in the refractive index should compensate for the beam spreading caused bydi�raction which is not a small e�ect. That is why to observe spatial solitons, much largernon-linearities are usually required, and very often such non-linearities are not of the Kerrtype (e.g. they saturate at higher intensities). This leads to the models of generalized non-linearities with the properties of solitary waves different from those described by theintegrable cubic NLS equation. For example, unlike the solitons of the cubic NLS equa-tion, solitary waves of generalized non-linearities may be unstable, they also show someinteresting features, such as fusion due to collision, inelastic interactions and spiralling in abulk, wobbling, amplitude oscillation, etc. Propagation distances usually involved in thephenomenon of beam self-focusing and spatial soliton propagation are of the order ofmillimetres or centimetres. As a result, the physics of spatial solitary waves is very rich andit should be understood in the framework of the theory of non-integrable models.

It is worth noting that such a distinction between spatial and temporal solitary waves isless clear for the so-called gap solitons, non-linear localized modes exciting in a frequencygap produced by the Bragg re¯ection of light on periodic structures such as ®bre gratings[34] or multilayered dielectric mirrors such as Bragg re¯ectors [35]). Because of a relativelylarge dispersion induced by the periodic variation of the refractive index, the soliton periodin ®bre gratings is about 1±10 cm, requiring much larger input powers (see, for example,[34] for an overview of the most recent results), and the propagation distances becomecomparable with those for spatial solitons. That is why, an experimental realization of theconcept of light bullets [36] unifying the spatial and temporal self-trapping, would requirematerials with large dispersion that can be achieved in the periodic structures.

2.2. Basic equationsTo describe spatial optical solitons in the framework of the simplest scalar model withnon-resonant non-linearities, we consider the propagation of a monochromatic scalarelectric ®eld E in a bulk optical medium with an intensity-dependent refractive index,n � n0 � nnl�I�, where n0 is the linear refractive index, and nnl�I� describes the variation inthe index due to the ®eld with the intensity I � jEj2. The function nnl�I� is assumed to bedependent only on the light intensity only, and it may be introduced phenomenologically.Solutions of the governing Maxwell's equation can be presented in the form

E�~R?; Z; t� � E�~R?; Z� exp�ib0Z ÿ ixt� � c.c. �1�574

YU. S. Kivshar

where c.c. denotes complex conjugate, x is the source frequency, and b0 � k0n0 � 2pn0=kis the plane-wave propagation constant for the uniform background medium, in terms ofthe source wavelength k � 2pc=x; c being the free-space speed of light. Usually, the spatialsolitons are discussed for two geometries. For the beam propagation in a bulk, we assumea �2� 1�-dimensional model, so that the Z-axis is parallel to the direction of propagation,and the X - and Y -axes are two transverse directions. For the beam propagation in a planarwaveguide, the e�ective ®eld is found by integrating Maxwell's equations over thetransverse structure de®ned by the waveguide con®nement, and therefore the modelbecomes �1� 1�-dimensional.

The function E�~R?; Z� describes the wave envelope which in the absence of non-linearand di�raction e�ects E would be a constant. If we substitute Equation 1 into the two-dimensional, scalar wave equation, we obtain the generalized non-linear parabolic equa-tion,

2ik0n0@E

@Z� @2E

@X 2� @

2E

@Y 2

� �� 2n0k20nnl�I�E � 0 �2�

In dimensionless variables, Equation 2 becomes the well-known generalized NLS equation,where local non-linearity is introduced by the function nnl�I�.

For the case of the Kerr (or cubic) non-linearity we have nnl�I� � n2I ; n2 being thecoe�cient of the Kerr e�ect of an optical material. Now, introducing the dimensionlessvariables, i.e. measuring the ®eld amplitude in the units of k0�n0jn2j�1=2 and the propa-gation distance in the units of k0n0, we obtain the �2� 1�-dimensional NLS equation in thestandard form,

i@W@z� 1

2

@2W@x2� @

2W@y2

� �� jWj2W � 0 �3�

where the complex W stands for a dimensionless envelope, and the sign �� is de®ned bythe type of non-linearity, self-defocusing (`minus', for n2 < 0) or a self-focusing (`plus', forn2 > 0).

For propagation in a slab waveguide, the ®eld structure in one of the directions, say Y , isde®ned by the linear guided mode of the waveguide. Then, the solution of the governingMaxwell's equation has the structure

E�~R?; Z; t� � E�X ;Z�An�Y � exp� ib�0�n zÿ ixt� � � � � �4�where the function An�Y � describes the corresponding fundamental mode of the slabwaveguide. Similarly, substituting this ansatz into Maxwell's equations and averaging overY , we come again to the renormalized equation of the form 3 with the Y -derivativeomitted, which in the dimensionless form becomes the standard cubic NLS equation

i@W@z� 1

2

@2W@x2� jWj2W � 0 �5�

Equation 5 coincides formally with the equation for the pulse propagation in dispersivenon-linear optical ®bres, and it is known to be integrable by means of the inverse scat-tering transform (IST) technique.

For the case of non-linearities more general than the cubic one, we should consider thegeneralized NLS equation,

575


i@W@z� @

2W@x2� f �jWj2�W � 0 �6�

where the function f �I� describes a non-linearity-induced change of the refractive index,usually f �I� / I for small I .

2.3. Nonresonant optical non-linearitiesThe generalized NLS equation 2 (or Equation 6) has been considered in many papers foranalysing beam self-focusing and the properties of spatial bright and dark solitons (see, forexample, [37±55]. All types of non-Kerr non-linearities discussed in relation to the exis-tence of solitary waves in non-linear optics can be divided, generally speaking, into threemain classes: (i) competing non-linearities, e.g. focusing (defocusing) cubic and defocusing(focusing) quintic non-linearity (see, for example, [39, 40, 43, 44, 50]) and also general-ization to a power non-linearity (for example, [53±55]); (ii) saturable non-linearities (see,for example, [47±49] and also [51, 52]), and (iii) transiting non-linearities (see, for example,[39±41, 46]).

Usually, the non-linear refractive index of an optical material deviates from the linear(Kerr) dependence for larger light intensities. Non-ideality of the non-linear opticalresponse is known for semiconductor (for example, AlGaAs, CdS, CdS1ÿxSex� waveguidesand semiconductor-doped glasses (see, for example, [56±58]). Larger deviation from theKerr non-linearity is observed for non-linear polymers. For example, recently the mea-surements of a large non-resonant non-linearity in single crystal PTS (p-toluene sulpho-nate) at 1600 nm [59, 60] revealed a variation of the non-linear refractive index with theinput intensity which can be modelled by competing, cubic-quintic non-linearities,

nnl�I� � n2I � n3I2 �7�This model describes a competition between self-focusing �n2 > 0�, at smaller intensities,and self-defocusing �n3 < 0�, at larger intensities. Similar models are usually employed todescribe the stabilization of wave collapse in the �2� 1�-dimensional NLS equation (forexample, [61] and references therein).

In a more general case, the models with competing non-linearities can be described by apower-law dependence on the beam intensity,

nnl�I� � npIp � n2pI2p �8�where p is a positive constant and usually npn2p < 0.

Models with saturable non-linearities are the most typical ones in non-linear optics. Forhigher powers, saturation of the non-linearity has been measured in many materials andconsequently the maximum refractive index change has been reported (see, for example,[62]). We do not linger on the physical mechanisms behind the saturation but merely notethat it exists in many non-linear media being described by phenomenological modelsintroduced more than 25 years ago (see, for example, [7, 8, 63, 64]). The effective gener-alized NLS equation with saturable non-linearity is also the basic model to describe therecently discovered �1� 1�-dimensional photovoltaic spatial solitons in photovoltaic±photorefractive materials such as LiNbO3 (see [51]). Unlike the phenomenological modelsusually used to describe saturation of non-linearity, for the case of photovoltaic solitonsthis model can be justi®ed rigorously [51, 52].

576

YU. S. Kivshar

Several models are usually considered to describe saturating non-linearities. From ageneral point of view, the function nnl�I� describing the saturating non-linearity should becharacterized by three independent parameters: the saturation intensity, Isat, the maximumchange in the refractive index, n1, and the Kerr coef®cient n2 which appears for small I . Inparticular, the phenomenological model

nnl�I� � n1 1ÿ 1

�1� I=Isat�p� �

�9�

satis®es these criteria, provided n2 � n1p=Isat. In the particular case p � 1, model 9reduces to the well-known expression derived from the two-level model, which is used veryfrequently. For the case p � 2, model 9 possesses localized solutions for bright and darksolitons in an explicit analytical form [48, 49].

Finally, bistable solitons introduced by Kaplan [39, 40] usually require a special type ofthe intensity-dependent refractive index which changes from one type to another one, e.g.it varies from one kind of the Kerr non-linearity, for small intensities, to another kindwith a different value of n2, for larger intensities. This type of non-linearity is known tosupport bistable dark solitons [41, 42] as well. One of the simplest models of suchtransiting non-linearities describes a change from one type of the Kerr dependence to theother one, i.e.

nnl�I� � n�1�2 I I < Icr

n�2�2 I I > Icr

(�10�

A smooth transition of this kind can be modelled by the function [41]

nnl�I� � n2If1� a tanh�c�I2 ÿ I2cr��g �11�where for I � Icr; nnl�I� ' n�1�2 I , where n�1�2 � n2�1ÿ a tanh2�cI2cr��, and for I � Icr,nnl�I� ' n�2�2 I , where n�2�2 � �1� a�. Unfortunately, examples of non-linear optical mate-rials with such dependencies are not yet known, but the bistable solitons possess attractiveproperties useful for their possible futuristic applications in all-optical logic and switchingdevices.

At last, we would like to mention the model of logarithmic non-linearity,n2�I� � n2

0 � � ln�I=I0�, that allows close-form exact expressions not only for stationaryGaussian beams (or Gaussons, as they were introduced in [65]), but also for periodic andquasi-periodic regimes of the beam evolution [66]. The main features of this model are thefollowing: (i) the stationary solutions do not depend on the maximum intensity (quasi-linearization) and (ii) radiation from the periodic solitons is absent (the linearized problemhas a discrete spectrum only). Such unusual properties persist in any dimension [65, 66].Similar localized solutions exist in a model of highly non-local media [67] allowing asimple analysis based on a linear equation for a quantum harmonic oscillator.

2.4. Validity of the NLS modelsAs we have discussed above, the scalar generalized NLS equation is a rather universalmodel for spatial optical solitons. It is derived from the ®rst principles on the basis of verygeneral assumptions about non-linear properties of an optical medium. However, the NLSmodel may fail in a number of cases (see also [68] for examples from other ®elds), andtherefore one should be aware of the validity limits of this simple model, introducing

577


instead more general and more appropriate models for describing self-guided beams. Herewe discuss two such generalizations.

First of all, as standard derivation of the NLS equation is based on the so-called multi-scale asymptotic technique, sometimes called the reductive perturbation method (forexample, [69, 70]). It assumes non-resonant non-linearities when the most importanteffects are described by an envelope of the ®eld of the fundamental frequency x prop-agating with the carrier wave number k. All higher-order harmonics, even if they havebeen excited a priori, are assumed to be very small and, therefore, they do not modify the®eld evolution of the main frequency which, in the case of the cubic non-linearity, isdescribed by the NLS equation. However, when some multiple frequencies are generated,they may strongly affect the wave propagation at the fundamental harmonic providedthe so-called matching conditions are satis®ed. For example, strong interaction betweenthe main frequency x and two other frequencies x1 and x2 occurs provided x � x1 � x2

and the phase mismatch Dk � k ÿ �k1 � k2� vanishes. This kind of three-wave mixing ispossible in a medium where the lowest-order non-linearity is quadratic. When themedium non-linearity is cubic, wave coupling is possible in the form of a four-wavemixing process. When any such resonance condition is satis®ed, the envelope of thefundamental ®eld becomes strongly coupled to a secondary ®eld (or more than one ®eld)and the single NLS equation is no longer valid. Instead, a coupling between the modesmay support multi-component solitary waves which differ drastically from the conven-tional solitons of the scalar NLS equation. Below we consider two such examples, thetwo-wave mixing solitons due to second-harmonic interaction (quadratic media) andthird-harmonic interaction (cubic media). Importantly, the intermode interaction pro-vides an ef®cient mechanism for non-Kerr non-linearities.The second class of problems when the NLS model should be generalized is closely

related to spatial optical solitons described by non-Kerr non-linearities. Indeed, it is wellknown that the NLS equation with non-linearity stronger than cubic, e.g. a power-lawfocusing non-linearity juj2qu, has localized solutions which blow-up, so that a singularityappears at ®nite z (see [6] for a review). This phenomenon occurs for negative values of thesystem Hamiltonian under the condition qD � 2, where q is the power of the non-linearityand D stands for the �D� 1�-dimensional model (see, for example, [71]). Blow-up(or collapse) in ®nite z means that the NLS model of this dimension fails as an envelopeequation since it breaks the scales on which it was derived in the framework of the multi-scale asymptotic technique. For spatial solitons this condition means that if D � 2, thenthe cubic non-linearity juj2u is already suf®cient to induce collapse. If D � 1, then oneneeds the quintic (or higher-order) non-linearity to induce collapse. Blow-up indicates alsothat the primary NLS model should be corrected, e.g. by taking into account the effects ofnon-paraxiality in the beam self-focusing [72, 73]. Because the catastrophic beam self-focusing is also corrected by any kind of saturable non-linearity, we do not discuss indetail the phenomenon of the wave collapse. Also, all solitons of a quadratic medium donot display collapse.

3. Existence of solitary waves3.1. Stationary soliton solutionsFrom the mathematical point of view, non-linear self-guided waves are described bystationary localized solutions of the corresponding partial di�erential equations. Scalarbright solitons have the form [2, 4, 16]

578

YU. S. Kivshar

E�~r; z� � E�~r; b� exp�ibz� �12�where ~r � �x; y� and E�~r; b� is real, and they depend on the propagation constant b.Propagation constant b de®nes both the shape and stability of the stationary solutions.The problem for the stationary pro®les E�~r; b� does not involves z and it reduces to asimple eigenvalue equation described by one or several coupled ordinary di�erentialequations (ODEs) in the case of soliton propagation in waveguides, for the �1� 1�-dimensional geometry, and in the case of the beams of circular symmetry, for the �2� 1�-dimensional geometry. Pro®les of bright solitary waves, i.e. those with vanishingasymptotics, can be found as separatrix trajectories which start from and return to acritical point corresponding to the vanishing boundary conditions, i.e. E�~r; b� ! 0 andrE�~r; b� ! 0 for j~rj ! 1. Therefore, the existence regions for solitary wave solutions canbe found by the analysis of the existence and types of critical points. For scalar solitons,the corresponding ODE can often be solved analytically, because the correspondingequation describes an effective system with one degree of freedom. Additionally, thesimple structure of the critical points does not allow the existence of complicated (e.g.multi-hump) localized waves.

Two-component solitons, or more general cases of vector solitons, were ®rst studied byBerkhoer and Zakharov [74], and later Manakov [75] who demonstrated the existenceof exact analytical solutions for the special kind of two-component solitary waves, theso-called Manakov solitons. Vector solitary waves were discussed for both temporal andspatial solitons [76±79] and they were also demonstrated experimentally [80, 81]. For thespatial case, a vector soliton brings a distinctly new concept. The vector soliton is a veryparticular case of a more general `dynamic soliton', which can be described as two coupledmodes of an effective waveguide that they induce [79].

When solitons consist of several interacting modes, the stationary waves aredescribed by a system of coupled (and generally non-integrable) ordinary differentialequations which may possess many exotic (including multi-hump) localized solutions[79, 82, 83], solitons with oscillating tails, etc. Usually, all these exotic solitons areunstable. The existence of such solitons is a direct consequence of the complex criticalpoints of the corresponding ODE dynamical system describing the stationary localizedmodes.

The structure of dark solitons is more complicated (see, for example, the recent reviewpaper on dark solitons [84]). From the physical point of view, dark solitons are re¯ec-tionless radiation modes of the waveguides they induce [85], they also have a localizedshape similar to bright solitons but with a complex envelope and non-vanishingasymptotics. For example, in the case of the �1� 1� geometry, a plane dark soliton hasthe form

E�x; z� � E�n; V � exp�ibz� exp�i/�n; V �� 13�where E�n; V � and /�n; V � are the real amplitude and phase of the wave, and n � xÿ Vz.This time, the propagation constant b is determined by the background ®eld, E�n; V � ! E0

for jnj ! 1. Parameter V de®nes the steering angle (or `velocity') of the dark soliton. Themain feature of a dark soliton is its non-trivial phase, so that the parameter V is related tothe total phase shift D/�V � � /��1� ÿ /�ÿ1�. The stationary pro®le of a dark soliton isdescribed by two coupled ODEs for E�n� and /�n�, and the corresponding separatrixtrajectory connects two critical points corresponding to non-vanishing boundary condi-

579


tions [84]. The structure of dark solitons of circular symmetry in �2� 1� dimension�, i.e.vortex solitons, is more complicated [84, 88],

E�~r;u; z� � E�r� exp�ibz� exp�iu� �14�where b is the background propagation constant, r � j~rj is the radius and u is the azi-muthal angle in the cylindrical system of coordinates. As above, the vortex pro®le isde®ned by the corresponding ODE for the function E�r�. Apart from the vortex soliton, noother spatial soliton is mathematically stable in a Kerr (cubic) non-linearity.

Therefore, the regions of the existence of solitary waves solutions with decaying (bright)or constant (dark) asymptotics can be determined by the analysis of the critical points ofthe corresponding system of ODEs. For scalar (bright and dark) solitons, the ODEs areintegrable (an effective mechanical system with one degree of freedom) and the structure ofthe critical points usually does not allow the existence of complicated (e.g. multi-hump)solutions. For vector solitons, e.g. two different orthogonal modes, the stationary wavesare described by a system of coupled (and generally non-integrable) ODEs which maydisplay many exotic solutions which are usually unstable. The existence of such solitons is adirect consequence of the complex structure of the critical points of the correspondingdynamical system describing the stationary localized modes.

3.2. Concept of an induced waveguideThe notion that a self-guided beam can be regarded as a mode of the waveguide it inducesis a useful tool for understanding spatial optical solitons. This concept, as a matter of fact,is somewhat self-evident, and it has been known for more than 30 years from the ®rstprediction of self-guided beams [2, 13, 14]. Recently, Snyder and co-workers [12, 79, 89]have suggested this concept for the so-called self-consistency method (also called the linearperspective approach) and interpretation of both stationary and even some non-stationaryself-guided beams.

The linear perspective concept is based on the fact that all spatial solitary waves arequalitatively the same. In particular, all stationary self-guided beams can be treated as themodes of a (linear) axially uniform waveguide. This waveguide is induced by the inter-action of light with the non-linear medium and is in general anisotropic. This elementaryconcept allows us to borrow physics and exact solutions for guided waves directly from thepages of waveguide theory providing a physical insight into the physics of guided waves.In the simplest case, a soliton is one mode of the waveguide it induces [2, 79]; moregenerally, it is any two (or more) modes of the induced waveguide [79] which explains thecoexistence of different classes of multi-component solitons such as dark and bright.Vector solitons are the special case when the modes are degenerate.

4. Solitons in v(2) media4.1. Models for quadratic solitonsWhen the fundamental frequency becomes phase matched with one of its harmonics, theNLS equation fails and the wave propagation should be described by some other models.In the case of quadratic (or the so-called v�2�) non-linearity, the simplest e�ect of resonant

* As spatially localized solutions of the defocussing cubic NLS equation, vortex solitons were introduced, for the

®rst time, in a paper by Ginzburg and Pitaevski [86] (see also [87]), as topological excitations in super ¯uids.

However the term `vortex' had been used much earlier for di�erent (linear) physical processes and for di�erent

de®ning equations.

580

YU. S. Kivshar

wave interaction is the generation of the second harmonics 2x by the fundamental fre-quency x, the process being a particular case of a more general process of three-wavemixing. Such an interaction is ef®cient provided the matching conditions between the wavepropagation constants are satis®ed. This is well known in the theory of the second-har-monic generation (SHG) (see, for example, [90, 91]). For example, in an anisotropicmedium, for any wave vector direction~k=k, two di�erent corresponding values of k�x� canbe found. In other words, for any direction of propagation there are two normal waves(which are called ordinary and extraordinary waves) which have different polarizations andtravel with di�erent phase velocities. For the ordinary wave the direction of wave vector~kcoincides with the direction of the Poynting vector~s (i.e. with the direction of energy ¯ow),whereas for the extraordinary wave the directions of ~k and~s do not coincide.

To derive the model of three-wave mixing in a di�ractive medium, one should considerparametric interaction between three stationary quasi-plane monochromatic waves withthe envelopes Ej (where j � 1; 2; 3) and assume x1 � x2 � x3 with the corresponding wavevectors to be nearly matched, i.e. k1�x1� � k2�x2� ÿ k3�x3� � Dk, where Dk � k3. If allthree vectors ~kj have the same direction, there is no phase velocity walk-o�. However, ifsome of the three waves are extraordinary then their energy ¯ows diverge and this shouldbe taken into account in the structure of the slowly varying envelopes Ej. Choosing thez-axis as the direction of~kj, and the x-axis being in the plane de®ned by~kj and the directionof the energy walk-o�, we consider the electric ®eld as a sum of three ®elds of the reso-nantly interacting frequencies. As a result, in the approximation of slowly varying enve-lopes, we can derive a system of equations that describes the type II SHG,

2ik1@E1

@zÿ 2ik1q1

@E1

@x� @

2E1

@x2� @

2E1

@y2� 8px2

1

c2~v�2�1 E3E�2 exp�ÿiDkz� � 0

2ik2@E2

@zÿ 2ik2q2

@E2

@x� @

2E2

@x2� @

2E2

@y2� 8px2

2


2ik3@E3

@zÿ 2ik3q3

@E3

@x� @

2E3

@x2� @

2E3

@y2� 8px2

3

c2~v�2�3 E1E2 exp�iDkz� � 0

�15�

Equations 15 describe the case when the spatial walk-o� of all waves occurs in the sameplain. Formally, this is true only for single-axis crystals, but the corresponding general-ization is trivial. For a slab waveguide, the structure of the linear guided modes in thedirection of the trapping provided by the waveguide is known. Then, using an approxi-mate separation of variables, Ej;Ej�x; y; z� � Fj�y�Ej�x; z�, and integrating out the depen-dencies in y, we can obtain the similar system of non-linear coupled equations but with thenormalized (scaled) coe�cients.

In the limiting case, when x1 � x2 � x3=2, only one characteristic frequency x0 � x1 isinvolved. This requires only one source of coherent radiation at the fundamental fre-quency x0 and a wave of the double frequency 2x0 is generated due to SHG phenomenon(type I SHG). In this case, we put E1 � E2 and, therefore, the spatial solitons due to type ISHG in a v�2� slab waveguide are described by the system of two coupled equations:

2ik0@E1

@z� @

2E1

@x2� 8px2

0


4ik0@E3

@zÿ 4ik0q3

@E3

@x� @

2E3

@x2� 32px2

0

c2~v�2�3 E2

1 exp�iDkz� � 0

�16�

581


where x0; k0 and E1 are the frequency, wave number and the electric ®eld intensity of the®rst harmonic wave, respectively; E3 and q3 are the electric ®eld intensity and the walk-o�angle for the second harmonic wave, and Dk � 2k0 ÿ k3 is the wave vector mismatch.

In the case of spatial solitons, we normalize Equations 16 measuring the transversecoordinate x in units of the input beam size r0, and the propagation coordinate z, in unitsof the di�raction length zd � r20k2 � r20=c1. Introducing the dimensionless ®elds,E1 � �v�jc1c2j�1=2=�2v1v2r40�

12 exp�ibz�� and E2 � �xjc1j=v1r20� exp�i�2b� D�z�, we ®nally

obtain

i@v@z� r

@2v@x2ÿ bv� wv� � 0

ir@w@zÿ id

@w@x� s

@2w@x2ÿ r�2b� D�w� v2

2� 0

�17�

where D � zdDk; d � d2r0=jc2j; r � sign�c1�; s � sign�c2� and r � jc1=c2j. The dimen-sionless parameter b is proportional to the non-linearity-induced phase velocity shift.

4.2. Solitons in the cascading limitIn the simplest case of no walk-o�, system 17 can be further normalized to scale out thepropagation constant b,

i@v@z� r

@2v@x2ÿ v� wv� � 0

ir@w@z� s

@2w@x2ÿ aw� v2

2� 0

�18�

where the dimensionless parameter a � 2r� rD=b includes the mismatch parameter.Equation 18 is the generic model of v�2� solitons in the absence of walk-o�. Its solutionshave been analysed by many authors, in the (1+1)-dimensional case [92±98] and in a moregeneral (2+1)-dimensional case [99, 100]. Solitary waves in the case of more general, non-degenerated three-wave mixing has also been investigated (see, for example, [101, 102]).Experimental observation of quadratic solitons has been reported for both waveguidegeometry [103] and for two-frequency beam propagation in a bulk [23, 104] (see also areview paper [11]).

It is straightforward to see why (and when) we expect to ®nd spatially localized solu-tions of Equations 18. Indeed, let us consider the limit of large a, which corresponds tolarge positive values of the mismatch Dk. In this case, the second equation of the system 18can be approximately reduced to the form w � v2=�2a�. The substitution of this expressioninto the ®rst equation of system 18 results in the standard NLS equation for the ®rstharmonic,

i@v@z� r

@2v@x2ÿ v� 1

2ajvj2v � 0 �19�

The NLS equation possesses stable bright (at r � �1) or dark (at r � ÿ1) soliton solu-tions. We will call the limit of large a the cascading limit. In this limit the effective Kerr-likebehaviour due to cascaded v�2� effects is clearly seen and the second harmonic componentw is much weaker than the ®rst harmonic v.

Using such a simple reduction to the NLS equation, we may look for a stationary (i.e.z-independent) localized solutions of Equations 18 in the form of an asymptotic series in

582

YU. S. Kivshar

the parameter aÿ1 and ®nd the real functions v�x� and w�x� in the form of asymptoticseries,

v�x� � 2a1=2 sech x� 4saÿ1=2 tanh2 x sech x� � � �w�x� � 2 sech2x� saÿ1�16 sech2xÿ 20 sech4x� � � � �

�20�

for bright solitons at r � �1, andv�s� � 21=2a1=2 tanh s� 21=2saÿ1=2�s sech2sÿ tanh s sech2s� � � � �w�s� � tanh2 s� saÿ1�2s tanh s sech2sÿ 4 sech2s� 5 sech4s� � � � �

�21�

where s � x=21=2, for dark solitons at r � ÿ1.The properties of Kerr solitons of Equation 19 are well known. Existence of the as-

ymptotic solutions 20 and 21 obtained in the cascading limit suggests that for a� 1 thesystem 18 should have stable bright solitons for r � �1; s � �1, and stable dark solitonsfor r � ÿ1; s � �1, similar to Equation 19. However, this conclusion is not satisfactory,because: (i) formal localized solutions 20 and 21 can be non-stationary for system 18 dueto their resonance with linear waves; (ii) in the case of dark solitons, the solutions 21 canalso be unstable due to parametric modulational instability, as has been ®rst demonstratedin [105, 106].

4.3. Families of bright solitonsAs follows from the NLS limit valid for large a, bright solitons of the two coupledequations 18 should exist for r � �1 in the form of one-hump localized pro®les for the realfunctions v�x� and w�x�. Such solutions were ®rst found by Buryak and Kivshar [95, 96, 98]using the numerical shooting technique, for any positive value of a.

Examples of two-wave localized solutions of Equations 18 are presented in Fig. 1a fora � 0:2 and a � 10:0. For a� 1 the maximum amplitude of the fundamental componentvmax is much larger than the similar value wmax for the second harmonic component andthis case corresponds to the asymptotic solution 20, v � �2a1=2 sech x; w � 2 sech2x. Theratio wmax=vmax characterizing the whole family is plotted in Fig. 1a where the ®lled circlecorresponds to the exact solution [92]

v�x� � 21=2w�x� � �3=21=2� sech2�x=2� �22�

that exists at a � 1, and the asymptotic (dashed) curve corresponds to the NLS limit,wmax=vmax � 1=a1=2.

The numerical analysis of the stability of this soliton family based on the direct inte-gration of Equation 18 and the eigenvalue analysis of the corresponding linearizedproblem has shown that both stable and unstable solitons exist, depending on the values ofsystem parameters a and r [107]. In the cascading limit a� 1, the solitons shown in Fig.1a are stable, whereas in the other limit �a! 0� solitons become unstable (e.g. left-bottomsoliton of Fig. 1a is unstable for r � 2:0). In spite of detected instability for a! 0, fora � 1 the parametric solitons are stable even under the action of very strong perturbations.These two-wave solitons can be generated from a rather broad class of initial conditions.Figure 1b shows the soliton generation from a ®rst harmonic sech-form input pulse. Dueto di�raction, the input pulse becomes broader, but it also generates the second harmonicsand, after a rather short transition period, a two-component bright soliton forms. This

583


Figure 1 (a) Characteristic pro®les of two-wave bright solitons at r � �1 and s � �1 as the functions of the

dimensionless mismatch parameter a. The ®lled circle at a � 1 corresponds to the exact solution 22. (b) Self-

trapping of an initial sech-pro®le input into a two-wave bright soliton. Shown are the peak intensities of the

fundamental and second harmonics.

584

YU. S. Kivshar

kind of behaviour is possible only due to the existence of the continuous family of stablebright solitons.

Additionally to the one-hump localized solutions described above, the numericalanalysis indicates the existence of continuous (in a) families of two-hump (and even multi-hump) bright solitons, which can be treated as bound states of one-hump solitons [108,109]. These solitons exist only for 0 < a < 1. At a! 1 the distance between the neigh-bouring solitons increases to in®nity. Numerical stability analysis indicates that thesemulti-hump bright solitons are unstable and either split into partial stable solitons ordisintegrate completely, for su�ciently small values of a where stable single solitons do notexist.

In spite of the fact that in the cascading limit the e�ective NLS equation 19 does notdepend on the sign s, localized solutions are very di�erent for s � �1 and s � ÿ1 inEquations 18. A simple analysis of the soliton tails indicates that for s � ÿ1 one-humpsolitons do not exist due to the resonance with linear waves. However, the numericalanalysis still allows one to ®nd bound states of such solitons existing as discrete sets oftwo- (and multi-)soliton radiationless states where radiation is suppressed outside, butexists between the solitons in the form of trapped oscillations. Such bound states ofsolitons in the presence of radiation may occur in other non-linear systems where singlesolitons in the presence of radiation may occur in other non-linear systems where singlesolitons do not exist (see, for example, [110] and references therein). Figure 2 gives theresults related to such two-soliton states as solutions of Equations 18 at r � �1 ands � ÿ1.

It is interesting, that in this case one solution is also known in an explicit analytical form[111]. It exists at a � 2 and has the form,

v�x� � 6�21=2� tanh x sech x w�x� � 6 sech2x �23�Solution 23 represents one member of the family of two-soliton bound states of an integerorder. Two-soliton bound states of the third and tenth orders are shown in the bottompart of Fig. 2. Because of a delicate balance between the solitons and radiation for suchstationary solutions to exist, all these bound states are unstable, they either split into singleradiating solitons, or disintegrate in a more complicated fashion.

4.4. Families of dark solitonsFollowing the preliminary results of the cascading limit when the e�ective NLS equation19 is valid, we expect to ®nd dark solitons in the case r � ÿ1, that corresponds to adefocusing e�ective cubic non-linearity. Indeed, the numerical results obtained by Buryakand Kivshar [95, 96, 98] indicate that single dark radiationless solitons exist forr � ÿs � �1 as localized solutions of Equations 18. In the case r � ÿs � ÿ1, a continuousfamily of parametric dark solitons exists for 0 < a <1 and in the interval 0 < a < 8 thesesolitons have non-monotonic radiationless oscillatory tails. Examples of these two-wavedark solitons are presented in Fig. 3 for a � 1:0 (nonmonotonic tails) and a � 10:0(monotonic tails).

For the cascading limit �a� 1� the solution can be presented in the asymptotic form 21.When a is not large, the asymptotic solution 21 fails (e.g. it does not describe oscillatorytails for a < 8), but the dark soliton family still exists for a > 0, and it can be charac-terized, e.g. by the minimum amplitude of the second harmonic wmin. The dependence ofwmin versus a is shown in Fig. 3. For large a it approaches the asymptotic dashed curve

585


wmin � 1=a which corresponds to the NLS-like solitons of the cascading limit. Dark sol-itons of Fig. 3 are stable if their backgrounds are modulationally stable (i.e. the stabilitydomain is 2 < a <1, and it does not depend on r).

Due to the existence of decaying oscillating tails, a dark soliton can trap another darksoliton to form a bound state, a twin-hole dark soliton [106]. This mechanism is well knownfor other types of solitons [112], and the v�2� dark solitons are just a particular example.

Figure 2 Discrete set of two-wave bright solitons for r � �1 and s � ÿ1 in the form of soliton pairs. These

solitons appear as two radiative bright solitons with radiation trapped between them. The ®lled circle at a � 2

corresponds to the exact solution 23.

586

YU. S. Kivshar

For a! 8 the distance between the neighbouring dark solitons in a bound state increasesto in®nity. To the best of our knowledge, this is the ®rst example when stable twin-holedark solitons have been identi®ed.The dark solitons presented above exist for the case r � ÿs � ÿ1 in Equations 18.

Recently other continuous families of dark solitons have been found for r � ÿs � �1 inthe interval 0 < a < 2 [109]. These dark solitons (similarly to the solitons shown in Fig. 3)

Figure 3 Characteristic pro®les of two-wave dark solitons for r � ÿ1 and s � �1 versus the dimensionless

mismatch parameter a. Oscillating tails of the dark solitons appear for a < 8.

587


possess oscillating tails and thus can form bound states. Stability of these solutions is stillan open problem.

Similar to the case of bright solitons, for r � s � ÿ1 in Equations 18 single darkradiationless solitons do not exist due to the resonance with linear waves. However, adiscrete set of two- (and multi-) soliton radiationless bound states can be found, thesesolutions appear due to trapping of radiation. Figure 4 gives the results related to two-soliton radiationless bound states of dark radiative solitons. One can see that, similar tothe cases discussed above, there exists an exact analytic solution [113]

v�x� � 21=2w�x� � 21=2 1ÿ 3

2sech2�x=2�

� ��24�

which is a two-soliton bound state of the ®rst order. The two-soliton bound states of thethird and sixth orders are shown in the bottom part of Fig. 4.

Analytical results [105, 106] indicate that all radiationless bound states of radiative darksolitons are unstable due to the development of parametric modulational instability. Thisresult has been con®rmed by the direct numerical simulations.

5. Solitons due to third-harmonic generation5.1. Model and motivationsIn the case of solitary waves of quadratic media, the coupling between the fundamentaland second harmonics produces an e�ective non-linearity that allows the beam self-trapping via mutual self-focusing. When the harmonics are decoupled, no solitary wavesexist due to solely quadratic non-linearity. Here we consider the qualitative di�erentexample. We consider the case of the Kerr-type non-linear response near the point of thethird-harmonic generation. As is well known (see, for example, [90, 91]), the third-ordercontribution to the polarization is made up of two components, a response at the fun-damental frequency of the beam (described through an intensity-dependent refractiveindex), and a response at the frequency 3x, which is known to lead to third-harmonicgeneration under the condition of phase matching. Then, launching a monochromaticbeam near the point of phase matching with the third harmonics can result in co-prop-agating beams at the fundamental and third-harmonic frequencies drastically modifyingthe plane waves [114] and solitons of Kerr non-linearity [115, 116]. Such a mechanism ofresonant wave coupling can be responsible for the generation of enhanced non-linearitiesand non-Kerr solitary waves even in the systems with a weak cubic response.

The process of the generation of non-Kerr solitons due to the third- harmonic inter-action can be regarded as another example of the cascading e�ect whereby e�ective higher-order non-linearities are generated [117], the mechanism responsible for the existence ofquadratic solitons. In a cubic medium, this mechanism drastically a�ects the propagationof spatial solitary waves of the fundamental frequency under the condition of phasematching with the third harmonic. The physics of this e�ect is rather simple. Indeed, it canbe shown [114] that a slightly mismatched process of third-harmonic generation leads toan e�ective quintic non-linearity and, therefore, to the cubic-quintic NLS equation. As aresult, we expect that solitary waves cease to exist when the effective quintic non-linearitybecomes strongly defocusing (see, for example, [55] and references therein, for the mostrecent comprehensive analysis of solitary waves of the cubic-quintic NLS equation).

To describe the solitons modi®ed by the third-harmonic generation, we follow theoriginal papers by Sammut et al. [115, 116] and consider resonant interaction between a

588

YU. S. Kivshar

linearly-polarized beam of frequency x and its third harmonic (assuming a slab waveguidegeometry) presenting the electric ®eld E in the form E � 1

2 �E1 exp�i�k1zÿ xt�� E3 exp�i�k3zÿ 3xt�� c.c., where kj � jxnj=c and nj � n�jx� for j � 1; 3. Each frequencycomponent of the ®eld then satis®es that scalar wave equation where we assume the cubicnon-linear response. For the slowly varying envelopes E1 and E3, we obtain

Figure 4 Discrete set of two-wave dark soliton pairs existing for r � ÿ1 and s � ÿ1. These solitons appear as

two dark solitons with radiation trapped between them. The ®lled circle at a � 1 corresponds to the exact

solution 24.

589


2ik1@E1

@z� @

2E1

@x2� v jE1j2 � 2jE3j2

� �E1 � E�21 E3 exp�ÿiDkz�

h i� 0 �25�

2ik3@E3

@z� @

2E3

@x2� 9v jE3j2 � 2jE1j2

� �E3 � 1

3E31 exp�iDkz�

� �� 0 �26�

where Dk � 3k1 ÿ k3 is the phase mismatch, the non-linearity parameterv � �3px2=c2�jv�3�j is de®ned to be always positive, whereas the sign of the Kerr non-linearity depends on whether the material is self-focusing (positive) or defocusing (nega-tive). These equations describe a special case of a more general four-wave mixing process(see, for example, [118] as an example of solitary waves).Equations 25 and 26 can be further normalized using the scales of a beam width z0 and

di�raction length zd � 2z20k1, and introducing the dimensionless ®elds U � 3�k1z20v�1=2E1

and W � �k1z20v�1=2 exp�ÿiDkz�E3. For the stationary solutions we then substituteU � ub1=2 exp�ibZ� and W � wb1=2 exp�i3bZ�, where b is the beam propagation constantwhich is de®ned by the beam total power and therefore can be treated as an externalparameter of families of solitary waves. As a result, the system of coupled equations forthe solitary wave pro®les can be rewritten in the following dimensionless form [115, 116]

i@u@z� @

2u@x2ÿ u� 1

9juj2 � 2jwj2

� �u� 1

3u�2w � 0 �27�

ir@w@z� @

2w@x2ÿ aw� �9jwj2 � 2juj2�w� 1

9u3 � 0 �28�

where z � bZ and x � b1=2X . Stationary beams are described by real solutions, u�x� andw�x�, de®ned by the system 27 and 28 with the z-derivatives omitted. These localized solu-tions depend only on a single dimensionless parameter, a � r�3� D=b� with two dimen-sionless parameters, D � 2k1Dkz20 and r � k3=k1, where r � 3 for the case of spatial solitons.

5.2. Multistability of bright solitonsThe basic structure of Equations 27 and 28 is qualitatively similar to the equations derivedabove for the case of parametric solitary waves supported by two-wave mixing in v�2�

media. Moreover, the de®nition of the e�ective mismatch parameter a is almost identicalto that case, in spite of the di�erent structure and physical meaning of the non-linearcoupling terms. We believe this is a general property of di�erent systems with cascadednon-linearities.

First of all, far away from the point of phase matching (i.e. for large a), the energyconversion from the fundamental to the third harmonic is small, i.e. jwj � juj. For a� 1,from Equation 28 we have approximately w ' u3=9a, and Equation 27 becomes the cubic-quintic NLS equation allowing the solutions for solitary waves in an explicit form (see, forexample, [55]). This suggests the general structure of an asymptotic expansion for thelocalized solutions of Equations 27 and 28 in powers of � � aÿ1, as was ®rst presented bySammut et al. [115]

u�x� � 3�21=2�cosh x

1ÿ 2� 2ÿ 1

cosh2 x

� �ÿ 2�2 70ÿ 29

cosh2 xÿ 12

cosh4 x

� �� O��3� �29�

w�x� � 6�21=2�cosh3 x

1ÿ 3� 1ÿ 10

cosh3 x

� �� O��3� �30�

590

YU. S. Kivshar

The asymptotic analytical solutions 29 and 30 are a limiting case of a family of localizedsolutions described by the real functions u�x� and w�x� which have been found numerically[115, 116]. All types of these solutions can be characterized by the normalized total power

Ptot �Z 1ÿ1�juj2 � 3rjwj2� dx �31�

which is a conserved quantity of the system 27 and 28.Figure 5 shows the normalized total power Ptot de®ned by Equation 31 as a function of

the mismatch parameter a, for di�erent types of two-wave localized solutions of the system27 and 28. The inset ®gure shows an expanded portion of the dependence Ptot�a� for therange 8.2 � a � 9.2. It can be seen from the form of Ptot�a� that, near the point of phasematching between the fundamental and third harmonics (i.e. at a � 9 when r � 3), thereexist three distinct types of localized solutions for bright solitary waves.

The most important class of two-wave bright solitons is described by a family of lo-calized solutions for coupled fundamental and third harmonic ®elds. The distribution ofpower between the two frequencies varies from being predominantly in the third har-monic, for smaller a, to predominantly in the fundamental, at larger values of a. In thislatter case, we can apply the cascading approximation to ®nd Ptot�a� �36ÿ 192�ÿ �27072=5��2 � O��3�, which is shown as a dashed line in Fig. 5.

For large values of the normalized mismatch parameter a, the amplitude of the beam atthe fundamental frequency grows whereas that of the third harmonic vanishes, in agree-ment with the prediction of the analytical results obtained in the cascading approximation.An example of this kind of solution is presented in Fig. 6a, it corresponds to the point A inFig. 5.

A simple analysis done by Sammut et al. [115, 116] shows that the family of two-frequency solitary waves bifurcates from the one-frequency solution for the third har-monic, w�x� � ��2a�1=2=3 �sech �a1=2x� and u�x� � 0, at the point of exact phase matching(i.e. a � 9 at r � 3 and D � 0). This family of the one-frequency solitary waves is char-acterized by the normalized power, Ptot � 4a1=2, and it is described by the standard cubicNLS equation which follows from Equation 28 at u � 0. It is clear that this type of solitarywave is possible only due to the self-phase modulation effect taken into account for thethird harmonic.

Figure 5 Variation of the normalized total power,

Ptot versus the dimensionless mismatch parame-

ter a for the three distinct families of solitary wave

solutions of Equations 27 and 28. The dashed

curve corresponds to the asymptotic expansion.

Lower curves merge at the bifurcation point

O �a � 9�. Points A to D indicate the particular

examples presented in Fig. 6a±d, respectively.

591


Finally, the third family of localized solutions shown in Fig. 5 includes the simplestanalytical solution [119, 120]. This solution exists only at a � 1 and it has the followingform us�x� � a sech x and ws�x� � bus�x�, where the parameter b is de®ned by a real root ofa cubic equation. In sharp contrast with the theory of v�2� parametric solitons, the ana-lytical solution of model 27 and 28 and the asymptotic solution of the cascading limit,a� 1, do not belong to the same family. Moreover, varying continuously the e�ectivemismatch parameter a along this family of localized solutions shows that this class ofsolitary waves corresponds to multi-hump solitary waves, as is shown in Fig. 6b for a � 8:2(point B in Fig. 5). Then, it is not surprising that all solutions of this family are unstable,the conclusion veri®ed by employing direct numerical simulations [116].The most interesting feature of this class of two-wave parametrically coupled solitary

waves manifests itself near the point of the phase matching. In the case of negative phasemismatch, corresponding to the part of the curve Ptot�a� from the left of the bifurcationpoint O shown in Fig. 5, for any ®xed value of the parameter a we reveal the simultaneousexistence of three localized solutions. Therefore, in this case the propagation character-istics of two-frequency coupled self-guided beams become multi valued. Two characteristicpro®les of the solutions in this region are shown in Fig. 6c and d corresponding to thepoints C and D in Fig. 5, respectively.

Therefore, in the problem of third-harmonic generation there exists more than onepossible propagation constant (and, consequently, more than one possible shape) of the

Figure 6 Examples of the fundamental (thin line) and third harmonic (thick line) pro®les for several solitary

wave solutions which belong to different families. Pro®les (a) to (d) correspond to points A to D in Fig. 5,

respectively.

592

YU. S. Kivshar

parametric spatial soliton for the same value of the total power Ptot de®ned by Equation31. This phenomenon is usually associated with soliton multi-stability predicted for non-Kerr non-linearities [39, 40, 121]. For the problem of the third-harmonic generation, anovel physical mechanism, resonant parametric wave mixing, can also lead to similare�ects indicating that e�ective non-Kerr non-linearities and multi-stable solitary wavesmay become possible even in a Kerr medium.

Similar results have been found by Sammut et al. [116] for two-frequency dark solitons,and it has been shown that these types of solitary waves are also drastically modi®ed nearthe point of phase matching. The most important effect discovered by Sammut et al. [116]is the existence of parametric modulational instability that strongly affects some types ofdark solitons existing on a non-vanishing background. Additionally, numerical simula-tions carried out by Sammut et al. [116] indicate that for the values of a selected close tothe phase-matching point a � ÿ9, the dark solitons display an instability similar to thatexisting for saturable non-linearities.

All these e�ects demonstrate that non-Kerr non-linearities arising from phase-matchedinteraction between the fundamental beam and its third harmonic can have a dramatice�ect on the spatial solitary waves. First, such induced non-Kerr non-linearities restrict theexistence region of allowed values of the beam power, and also they lead to multi-stablebeam propagation when more than one possible beam pro®le and propagation constantexist for a ®xed value of power near the phase matching. For dark solitary waves, para-metric coupling generates two types of instabilities, parametric modulational instability ofthe background and also inherent instability of black solitons, similar to that earlierpredicted for scalar dark solitons of saturable non-linearities.

6. Soliton internal modesAs has been shown above, di�erent types of spatial optical solitons require the models ofnon-Kerr non-linearities which are generally non-integrable. Soliton instability is one of themajor novel features of such solitary waves (see Section 7 below). Then, for the case ofstable non-Kerr solitons, the question is: What kind of novel physical properties can beexpected for solitary waves of non-integrable models? For many years, it was believed thatsolitary waves of non-integrable models differed from solitons of integrable models only inthe character of soliton interactions: unlike proper solitons, interaction of solitary waves isaccompanied by radiation. However, the recent analysis of solitary wave solutions of thegeneralized NLS models (see, for example, [122±124]) has revealed that a small pertur-bation to an integrable model may create an internal mode of a solitary wave. This effect isbeyond a regular perturbation theory, because solitons of integrable models do not possessinternal modes. But in non-integrable models such modes may introduce qualitatively newfeatures into the system dynamics being responsible for long-lived oscillations of thesolitary wave amplitude and resonant soliton interaction.

Internal modes have been earlier analysed only for the so-called kink solitons, topo-logical solitary waves of the Klein±Gordon type models (see, for example, [125±127]). Theinternal modes of kinks, usually called `shape modes', are known to modify drastically thekink dynamics because they can temporarily store energy taken away from the kink'stranslational motion and later restore the energy back. This mechanism gives rise toresonant structures in the kink-antikink collisions [125, 126].

The important common feature of non-Kerr solitons observed in numerical simulationsis that they display long-lived persistent oscillations of their amplitude [37, 107, 128, 129].

593


Many of such features observed numerically for di�erent types of envelope solitons can benaturally explained in the framework of the concept of the soliton internal mode, gener-ically similar to the kink's shape mode. Therefore, the existence of internal modes is acommon feature of solitary waves in many different non-integrable non-linear models.

First of all, we demonstrate that any small perturbation of the integrable NLS equationmay lead to an internal mode of a solitary wave. We follow the original work by Kivsharet al. [124] and consider the perturbed NLS equation 6 in a slightly modi®ed form

i@W@z� @

2W@x2� 2jWj2W� �g�jWj2�W � 0 �32�

where, in general, g�:� is an operator describing corrections due to non-Kerr non-linear-ities. A localized solution of Equation 32 for solitary waves can be found in the formW�x; z� � U�x� exp�iz�, where for simplicity we put b � 1 that may always be scaled out.The real function U�x� is expressed asymptotically as U�x� � U0�x� � �U1�x� � O��2�,where U0 � sech x is the soliton of the cubic NLS equation, and U1 is a localized cor-rection de®ned from Equation 32. The linearized problem for the perturbed NLS equationarises upon the substitution W�x; z� � fU�x� � �U�x� ÿ W �x�� exp�iXz� � �U��x� � W ��x��exp �ÿiXz�g exp�iz� and it has the form,

d2Udx2� 6

cosh2 xÿ 1

� �U � XW � �f1�x�U � 0 �33a�

d2Wdx2� 2

cosh2 xÿ 1

� �W � XU � �f2�x�W � 0 �33b�

where f1�x� � g�U20� � 2U2

0g0�U2

0� � 12U0U1 and f2�x� � g�U20� � 4U0U1.

The linear eigenvalue problem 33 can be solved exactly at � � 0 (see, for example,[130]). Its spectrum consists of two (symmetric) branches of the continuous modes withthe eigenvalues X � �X�k� � ��1� k2�, and discrete spectrum modes corresponding tothe degenerated eigenvalue X � 0. In the presence of a perturbation, the eigenvalueproblem 33 can possess an additional discrete eigenvalue that appears due to a bifur-cation from the continuum spectrum band. To ®nd that bifurcating eigenvalue, weassume, for de®niteness, that the cut-o� frequencies Xmin � �1 are not a�ected by theperturbation. Then, the internal mode frequency can be presented in the form,X0 � 1ÿ �2j2. The analytical approach developed in [124] allows one to obtain theexpression for the parameter jjj,

jjj � 1

4sign��

Z 1ÿ1

U�x; 0�f1�x�U�x; 0�n

� W �x; 0�f2�x�W �x; 0�odx �34�

where the non-oscillatory eigenfunctions are de®ned in the limit k ! 0, i.e.U�x; 0� � 1ÿ 2 sech2x and W �x; 0� � 1. The positiveness of the parameter j gives thecriterion for the existence of the soliton internal mode.

It is interesting to note that this criterion for the existence of a soliton internal mode hasan analogy with the famous Peierls problem in quantum mechanics [131] stating that aone-dimensional potential well always possesses at least one discrete eigenvalue. For thecase of the soliton internal mode, an additional eigenvalue appears with no thresholdprovided �j > 0, due to a deformation of the re¯ectionless soliton potential.

594

YU. S. Kivshar

In a general case, we should solve the eigenvalue problem 33 numerically. The case ofthe cubic-quintic non-linearity, where the soliton solutions are known in an explicit form[55], was analysed in [123]. For this kind of non-linearity, the non-linear refractive index inEquation 6 is modelled by the function f �jWj2� � 4jWj2 � 3rjWj4, where the quintic termdescribes a higher-order correction to the Kerr law, focusing �r � �1� or defocusing�r � ÿ1�. The eigenvalue problem 33 was solved numerically by means of the shootingtechnique.

A simple analysis of the bifurcation criterion 34 shows that for r � �1 an additional(discrete) eigenvalue X � X0 appears (with its symmetric counterpart, X � ÿX0�, and itlies inside the spectrum gap, X0 < b. The bifurcation theory allows one to ®nd thiseigenvalue near the spectrum band edge,

X0 � b 1ÿ 1

36b2 � O�b4�

� ��35�

The eigenfunctions corresponding to this eigenvalue decay exponentially for large x. Thus,in the cubic-quintic NLS equation, any small soliton perturbation should transform intothree components: localized correction to the stationary shape of the solitary wave andradiation (those are the same as for the Kerr solitons), and the third component, which wecall the soliton internal mode, that manifests itself as oscillations of the soliton width andamplitude. This mode is associated with a novel (discrete) eigenvalue of the linearizedproblem 33.

In contrast, for r � ÿ1 the bifurcation analysis does not predict any additional discreteeigenvalue of the linear problem 33. Therefore, the soliton internal mode does not appear,so that in this case long-lived oscillations of the soliton width and amplitude are notexpected.

To illustrate these conclusions, Fig. 7a and b shows two distinct types of the evolution ofthe soliton peak intensity when the soliton amplitude is perturbed by 10% of its value. Forr � �1, this perturbation excites a long-lived periodic variation of the soliton peak in-tensity and width, see Fig. 7a. However, no soliton internal mode exists for r � ÿ1, that iswhy the initial oscillations decay rapidly for any value of b. An example is shown in Fig. 7bfor b � 0:8, where also a shift of the soliton peak intensity due to the perturbation can beclearly seen.

We would like to emphasize an important consequence of the existence of the solitoninternal mode, which can be probably observed experimentally, as has been suggested in[123]. Indeed, because the period of the internal soliton oscillation depends on the totalpower of the beam, the output width of the solitary wave created by an input beam at a®xed propagation distance depends non-monotonously on the input power. Figure 8 showstwo examples of the dependence of the output beam width as a function of the input powerfor the scaled soliton, i.e. the input beam of the form W�x; 0� � AU�x� with A 6� 1. It is clearthat this dependence becomes oscillatory for the case when the soliton internal mode existsbeing excited by the input beam (thin solid, Fig. 8), whereas it is an almost straight line forthe other case, when the soliton internal mode does not exist (thick solid, Fig. 8).

7. Soliton stability: a general overviewSpatial optical solitons are of both fundamental and technological importance if they arestable under propagation. Existence of stationary solutions of the di�erent models of non-Kerr non-linearities, in particular those presented above, does not guarantee their

595


stability. Therefore, the soliton stability is a key issue in the theory of self-guided opticalbeams.

For temporal solitons in optical ®bres, non-linear e�ects are usually weak and the modelbased on the cubic NLS equation and its deformations is valid in most of the cases [31, 32].Therefore, being described by integrable or nearly integrable models, solitons are alwaysstable, or their dynamics can be a�ected by (generally small) external perturbations whichcan be treated in the framework of perturbation theory.

As has been discussed in Section 2.1, much higher powers are usually required forspatial solitons in waveguides or bulk media, so that real optical materials demonstrate

Figure 7 Dynamics of a perturbed soliton in the cubic-quintic NLS model: (a) r � �1 and b � 5 (soliton

internal mode exists and its width is of the order of the soliton width); (b) r � ÿ1 and b � 0:8 (mode does not

exist). Dashed lines in (a) and (b) show the corresponding unperturbed values of the soliton intensity.

596

YU. S. Kivshar

essentially a non-Kerr change of the non-linear refractive index with increasing lightintensity. Generally speaking, the non-linear refractive index always deviates from Kerrfor larger input powers, e.g. it saturates. Therefore, models with a more general intensity-dependent refractive index are employed to analyse spatial solitons and, as we discussbelow, solitary waves in such non-Kerr materials can become unstable. Importantly, veryoften the stability criteria for solitary waves can be formulated in a rather general formusing the system invariants.

7.1. Stability of one-parameter solitons7.1.1. Bright solitonsStability of bright solitons of the NLS equation with a generalized non-linearity has beenextensively investigated for many years, and the criterion for the soliton stability has beenderived by di�erent methods (see, for example, [132±137]). Stability of bright solitons inthe generalized NLS equation of any dimension is given by the simple integral criterion[132]

dPdb� d

db

ZVjE�~r; z�j2 d~r

� �> 0 �36�

Figure 8 Output beam width wout, normalized to the input beam width w0, versus the input beam power P,

normalized to the soliton power P0, both measured at z � 10. Thin solid: the case when the soliton internal

mode exists �r � �1; b � 2�; thick solid: the case when the soliton internal mode does not exist

�r � ÿ1; b � 0:6�.

597


where P is the total beam power and b is the soliton propagation constant. The validity ofthe Vakhitov±Kolokolov criterion 36 is based on the speci®c properties of the eigenvalueproblem that appears after linearizing the NLS equation 6 near the solitary wave solutions[134, 135], for example, the condition 36 is associated with the existence of only onenegative eigenvalue of that problem. If this latter condition is not ful®lled, the stabilitycriterion may not be directly formulated in terms of the beam power P , as we have in thecase of non-linear guided waves. Indeed, it has already been established that stability ofself-guided waves in non-linear waveguide structures can be given in some cases by thesame criterion (see, for example, [137±139]) but, in general, it is more complicated anddepends on a particular structure and the type of non-linearity.

Criterion of the soliton stability 36 is usually valid for the bright solitons (and non-linearguided waves) which constitute a one-parameter family of localized solutions, i.e. theirshape is solely de®ned by the beam propagation constant b. The similar criterion is valideven in the case of two-component solitons governed by the only power invariant. Forexample, it has been shown that a similar criterion applies for two-wave solitons in v�2�

materials [107].Linear stability analysis does not allow one to predict the subsequent evolution of

unstable solitons. Di�erent scenarios of the instability-induced dynamics of bright solitonshave been found numerically (see, for example, [133] and references therein). Recently,Pelinovsky et al. [53, 54] have developed an asymptotic analytical method to describe thedynamics of unstable solitons (e.g. their diffraction-induced decay, collapse, or switchingto a novel stable state). Some of these results are summarized below in Section 8.

7.1.2. Dark solitonsIn contrast to bright solitons, the stability criterion for dark solitons of the generalizedNLS equation has not been understood until recently and, even more, this issue created alot of misunderstanding in the past. In particular, we notice some e�orts to employ, byanalogy with bright solitons, the soliton complementary power (see, for example, [41]) andto use this invariant for analysing the soliton bistability [42, 46], a statement that a blackdark soliton (a dark soliton with zero intensity at its centre) is always stable [137], etc. Aswas noticed in [50] after analysing the results of numerical simulations, the complimentarypower does not de®ne the stability of dark solitons.

From a historical point of view, the ®rst e�orts to analyse the stability of dark solitonswere stimulated by numerical simulations done by Barashenkov and Kholmurodov [140]who observed instability of the so-called `bubbles', localized waves of rarefaction of theBose gas condensate. These non-topological solitary waves belong, in the �1� 1�-dimensional case, to the family of dark solitons of the NLS equation with the cubic-quinticnon-linearity and they survive in higher dimensions [141]. Although quiescent bubbleswere found to be always unstable regardless of dimension [142], numerical simulationsrevealed that moving bubbles can be stabilized at non-zero velocities [140]. Later Bogdanet al. [143] (see also [144]) explained this phenomenon through the multi-valued depen-dence of the bubble's energy versus the remormalized bubble's momentum.

However, it was believed for a long time that kink-type dark solitons (in particular,black solitons) are always stable (see, for example, [137]). Instability of black solitons wasobserved for the ®rst time by Kivshar and Krolikowski [145] in numerical simulations ofthe NLS equation with the saturable non-linearity 9 at p � 2. These authors used thevariational principle to link the instability to the existence of multivalued solutions in

598

YU. S. Kivshar

terms of the system invariants, and also derived the instability threshold condition (i.e. themarginal stability condition) by using the multi-scale asymptotic expansions.

Eventhough it was known for some time that the stability criterion for dark solitonsshould be de®ned through the renormalized soliton momentum,

dMr

dv� d

dvi

2

Z �1ÿ1

u@u�

@xÿ u�

@u@x

� �1ÿ u2

0

juj2 !

dx

( )> 0 �37�

and this was shown to be consistent with the results of numerical simulations [140, 142,146] and the variational principle for bubble-type [143] and kink-type [145] dark solitons,the rigorous proof of this stability criterion was presented only recently by Barashenkov[147], with the help of the Lyapunov function, and Pelinovsky et al. [54], by using theasymptotic expansion near the instability threshold. The ®rst approach does not allow oneto describe the instability itself but it is more general to prove the global stability if itexists, whereas the second method is valid in the vicinity of the instability threshold beingsuf®cient to determine the instability domain.

All basic models of optical non-linearities, i.e. the models with saturable, transiting andcompeting non-linearity discussed in Section 2.3, display instabilities of dark solitons insome regions of the parameters. For example, in a saturable medium dark solitons becomeunstable provided the saturation intensity is below a certain threshold value [54, 146].Instability displayed by dark solitons is qualitatively different from the instability of brightsolitons, it is a drift instability (see [146]).

7.2. Stability of two-parameter solitonsMany of the `novel' soliton states demonstrated for case of two interacting modes areunstable. Stability of bright and dark vector solitons (see, for example, [12, 76±78]),rotating or dynamic solitons [79] is still an open problem. However, as has been demon-strated for the ®rst time for the case of non-degenerated three-wave mixing in a diffractivemedium [101], the threshold of the instability for two-parameter solitons (marginal sta-bility condition) is given by the following criterion

J b1; b2� � � @ F1;F2� �@ b1; b2� � �

@F1

@b1

@F2

@b2

ÿ @F1

@b2

@F2

@b1

� 0 �38�

where Fj �j � 1; 2� are two invariants of the system (total and complimentary powers, forthe problem of three-wave mixing) describing two-parameter solitary waves, and bj aretwo independent parameters of the stationary localized solutions. The stability criterionitself, i.e. the sign of the function J, depends on the model under consideration.

Result 38 has ®rst been derived by the asymptotic expansion technique [101] and thenveri®ed by the analysis of the global structure of the system invariants [102, 148]. It is adirect consequence of the topology of the invariant surface H�F1;F2�, and it seems tobe valid for di�erent types of vectorial and coupled solitons described by two indepen-dent parameters introduced by two non-trivial invariants of the model. For example, inthe case of coupled bright-dark solitons, F1 is the power of the bright component P andb1 is the propagation constant of the bright component, whereas F2 is the totalmomentum M and b2 is the soliton velocity V [149]. For incoherently interacting brightsolitons, the invariants Fj are two powers corresponding to two scalar components[150]. The same result holds for the stability of solitons in the models with the absence of

599


the Galilean invariance, such as two-wave parametric solitons with the walk-off effect. Inthis latter case the second parameter is the soliton velocity V and the second invariant isthe soliton momentum M ; the same criterion has been recently demonstrated for theso-called walking solitons [151].

In general, the marginal stability criterion 38 is valid provided the conditions similar tothose for the validity of the Vakhitov±Kolokolov criterion 36 are satis®ed, namely theinstability is associated with a kind of translational bifurcation of localized eigenmodes ofan associated linear eigenvalue problem when the value k2, where k is an eigenvalue,remains real but it changes its sign passing zero. For more complicated models, theinvariant criterion may be not valid and other types, e.g. oscillatory instabilitiesmay occur,as has been recently demonstrated for gap solitons in a non-integrable deformation of theThirring model [152]. Another example is given by a system where there exist, for the samevalues of the system parameters, a number of different soliton families corresponding tobifurcations of invariant surfaces. An example of this kind has been recently found for theproblem of non-degenerated four-wave mixing where stable multi-colour solitonscorrespond to the lowest invariant surface [153].

7.3. Symmetry-breaking and transverse soliton instabilitiesThe soliton criteria discussed above are a special case of more general wave instabilitiesthat may occur due to non-linear properties of physical systems [154]. The simplest type ofsuch an instability is modulational instability characterizing a breakup of a continuous-wave (or plane wave) ®eld of a large intensity. Modulational instability was ®rst predictedand analysed in the context of non-linear wave propagation in ¯uids [155, 156] and, evenearlier, in non-linear optics [157, 158]. One of the important physical processes associatedwith the development of modulational instability is the generation of a train of localizedwaves (beams or pulses) [159], the effect observed experimentally in different physicalsystems, e.g. in optical ®bres [160].

Modulational instability can be viewed as the simplest case of the so-called symmetry-breaking instability, when a solution of a non-linear system of a certain dimension (e.g.a plane wave or soliton) is subjected to a broader class of perturbations. An example of thesame type of instability is a breakup of low-dimensional solitary waves under the action ofperturbations involving higher dimensions. This is a typical case of the so-called transverseinstability of plane solitary waves, ®rst discussed for long-wave solitons of the Korteveg±de Vries (KdV) equation [161] and envelope solitons of the NLS equation [162, 163], andthen investigated by different methods for a variety of non-linear soliton-bearing models(see, for example, [164±170]), with demonstrations in numerical simulations [171, 172].Many of these results are now well-known in the soliton theory (see, for example, [173]).

In application to spatial optical solitons this type of soliton instability has attractedspecial attention and discussion these days. First of all, this is because of the recentprogress in developing and employing non-linear optical materials, that led to severalimportant discoveries of self-focusing, self-trapped beam propagation, and solitary wavesin di�erent non-linear materials, including photorefractive [17, 18], quadratic [23], andsaturable non-Kerr [21] media. It is expected that the development of novel band-gapmaterials based on quadratic or cubic non-linearities will eventually lead to the experi-mental observation and manipulation of the so-called light bullets, self-focused states oflight localized in both space and time, the building blocks of the future all-opticalphotonics devices.

600

YU. S. Kivshar

Secondly, di�erent types of symmetry-breaking instabilities have been recently describedtheoretically and even observed experimentally in non-linear optics. This includes theobservation of breakup of bright soliton stripes in a bulk photorefractive medium due totransverse modulational instability and formation of two-dimensional bright solitonbeams [174, 175]; the theoretical study [176±179] and experimental observation [180, 181]of generation of pairs of optical vortex solitons due to the transverse instability of dark-soliton stripes; the theory and experimental demonstration of spatial modulationalinstability in quadratically non-linear v�2� optical media [104, 182]; a decay of ring-shapeoptical beams with non-zero angular momentum into higher-dimensional solitary waves,observed experimentally in rubidium vapour (a saturable defocusing medium) [21] andinvestigated theoretically as a general solitonic e�ect [183]. As a result of the developmentof the soliton transverse instability, the �1� 1�-dimensional soliton stripe decays eitherinto solitary waves of higher dimensions (e.g. �2� 1�-dimensional bright or dark solitonsof radial symmetry) or it decays into radiation. For the case of quadratic solitons, bothsuch scenarios have been discussed theoretically [182] and also observed experimentally[104].

8. Nonlinear theory of soliton instabilitiesLinear stability analysis does not allow one to predict the subsequent evolution of unstable(bright and dark) solitons. However, recently the so-called non-linear theory of solitoninstabilities has been suggested and demonstrated for several soliton bearing non-linearmodels [53, 54, 107]. This theory is based on the multi-scale asymptotic technique and itallows one to describe the soliton evolution near the stability threshold (i.e. the marginalstability curve).

According to this approach, unstable bright solitons display three general scenarios oftheir evolution [53], i.e. they diffract, collapse, or switch to a stable state with long-livedoscillations of their amplitude. These oscillating solitons are possible due to the existence ofsoliton internal modes [122±124], one of the major properties that differentiate integrableand non-integrable models. Unstable dark solitons display much simpler types of evolu-tion which is always accompanied by emission of radiation which can generate new darksolitons (soliton splitting). Generally, this leads to a drift of a dark soliton that becomes`greyer' [54, 146].

In this section we present a brief summary of analytical results of the asymptotic theorythat describes not only linear instabilities but also the non-linear long-term evolution ofthe unstable solitons. This approach is based on a non-trivial modi®cation of the solitonperturbation theory [33] near the instability threshold. We consider the instabilities ofbright solitons in a non-Kerr medium described by the generalized NLS equation 6 andtwo-component bright solitons supported by resonant parametric interactions in a v�2�

optical material, and also the drift instability of dark solitons in a non-Kerr defocusingmedium. All cases brie¯y considered below are for the �1� 1�-dimensional solitary waves,but the analysis can be readily extended to cover similar problems for solitary waves ofhigher dimensions.

8.1. Bright solitons in non-Kerr mediaWe consider the generalized NLS equation 6 where the function f �jWj2� characterizes anon-linear correction to the material refractive index. The model (equation 6) has beenintensively investigated in the context of the existence and stability of bright optical

601


solitons which are stationary solutions of Equation 6 of the formWs�x; z� � U�x; b� exp �ibz�, where the real function U�x; b� vanishes for jxj ! 1, and b isthe non-linearity-induced shift of the propagation constant. As has been already discussedabove, for generalized non-linearities spatial solitons may become unstable, and a standardapproach is to analyse the soliton instability using the linear stability analysis. For the caseof bright solitons of the generalized NLS equation 6, the soliton stability is given by theVakhitov±Kolokolov criterion [132]. However, the linear stability analysis does not allowone to understand the subsequent evolution of unstable solitons when the linearizedequations, describing exponentially growing perturbations on the soliton pro®le, becomeinvalid, and numerics is used for every particular problem.

The standard soliton perturbation theory [33] is usually applied to analyse the solitondynamics under the action of external perturbations. Here we deal with a qualitativelydi�erent physical problem when an unstable bright soliton evolves under the action of its`own' perturbations. As a result of the development of the instability, the soliton frequencyb varies in time, b � b�z�. The linear stability of bright solitons is determined by the slopeof the derivative dPs=db, where Ps�b� is the soliton energy

Ps�b� � 1

2

Z �1ÿ1

U2�x; b� dx

Near the instability threshold, when the derivative P 0s�b� vanishes, the growth rate is small,and we can assume that the instability-induced evolution of the perturbed soliton is, ®rst,slow in z and, second, almost adiabatic (i.e. self-similar). Therefore, we can develop anasymptotic theory representing the solution to the original model (Equation 6) asW � /�x; b; Z� exp�ib0z� i�

R T0 X�Z 0� dZ 0�, where b � b0 � �2X�Z�, Z � �z, and �� 1. Here

the constant value b0 is chosen near the critical value bc where the derivative dPs�b�=dbvanishes. Then, using the asymptotic multi-scale expansion for /�x; b; Z�,/ � U�x; b� � �3/3�x; b; Z� � O��4�, we obtain the following equation for the solitonpropagation constant X (details can be found in [53]),

Ms bc� �d2XdZ2� 1

�2dPs

db

��b�b0

X� 1

2

d2Ps

db2

��b�bc

X2 � 0 �39�

where Ms�b� is calculated through the stationary soliton solution,

Ms �Z �1ÿ1

1

U�x; b�Z x

0

U�x0; b� @U x0; b� �@b

dx0� �2

dx > 0

Thus, the remarkable result of this asymptotic analysis is the following. In the gener-alized NLS equation 6 the dynamics of solitons near the instability threshold can bedescribed by Equation 39 which is equivalent to the equation for motion of an e�ectiveinertial and conservative particle of the mass Ms�bc� and coordinate X moving under theaction of a potential force which is proportional to the difference P0 ÿ Ps�b�, whereP0 � Ps�b0�.

The ®rst two terms in Equation 39 cover the result of the linear stability analysis [132,134, 135, 137], i.e. the soliton of Equation 6 is linearly unstable provided dPs�b0�=db < 0.The non-linear term in Equation 39 allows us to consider not only linear but also long-term (non-linear) dynamics of unstable solitons and, moreover, to identify all scenarios ofthe soliton instability dynamics.

602

YU. S. Kivshar

As an example, we consider the generalized NLS equation with two power-law non-linearities, f �I� � ÿIp=2 � cIp, which possesses an explicit soliton solution for any p > 0[55]. For 1 < p < 2 the function Ps�b� always has a minimum, and the derivatived2Ps�bc�=db2 is positive (see Fig. 9a). It is convenient to analyse the dynamical system 39on the phase plane �X; _X� or, equivalently, on the plane �b; _b�. When the value P0 exceedsthe extremum value of Ps�xc�, the corresponding phase plane is presented in Fig. 9b.

The standard initial-value problem to the generalized NLS model (Equation 6) corre-sponds to the initial condition for equation 39 lying on the axis _b�0� � 0. For such initialconditions we reveal three di�erent regimes of the soliton dynamics. They are all depictedby the curves 1, 2 and 3 in Fig. 9a and b.

If the amplitude of the initially perturbed soliton is taken to be smaller than the am-plitude of the (unstable) stationary solution (i.e. b�0� < b0, curves 1 in Fig. 9a and b), theinstability leads to a decrease of b (proportional to the soliton amplitude), and this processresults in the soliton spreading into small-amplitude waves which ®nally decay into linearwaves due to di�raction.

On the other hand, if the initially perturbed unstable soliton has the amplitude slightlybigger than that of the stationary soliton solution (curves 2 in Fig. 9b), the exponentiallygrowing instability `pushes' the soliton into the stability region where there exists a stablestationary soliton solution with b � bf, corresponding to the same value of the powerinvariant P0 (see Fig. 9a). However, due to the inertial nature of the soliton evolution, thetransition from an unstable to stable state is accompanied by the long-lived periodicoscillations of the soliton amplitude. These oscillations can be explained by the existenceof a non-trivial internal mode of a bright soliton in the generalized NLS model (Equa-tion 6), recently observed numerically and discussed analytically for the solitons of dif-ferent types of non-linear response [122±124, 129].

We note that, according to our analytical theory, the periodic oscillations of the solitonamplitude must disappear for larger deviations from the stable equilibrium state(b�0� > b1, curves 3 in Fig. 9a). In spite of the fact that larger deviations of the solitonamplitude do not remove the soliton from the stability region, the evolution of pertur-

Figure 9 (a) Characteristic dependence of the soliton power Ps�b� in the model (Equation 6) with

f �I� � ÿIp=2 � cIp for 1 < p < 2. Minimum point bc corresponds to the instability threshold, b0 and bf, to the

unstable and stable solitons at Ps�b0� � P0 � 1:96, respectively. (b) Phase plane �b; _b� of the asymptotic

model describing dierent regimes of the soliton dynamics: soliton decay (curves 1 and 3) and amplitude

oscillations (curve 2).

603


bations is predicted to lead again to decreasing of the soliton amplitude. As soon as thesoliton enters the unstable region, it ®nally spreads out due to di�raction.

All predicted regimes of the soliton instability have been found in numerical simulations[54]. Figure 10 shows several examples of the soliton evolution for the soliton propagationconstant b�z� determined as the ®rst-order derivative of the soliton phase calculatednumerically at the pulse maximum. The results are in an excellent agreement with thepredictions of the analytical model (Equation 39) (cf. curves in Fig. 10 with the trajectorieson the phase plane shown in Fig. 9b).

8.2. Two-wave solitons in v(2) mediaAs has been mentioned above in Section 2.4, the NLS equation becomes invalid nearresonances with the higher harmonics excited when the phase matching condition is sat-is®ed. In a medium with quadratic response, non-trivial e�ects can be observed already fortwo interacting waves due to parametric two-wave mixing in a di�ractive medium. InSection 4, we have discussed di�erent types of two-wave parametric solitons which havebeen recently observed experimentally in non-linear planar waveguides [103]. Here wediscuss brie¯y the stability of bright two-component solitons in such systems.

We recall that resonant interaction between the fundamental and second harmonics in adi�ractive v�2� medium can be described by the coupled equations for the dimensionlessvariables (see also Section 4.1)

i@v@z� @

2v@x2ÿ v� v�w � 0

ir@w@z� @

2w@x2ÿ aw� 1

2v2 � 0

�40�

where z is the propagation distance, x stands for the beam transverse coordinate. Theparameter r describes the ratio of the wave vectors (for spatial solitons, r � 2), and weconsider it positive. The parameter a can be presented as a � 2rÿ D, where D is pro-portional to the wave vector mismatch Dk � k2 ÿ 2k1 between the harmonics.

Figure 10 Results of numerical simulations of the soliton instabilities in the generalized NLS equation with

f �I� � ÿIp=2 � cIp at p � 1:35. (a) Initial conditions are selected near the unstable soliton with b � b0. (b) Initial

conditions are selected near the stable soliton with b � bf.

604

YU. S. Kivshar

Di�raction described by the second-order derivatives in Equations 40 is crucial for theexistence of two-component soliton solutions. The stationary soliton solutions of Equa-tions 40 are presented by real functions v � v0�x; a� and w � w0;a�x� which are independentof z. For the particular value a � 1 these solutions were ®rst found by Karamzin andSukhorukov [92]. Numerical analysis done in [95, 96] revealed the existence of the solitonsolutions of system 40 for any positive value of a.

To analyse the stability of the soliton solutions v0 and w0, we linearize Equations 40 onthe soliton background as the following,

v�x; z� � v0�x� � Vr�x� � iVi�x�� exp�kz� w�x; z� � w0�x� � Wr�x� � iWi�x�� exp�2kz�and investigate the corresponding linear eigenvalue problem for the corrections �Vr; Vi�and �Wr;Wi�. Similar to the case of bright NLS solitons discussed above in Section 8.1, thisproblem can be solved by the asymptotic method for non-zero but small k. Near the in-stability threshold, such solutions are expected to exist only for special values of theparameters a near the critical curve a � ac�r�. The instability threshold, as well as thegeneral dependence k�a; r�, can be found from the corresponding solvability conditions tothe linear problem. This analysis was ®rst presented in [107], and it was shown that theinstability threshold is given by the generalized Vakhitov±Kolokolov criterion,@Q=@b � 0, where b�a; r� � �2rÿ a�ÿ1 is the renormalized soliton propagation constantand

Q � 1

2�2rÿ a�3=2Z �1ÿ1

v20�x; a� � 2rw20�x; a��

dx

is the renormalized power (Menley±Rowe) invariant of Equations 40. In all these formulasthe parameter r is considered as an arbitrary parameter. Using this criterion and thenumerical results on the stationary soliton solutions, the instability threshold curvea � ac�r� has been calculated in [107]. Moreover, it has been shown that the two-waveparametric solitons are unstable for a < ac.

The important physical question is the development of linear instability in the subse-quent dynamics of the two-wave solitons. In order to describe this analytically, we shouldtake into account non-linear e�ects. We select a � a0 close to ac, so that the smallparameter � characterizes the deviation �a0 ÿ ac� � O��2�. Then, it follows from the lineartheory that the growth rate has the order O�� and, therefore, the unstable linear per-turbations grow on the `slow' scale Z � �z. This allows us to introduce the slowly varyingcomplex phase S � S�Z� and look for the perturbed solutions of Equations 40 in the formof asymptotic series v � �v0�x; a� � �2XVr�x; a� � O��3�� exp�i�S�, w � �w0�x; a��2XWr�x; a� � O��3�� exp�2i�S�, where the functions Vr and Wr are the solutions of thelinearized problem, and X � �2rÿ a� dS=dZ describes a correction to the soliton propa-gation constant. Using the asymptotic multi-scale technique we ®nd the non-linearequation for the function X,

Ms ac� � d2X

dZ2� 1

�2@Q@b

��a�a0

X� 1

2

@2Q

@b2

��a�ac

X2 � 0 �41�

where the renormalized soliton mass is de®ned as

Ms � �2rÿ a�ÿ1=2Z �1ÿ1

w20

Wi

w0

� �2

x�v20

Vi

v0

� �2

x� 1

2w0v0Wi ÿ 2w0Vi� �2

" #dx > 0

605


where the index x stand for the corresponding partial derivatives, and the functions Vi andWi are the solutions of the linearized problem. We mention that the derivative@2Q=@b2ja�ac is always positive for the two-wave parametric solutions.It turns out that the asymptotic theory applied to the two-wave v�2� solitons gives, in a

renormalized form, essentially the same equation of motion of the equivalent particle asEquation 39 for the (one-component) bright solitons in non-Kerr optical materials. As wediscussed above, this equation describes two main scenarios of the instability of brightsolitons, either long-lived periodic amplitude oscillations or soliton decay.

Indeed, according to Equation 41, the exponential growth of linear perturbations withX�0� > 0 is stabilized by non-linearity leading to oscillations around a novel stable equi-librium state. This equilibrium state corresponds to a stationary soliton which can also bedescribed by Equations 40 but for a renormalized parameter a lying inside the stabilityregion. Therefore, for a slightly increased amplitude of an unstable soliton our analyticalmodel (Equation 41) predicts in-phase pulsations of the fundamental and second har-monics around a novel stable soliton, and this exactly corresponds to the evolutionobserved numerically, see Fig. 11. For X�0� < 0, according to Equation 41, such a sta-bilization is not possible and, as a result, a slightly decreased amplitude of an unstablesoliton gradually decreases further.

Figure 11 Periodic oscillations of the two-wave soliton of the model 40 after the development of instability

�r � 2; a � 0:05�. (a) Evolution of the harmonics amplitudes vm � jv�0; z�j (solid) and wm � jw�0; z�j (da-

shed). (b), (c) Propagation of the soliton components v and w, respectively.

606

YU. S. Kivshar

8.3. Dark solitons in non-Kerr mediaThe asymptotic analysis presented above for one- and two-component bright solitons hasbeen also developed for dark solitons [54]. Here we consider the dark solitons described bythe generalized NLS equation 6 and derive an asymptotic equation governing their non-linear dynamics near the instability threshold.

The dark soliton of Equation 6 can be presented in the form,

Ws � Us�n; v; q� exp�if �q�z� Us � U exp�ih�where n � xÿ vz, dh=dn � 1

2 v�1ÿ q=U2�, and the real function U�n; v; q� tends to the non-zero boundary conditions at in®nity, U2 ! q as jnj ! 1. We consider, for simplicity, thecase when U2 < q for any n. Thus, the dark soliton depends on two parameters, thevelocity v and the background intensity q. For v! c, where c � �ÿ2qf 0�q��1=2 is the phasevelocity of linear waves propagating on the constant background jWj2 � q, the dark sol-iton has small amplitude, while for v! 0 the soliton amplitude (or `darkness') reaches itsmaximum (but always Imin > 0).

In spite of the fact that the background intensity is determined by the boundary con-ditions, its local variations are still possible, e.g. due to radiation emitted by the perturbeddark soliton during its non-stationary evolution. Therefore, the dynamics of unstable darksolitons is much more complicated than that of bright solitons. However, we show belowthat even in this case, but near the instability threshold, the effective analytical model forthe non-linear dynamics of the dark solitons can be derived with the help of the asymptoticmethod.

As has been already mentioned in Section 7.1.2, the renormalized ®eld momentumde®ned as

M s�v; q� � 2 Im

Z �1ÿ1

@W�s@n

Ws 1ÿ g

Wsj j2 !

dn � ÿvZ �1ÿ1

U2 ÿ qÿ �2

U2dn �42�

plays a key role in the formulation of the variational principle for dark solitons, and also itde®nes the criterion for their linear instability, given by the slope of the derivative dMs=dv.Thus, both linear and non-linear stability analysis for dark solitons is based on therenormalized momentum (Equation 42).

Following the approach outlined above for bright solitons, we present the perturbeddark soliton of the generalized NLS equation 6 in the form W � /�n; v; q; X ; Z�exp�if �q�z� �R�X ; Z��, where n � xÿ v0zÿ

R Z0 V �Z 0� dZ 0, v � v0 � �V �Z�, X � �x, Z � �z,

and �� 1. The constant value v0 stands for the velocity of the (unstable) unperturbeddark soliton which is selected near the critical value vc where the derivative dMs=dv van-ishes. Then, we expand / in the asymptotic series,

/ � Us�n; v; q� � �2/2�n; v; q; X ; Z� � O �3ÿ �

and reduce the generalized NLS equation 6 to a sequence of linear inhomogeneousequations, similar to the case of bright solitons. However, the important di�erence be-tween the asymptotic procedure for bright and dark solitons is the existence of the linearcorrection term to the dark soliton which does not vanish at in®nity, �1=�2��j/j2 ÿ 1� ! u�

as n!1. This non-localized part of /2 in the region of the soliton �n � O�1�� indicatesthe generation of the radiation ®eld escaping the perturbed dark soliton. This radiation®eld can be found as a solution to the generalized NLS equation 6 in the outer regionX � O�1� where jWj2 � q� �2U�r �X ; Z� � O��3�. Analysis shows that, in the leading order,

607


the radiation propagates to the right and to the left with the limiting phase velocity c,U�r � U�r �X � cZ�. At the more slower time scale, the evolution of the radiation ®eldobeys the KdV equations [179],

�4c@U�r@f� aU�r

@U�r@X� @

3U�r@X 3

� 0 �43�

where f � �3z and a � 4�3f 0�q� � qf 00�q��. Then, the matching conditions produce thepro®le of the radiation ®eld generated by the perturbed dark soliton, U�r � u� asn! �1, X ! 0, and Z � X=v0.

After such a technical procedure, in the third-order approximation, we can derive thefollowing di�erential equation for V �Z�

ls vc� � dVdZ� 1

�

@Ms

@v

��v�v0

V � 1

2

@2Ms

@v2

��v�vc

V 2 � 0 �44�

where the positive coe�cient ls�v; q� has the form,

ls�v; q� �qc

@Ss

@v

� �2

� 2cq

@Ps

@v

� �2

> 0

Here

Ss�v; q� � vZ �1ÿ1

1ÿ q

U2

� �dn

and

Ps�v; q� � 1

2

Z �1ÿ1

U2 ÿ qÿ �

dn

are proportional to the total phase shift across the dark soliton and its complementarypower, respectively. Thus, due to the radiation the dark solitons near the instabilitythreshold can be described by an equation for motion of an e�ective inertial and dissipativeparticle of mass ls and velocity V under the action of the frictional force which is pro-portional to the difference M0 ÿMs�v�, where M0 � Ms�v0�.

As follows from Equation 44, the dark soliton becomes unstable for @Ms=@vjv�v0 < 0and the instability dynamics essentially depends on the sign of the perturbation. Usually,the instability is observed for small velocities so that @2Ms=@v2jv�vc > 0. Therefore, if wedecrease the intensity of the unstable dark soliton, the instability results in a monotonictransition from an unstable to a stable soliton realized at the same value of the renor-malized momentum M0. This scenario of the soliton instability is described asymptoticallyby the following solution of Equation 44,

V � V0Vf

Vf ÿ V0� � exp�ÿkZ� � V0�45�

where V0 is the initial velocity of the dark soliton, and the instability growth rate k and the®nal soliton velocity Vf are de®ned through the coe�cients of Equation 44. For the othersign of the perturbation, the instability cannot be described by bounded solutions. Thisimplies that an increase of the intensity of the dark soliton cannot be suppressed by weaknon-linear e�ects described by Equation 44, and it can lead to an essential transformationof the unstable dark soliton. As a result of this transformation, the intensity of the dark

608

YU. S. Kivshar

soliton may fall to zero at the minimum point with the subsequent evolution depending onthe global non-linear properties of the particular case of the generalized NLS model.

The radiation ®eld excited due to the transition of the dark soliton from unstable tostable states can be also found,

U�r � �kDV

4c�c� v�q2

@Ss

@v� c

@Ps

@v

� �sech2�kX=2v�

Since the radiation ®eld develops into a soliton of the KdV equation 43 only for negativeamplitude, the secondary dark soliton is formed in the wave Uÿr which propagates in theopposite direction with respect to the original perturbed soliton. On the other hand, theradiation in a co-propagating wave U�r decays into quasi-linear dispersive waves.

To con®rm the analytical theory of the dark soliton instability, the generalized NLSequation 6 with the non-linearity in the form, f �I� � I ÿ cI2 was integrated numerically[54]. Figure 12 presents two scenarios of the instability development, namely, the decay ofthe unstable dark soliton into two stable solitons of smaller amplitudes which propagate inthe opposite directions (Fig. 12a) and the `collapse'-type dynamics of a dark soliton with aformation of two kink-like fronts (Fig. 12b). Thus, the basic predictions of the asymptoticanalytical theory have been shown to be in good agreement with the results of thenumerical simulations.

9. Concluding remarksWe have discussed di�erent aspects of the theory of spatial optical solitons. From themathematical point of view, these kinds of non-linear waves demonstrate a number of very

Figure 12 Evolution of an unstable dark soliton in the model 6 with f �I� � I ÿ cI2. (a) Transformation to a stable

dark soliton via decay of an unstable soliton into two stable ones. (b) Collapse of an unstable dark soliton into

two kink-type fronts. Top: initial (solid) and ®nal (dashed) states; bottom: the corresponding contour plots.

609


interesting properties which seem generic to spatially localized solutions of non-integrablenon-linear models, e.g. the existence of soliton instabilities, the long-lived oscillations ofthe soliton amplitude due to the soliton internal modes, etc. From the physical point ofview, it seems important that in many cases spatial optical solitons cannot only bedescribed analytically and numerically for a variety of nonlinear models, but, and this isthe most amazing fact, they can also be veri®ed, with relatively good accuracy, by directmeasurements of the beam propagation through non-linear media. This suggests thevalidity of di�erent phenomenological models for describing self-guided beams in opticalmedia with non-resonant and resonant non-linearities.

We would like to emphasize that for many years the properties of non-linear wavesin non-linear models of di�erent physical context have been described by reducing theoriginal and complex model equations to integrable ones. The concept of spatial sol-itons does not allow, in principle, such a simpli®cation. As a result, in presenting theanalytical results, we have tried to avoid the traditional restrictions associated withconsideration of only the cases of integrable models (i.e. the cubic NLS equation forscalar solitons, or the Manakov equations for vector solitons) and their deformations,or even analytically solvable models, as has been done in many previous review papersand books on optical solitons. Instead, we have emphasized the physics of theunderlying phenomena and general concepts associated with more realistic (and usuallynon-integrable) physical models. This approach has involved discussion of solitonstability and instability-induced dynamics of non-Kerr solitons, since in non-integrablemodels solitary waves can become unstable. We believe that the concept of spatialoptical solitons gives us one of the examples where, even dealing with a variety ofnon-integrable models, we are able to develop more general physical approaches andanalytical tools for explaining, at least qualitatively, a number of e�ects observed inexperiments.

As for the future application of spatial solitons, many problems involving the soliton-based concept of light guiding light are still not solved experimentally, and this de®nitelywill require more e�ort to make this concept a practical reality. Searching and employingnew materials with strong non-linear properties may su�ciently speed up this process.However, even now we can say that the rapid progress of the physics of spatial opticalsolitons has already resulted in the development of novel mathematical techniques andconceptual approaches to deal with solitary waves of non-integrable systems, and thesetools are expected to be useful for many other ®elds where non-linear properties of wavepropagation become important.

AcknowledgementsI would like to thank Professor G. Stegeman for his kind invitation to present my view ofthe recent advances in the theory of non-Kerr spatial solitons in this special issue. Di�erentparts of this paper are based on the original research I have been carrying out during thelast years in collaboration with my colleagues and PhD students, including Dr V. V.Afanasjev, Dr O. Bang, Dr A. V. Buryak, Dr D. E. Pelinovsky, Professor R. A. Sammut,Mrs V. V. Steblina, and Dr S. Trillo. I thank them all for a very productive collaboration.I also wish to acknowledge many useful and encouraging discussions with ProfessorB. Luther-Davies, Professor M. Segev, and Professor A. W. Snyder, who contributedenormously into my understanding of the properties of self-guided beams in non-Kerroptical media.

610

YU. S. Kivshar

This work was supported by the Australian Photonics Cooperative Research Centre, theAustralian Research Council, the Department of Industry, Sciences, and Tourism, theAustralian Academy of Sciences, and the Research School of Physical Sciences andEngineering, through research projects and research grants on nonlinear waveguides,spatial optical solitons, and light guiding light.

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