Collapse in the nonlocal nonlinear Schrödinger equation

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IOP PUBLISHING NONLINEARITY

Nonlinearity 24 (2011) 1987–2001 doi:10.1088/0951-7715/24/7/005

Collapse in the nonlocal nonlinear Schrödingerequation

F Maucher1, S Skupin1,2 and W Krolikowski3

1 Max Planck Institute for the Physics of Complex Systems, 01187, Dresden, Germany2 Friedrich Schiller University, Institute of Condensed Matter Theory and Optics, 07743 Jena,Germany3 Laser Physics Centre, Research School of Physics and Engineering, Australian NationalUniversity, Canberra, ACT 0200, Australia

E-mail: fabian@pks.mpg.de

Received 14 September 2010, in final form 29 April 2011Published 24 May 2011Online at stacks.iop.org/Non/24/1987

Recommended by M J Peyrard

AbstractWe discuss spatial dynamics and collapse scenarios of localized waves governedby the nonlinear Schrödinger equation with nonlocal nonlinearity. Firstly, weprove that for arbitrary nonsingular attractive nonlocal nonlinear interaction inarbitrary dimension collapse does not occur. Then we study in detail the effectof singular nonlocal kernels in arbitrary dimension using both Lyapunoff’smethod and virial identities. We find that in the one-dimensional case, i.e. forn = 1, collapse cannot happen for nonlocal nonlinearity. On the other hand,for spatial dimension n � 2 and singular kernel ∼1/rα , no collapse takes placeif α < 2, whereas collapse is possible if α � 2. Self-similar solutions allowus to find an expression for the critical distance (or time) at which collapseshould occur in the particular case of ∼1/r2 kernels for n = 3. Moreover,different evolution scenarios for the three-dimensional physically relevant caseof Bose–Einstein condensates are studied numerically for both the ground statesoliton and higher order toroidal states with, and without, an additional localrepulsive nonlinear interaction. In particular, we show that the presence of localrepulsive nonlinearity can prevent collapse in those cases.

Mathematics Subject Classification: 35Q55, 60G18, 35B44

PACS numbers: 42.65.Jx, 03.75.Kk, 52.35.Mw

(Some figures in this article are in colour only in the electronic version)

0951-7715/11/071987+15$33.00 © 2011 IOP Publishing Ltd & London Mathematical Society Printed in the UK & the USA 1987

http://dx.doi.org/10.1088/0951-7715/24/7/005mailto: fabian@pks.mpg.dehttp://stacks.iop.org/no/24/1987

1988 F Maucher et al

1. Introduction

Spatial and/or temporal dynamics of waves in many nonlinear systems are often governedby the famous nonlinear Schrödinger (NLS) equation [1]. This universal equation appears inmany diverse physical settings including, for instance, nonlinear optics [2], Bose–Einsteincondensate (BEC) [3–5], water waves [6] and is even discussed in the context of roguewaves [7]. A particular solution to this equation—a soliton, represents a nonlinear localizedbound state that does not change its shape when propagating or evolving in time and/orspace [8]. The existence of solitons is brought about by the balance between the natural effectof dispersion or diffraction which tends to spread the wave and the self-focusing, provided bythe nonlinear response of the medium induced by the wave itself. In the case of optics thenonlinearity represents the self-induced refractive index change, while, e.g. in BECs it is justthe nonlinear potential of bosonic interaction. In both cases the presence of the nonlinearitymay lead to self-focusing and consequently, self-trapping of light or condensate. When thisnonlinear potential induced by the wave itself is strong enough for the self-focusing to dominateover the effects of dispersion/diffraction, the amplitude of the wave may grow to infinity(blow-up) within finite time or distance. This phenomenon is often referred to as a collapse.Collapse is a well-known effect associated with dynamics of finite waves in nonlinear systems.It has been studied in several physical settings including, e.g. plasma waves and nonlinearoptics [9], BEC [10], as well as collapse of very heavy stars into black holes [11]. In themajority of works devoted to collapse the nonlinear potential induced by the wave is of a localcharacter, i.e. the system response in the particular spatial location is determined solely by thewave amplitude (or intensity) in the same location.

Here we will discuss the dynamics of waves and collapse in systems exhibiting a spatiallynonlocal nonlinear response. In nonlocal media the nonlinear response at the given pointdepends on the wave intensity in an extended neighbourhood of this point. The extent ofthis neighbourhood in comparison with the spatial scale of the localized wave determines thedegree of nonlocality [12]. It appears that spatial nonlocality is a generic property of manynonlinear systems and is often a result of an underlying transport process, such as ballistictransport of atoms [13] or atomic diffusion [14] in atomic vapours, heat transfer in plasma [15]and thermal media [16, 17], or drift of electric charges as in photorefractive crystals [18]. It canalso be induced by a long-range particle interaction as in nematic liquid crystals which exhibitorientational nonlocal nonlinearity [19] or in colloidal suspensions [20]. Recent increasedinterest in nonlocal nonlinearities has been stimulated by the research on BECs where theinter-particle nonlinear interaction potential is inherently nonlocal [5, 21, 22].

In contrast to local attractive nonlinearity, which leads to collapse in more than onedimension, nonlocal nonlinearities may actually prevent collapse. Turitsyn was the first toprove analytically the absence of collapse in the nonlocal NLS equation for three differentsingular nonlocal kernels in [23], including the singular gravity-like 1/r-kernel. In a later workPerez-Garcia et al investigated the collapse dynamics of the nonlocal Schrödinger equationusing variational techniques and numerical simulations [24]. They showed, in particular, thatfor the nonsingular nonlocal nonlinearity collapse is always arrested. The absence of collapsefor a singular 1/r-response and form stability of higher order states, such as dipole and torus,have recently been verified numerically in [25]. More general treatment of the NLS and ensuingdynamics and collapse is presented in [26, 27]. As far as collapse in the general case of thenonlocal Schrödinger equation is concerned, Bang et al showed rigorously that nonlocalityarrests the collapse as long as the Fourier spectrum of the nonlocal nonlinear potential isabsolute integrable [28]. Rigorous recent studies of collapse in three dimensions in the case ofa particular singular form of the nonlocal nonlinear potential have been given in [29–31]. Here,

Collapse in the nonlocal nonlinear Schrödinger equation 1989

we will generalize those previous findings to arbitrary number of transverse dimensions, andshow that for the nonlocal nonlinearity represented by singular kernels ∼1/rα , α = 2 thereis a sharp threshold for collapse. By means of (3 + 1)-dimensional numerical simulations weelucidate different evolution scenarios, and show that an additional local repulsive nonlinearinteraction may prevent collapse.

2. Model

In this paper we will consider the self-trapped evolution of the wave function ψ(x, t) describedby the nonlocal nonlinear Schrödinger equation (NNLS), given by

i∂tψ(x, t) = −�ψ(x, t) −(∫

A(x − y)|ψ(y, t)|2 dy)

ψ(x, t), (1)

where x, y ∈ Rn with n being the spatial dimension. A(x) is the so-called nonlocal responseor kernel which represents the nonlocal character of the nonlinearity of the medium. Its widthdetermines the degree of nonlocality [12]. In particular, for A(x) ∼ δ(x) the above equationdescribes the standard Kerr-type local medium. While this is only a phenomenological modelit nevertheless describes very well general properties of nonlocal media. Moreover, for certainforms of the nonlocal response function this model represents actual experimentally realizablephysical systems. This is the case of nematic liquid crystals [19, 32] or thermal nonlinearityin liquids [17] and plasma [15] where, in one-dimensional case A(x) ∼ exp(−|x|) and intwo dimensions A(x) is proportional to the modified Bessel function of the second kind [33].In addition, equation (1) (also known as the Gross–Pitaevskii equation) governs the spatio-temporal dynamics of BECs condensates where the analytical form of the nonlocal Kernel is aconsequence of the dipolar character of the long-range particle–particle interaction [22, 34–36].While, in general, the nonlocal interaction potential in condensates is of rather complex andanisotropic character, under certain experimental conditions it may acquire ‘gravity-like’ 1/rform [37].

3. Analytical estimates for possible evolution scenarios

In a vivid picture, collapse refers to a phenomenon when the amplitude of the wave functionψ goes to infinity within finite time or distance (for stationary processes). However, in themathematical context collapse is usually defined by means of a diverging gradient norm ‖∇ψ‖2(see appendix A for notations). For practical considerations, this is essentially the same,because a wave function maintaining a finite L2 norm, ‖ψ‖2, attains infinite amplitude whenits gradient norm ‖∇ψ‖2 diverges [38].

3.1. Lyapunoff methods

In this subsection, we will use Lyapunoff’s method [39], to discuss collapse scenarios. Theidea is, roughly speaking, to show that the Hamiltonian

H =∫

|∇ψ |2 dx − 12

∫ ∫|ψ(x)|2A(x − y)|ψ(y)|2 dx dy (2)

for equation (1) is bounded from below. Then, since in our case the L2 norm ‖ψ‖2 and theHamiltonian H are conserved quantities, it is possible to show that the gradient norm ‖∇ψ‖2is bounded from above, which implies that it cannot diverge. Moreover, we can expect theexistence of at least one stable soliton solution when the Hamiltonian is bounded (see, e.g. [40]).

1990 F Maucher et al

We can decompose a general kernel A = A+ +A−, A+ � 0 and A− � 0. Then, the kernelsA+ and A− represent attraction (in BEC) or focusing (in nonlinear optics) and repulsion ordefocusing, respectively. Since the amplitude of the wave function has to become infinity ata certain point, one can expect that a singular kernel A+ promotes collapse. Using Young’sconvolution theorem, one can immediately infer the following related fact. Without loss ofgenerality, we restrict our following considerations to solely attractive kernels A = A+.Lemma 1. Let A ∈ L∞(Rn), ψ(t = 0, x) ∈ H 1(Rn). Then no collapse occurs at all times,since ‖∇ψ‖22 is bounded from above.

Proof.

H � ‖∇ψ‖22 − 12C‖ψ‖42‖A‖∞. (3)Here, C = 1 is a sharp constant (see, e.g., [41]). Since ‖ψ‖2 and the Hamilton H itself areconserved quantities under the flow of time (or propagation), one can read inequality (3) asbounding ‖∇ψ‖22 from above in terms of conserved quantities. �

Let us try to connect inequality (3) with the earlier results of the stability of finite beamsgoverned by the NNLS equation (1) [28]. This work states that in order to arrest collapse thespectrum of the nonlocal kernel A should be absolutely integrable. Since ‖A‖∞ � C‖Ã‖1,where à denotes the Fourier transform of A (see appendix A), inequality (3) clearly containsthe result of [28] and also explains earlier reports on collapse arrest for nonsingular kernelsobserved in numerical simulations [24].

Further on, inequality (3) shows that collapse can only be relevant if the kernel A divergesat a certain point (usually at the origin x = 0). Then the assumption A ∈ L∞(Rn) is violated,as is the case, e.g., of local interaction [A ∼ δ(x)]. However, a diverging kernel A does notnecessarily imply collapse. In order to ease the restrictions on the nonlocal kernel which ensurethe collapse arrest we have to restore to the gradient norm ‖∇ψ‖2 apart from the L2-norm‖ψ‖2. As we will show below one can weaken the sufficient restriction A ∈ L∞ by employingthe Hardy–Littlewood–Sobolev inequality (see appendix B) and the Gagliardo–Nirenberg–Sobolev inequality (see appendix C).

Lemma 2. Let x ∈ Rn, A = f (x)/rα , r = |x| with 0 � f (x) < ∞, α < min(2, n),ψ(t = 0, x) ∈ H 1(Rn), κ := maxxf < ∞. Then collapse cannot occur.

Proof.

H � ‖∇ψ‖22 −1

2κ

∫ ∫ |ψ(x)|2|ψ(y)|2|x − y|α dy dx

� ‖∇ψ‖22 −1

2κC‖ψ‖42n/(n−α/2) = ‖∇ψ‖22 −

1

2κC

(∫|ψ |2n/(n−α/2) dx

)4 n−α/22n. (4)

In the second step we just applied the Hardy–Littlewood–Sobolev inequality (B.1). Introducingσ = α/(2n − α), we can rewrite the last expression in (4) and find:

H � ‖∇ψ‖22 −1

2κC

(∫|ψ |2n/(n−α/2) dx

)4 n−α/22n= ‖∇ψ‖22 −

1

2κC

(∫|ψ |2+2σ dx

) αnσ

� ‖∇ψ‖22 −1

2κC

[K

(∫|∇ψ |2 dx

) σn2

(∫|ψ |2dx

) 2+σ(2−n)2

] αnσ

= ‖∇ψ‖22 −1

2C̃‖∇ψ‖α2 ‖ψ‖4−α2 . (5)

Collapse in the nonlocal nonlinear Schrödinger equation 1991

Here, we employed the Gagliardo–Nirenberg–Sobolev inequality (C.1), and C̃ := κCKα/nσis some positive constant independent of ψ . Since α < 2 and because the Hamiltonian H andthe L2-norm ‖ψ‖2 are conserved quantities, inequality (5) can be read as a uniform bound ofthe gradient norm ‖∇ψ‖2. �

For dimension n = 1, the convolution term in the Hamiltonian H and expression (4)diverge without regularization for singular kernels with α � 1, and hence singularity cannotbe strong enough to promote collapse. Generally, when regularizing the kernel by, e.g.,introducing a cut-off, again the wave function does not collapse since the kernel is thennonsingular.

If we apply the lower bound for the Hamiltonian (5) for the particular exponent α = 2we find

H � ‖∇ψ‖22(1 − 12 C̃‖ψ‖22). (6)Because we know the constant C̃ := κCKα/nσ [κ = 1, C = C(n, α) according toequation (B.2), K = K(σ, n) according to equation (C.2)] we can use inequality (6) to computea lower bound for the ‘critical mass’ ‖ψ‖22 = 2/C̃ for collapse. In the physically relevantcase of n = 3, we find this lower bound numerically as 2/C̃ = 2.80,4, which is slightlylower than the numerical result for the mass of the ground state soliton, ‖ψGS‖22 = 2.85 (seesection 4.1). This finding may seem surprising, because for the local nonlinear Schrödingerequation the exact soliton mass can be computed from sharp constants in the lower bound ofthe Hamiltonian [42, 43]. However, a closer look at the Hardy–Littlewood–Sobolev inequality(appendix B) reveals that equality can be expected only for wavefunctions ψ of the form(B.3). Since our ground state solitons ψGS(x) are not of the form (B.3)5, we have toexpect 2/C̃ < ‖ψGS‖22. Finally, we verified numerically that for the ground state soliton(B.1) is indeed an inequality. Interestingly, then the ratio of rhs and lhs of (B.1) is about2.85/2.80 = ‖ψGS‖22 × C̃/2.

3.2. Virial identities

The idea to use virial identities to derive sufficient conditions for collapse was first establishedin the context of the local two-dimensional nonlinear Schrödinger equation by Vlasov et al[44], and the resulting condition H < 0 is often referred to as the VPT criterion. The originalidea behind the VPT criterion has been extended to various modifications of the original NLSequation, e.g., the Gross–Pitaevskii equation for a BEC with dipole–dipole interactions [29]or, more general, the NLS with 1/rα nonlocal response in three dimensions [31]. Followingthe calculations presented in [31], one may observe that the derived virial identities hold forarbitrary dimension. The virial is given by V (t) := ∫ r2|ψ |2 dx with r = |x|.Lemma 3. Consider a kernel A = 1/rα , with x ∈ Rn and α < n. Then the second timederivative of the virial in arbitrary dimension n is given by

∂ttV = 8(∫

|∇ψ |2 dx − α4

∫ ∫ |ψ(y)|2|ψ(x)|2|x − y|α dy dx

)

= 8H − 2(α − 2)∫ ∫ |ψ(y)|2|ψ(x)|2

|x − y|α dy dx. (7)

4 Note that the constant K = K(σ, n) involves the quantity ‖R‖2σ2 , which has to be computed numerically.5 limr→∞ ψGS ∼ exp(−r) � r−4, with |x| = r .

1992 F Maucher et al

Proof. This task was already done for three dimensions in [31]. Therefore we only brieflysketch the derivation of this virial identity here. Apparently,

∂tV = 4�∫

ψ∗x∇ψ dx. (8)

Using the equation of motion, a longer but straightforward calculation leads to (see also [30])

∂ttV = 8∫

|∇ψ |2 − 4n∫ |ψ(x)|2|ψ(y)|2

|x − y|α dy dx + 4∫

x∇(|ψ(x)|2)∫ |ψ(y)|2

|x − y|α dy dx.(9)

Integrating the last term by parts and rewriting it in a more favourable way gives∫x∇(|ψ(x)|2)

∫ |ψ(y)|2|x − y|α dy dx (10)

= n∫ ∫ |ψ(x)|2|ψ(y)|2

|x − y|α dy dx −α

2

∫ ∫x

(x − y)|ψ(x)|2|ψ(y)|2|x − y|α+2 dy dx

−α2

∫ ∫y

(y − x)|ψ(x)|2|ψ(y)|2|x − y|α+2 dy dx. (11)

Plugging this into equation (9), the first term cancels out, and the two symmetric terms withrespect to x and y add up and give the desired expression. �

The restriction α < n guarantees applicability of the Hardy–Littlewood–Sobolevinequality, and hence finiteness of the convolution term. In particular, lemma 3 implies∂ttV = 8H for A = 1/r2 (or α = 2) in arbitrary dimension. Together with a particularform of Hölder’s inequality∫

r|ψ ||∇ψ | dx �(∫

r2|ψ |2 dx)1/2 (∫

|∇ψ |2 dx)1/2

, (12)

we can derive a sufficient criterion for collapse.

Lemma 4. Let x ∈ Rn, n > 2, A = 1/r2, ψ(t = 0, x) ∈ H 1(Rn), and the HamiltonianH < 0. Then the solution ψ undergoes collapse.

Proof. As seen above, the second time derivative of the virial V equals eight times theHamiltonian. The virial is positive for t = 0 and has the apparent meaning of the mean widthof ψ . Hence, for a negative Hamiltonian, the virial vanishes at a certain finite time t .

Using partial integration and Hölder’s inequality (12), one can infer that a vanishing virialleads to collapse due to conservation of the L2-norm ‖ψ‖2 of the wavefunction and∫

|ψ |2 dx � C(∫

r2|ψ |2 dx)1/2 (∫

|∇ψ |2 dx)1/2

, (13)

where the positive constant C does not depend on ψ . �

In other words, if the mean width or the virial becomes zero, the gradient norm has todiverge in order to keep the L2-norm constant. Closer inspection of lemma 3 reveals that anegative Hamiltonian also implies collapse for α > 2, as already pointed out in [31]. Togetherwith lemma 2, we see that α = 2 is a sharp boundary for collapse in arbitrary dimension n > 2.

Collapse in the nonlocal nonlinear Schrödinger equation 1993

3.3. Self-similar solutions for the 1/r2 kernel

Let us now have a look at the special case A = 1/r2. As we have seen in the previous section,this singular kernel marks the boundary between collapsing and noncollapsing solutions. Thisis somewhat similar to the two-dimensional local nonlinear Schrödinger equation, and we willfind many aspects of this famous equation here. However, in the nonlocal case the collapsethreshold does not appear as a critical dimension, but as a critical exponent of the singularityof the kernel.

The NNLS equation with kernel A = 1/r2 appears as equation of motion of the followingLagrangian density,

L := i2(ψ∂tψ

∗ − ψ∗∂tψ) + |∇ψ |2 − 12|ψ |2

∫1

|x − y|2 |ψ(y, t)|2 dy. (14)

To simplify the considerations and following [9], let us assume radial symmetry. A possibleansatz for a self-similar solution (or similariton [45]) is

ψ = B(t)R(r/a(t))eiθ(r,t) (15)with r = |x|. Here, R(t/a(t)) is a real profile and a(t) is the wave radius. Due to theconservation of the L2-norm ‖ψ‖2 and the related continuity equation, which comes aboutfrom the global U(1) invariance of the Lagrangian, one finds the interdependence between Band a:

|B|2an =: P = const. (16)An appropriate ansatz for the radially symmetric phase is

θ(r, t) = θ2(t)r2 + θ0(t). (17)Using θ and P , one may express the spatially integrated Lagrangian density as follows:

L = −|B|2an[κ0∂tθ0 + κ2a

2(∂t θ2 + 4θ22 ) +

γ

a2

]− β|B|4a2n−2, (18)

where we introduced

κm :=∫

rmR2(r) dx, β := 1/2∫

R2(|y|)R2(r)|x − y|2 dx dy, γ :=

∫(∂rR(r))

2 dx. (19)

The variational problem δL = 0 with respect to B, θ0, θ2 and a gives the equations of motionfor these parameters. Using P = |B|2an and θ2 = (∂ta)/4a, one may rewrite the equation ofmotion for a in the form

∂tta = 4γκ2

1

a3− 4βP

κ2

1

a3. (20)

Elementary integration with the initial conditions a0 := a(t = 0), and ∂ta(t = 0) = 0 gives

a(t) = a0√

t2(

4γ

κ2a40

− 4βPκ2a

40

)+ 1. (21)

Now one can estimate the critical time tcr, where a → 0 as

tcr =(

4βP

κ2a40

− 4γκ2a

40

)−1/2. (22)

In terms of P , one may express the condition for collapse as P > γβ

. Compared with thevirial results from the previous section, this threshold value indicates that the Hamiltonian Hbecomes negative.

1994 F Maucher et al

It is important to stress here that the variational approach presented above strongly relies onthe ansatz equations (15) and (17). A priori, there is no guarantee that variational results showeven qualitative agreement with the exact solutions. A classical example of unsuccessfulapplication of the variational method is given in [46], where the authors show that thedescription of solitons interacting with radiation in the one-dimensional local NLS equationfails. However, as far as collapse in the two- and three-dimensional local NLS is concerned, theabove ansatz can give useful estimates [9, 47, 48], but also here counter-examples exist [49]. Aswe will see in the following sections, in our case variational predictions show some qualitativeagreement with rigorous numerical simulations in the physically relevant case of n = 3. Inthe light of [48], this is indeed the best we can expect.

4. (3 + 1)-dimensional numerical investigation of collapse dynamics and collapse arrestfor the 1/r2 kernel

In this section we will investigate the dynamics of finite wave packets using numericalintegration of the NNLS equation (1). We start by using variational methods to find familiesof approximate localized solutions to the stationary version of this equation, and comparethem with their numerically exact counterparts computed by imaginary time evolution [50]or iteration technique [33]. Both of them will be subsequently used as initial conditions innumerical simulations in order to illustrate possible scenarios of time evolution of the wavefunction in the case of the nonlocal 1/r2 kernel. Our (3+1)-dimensional simulation code isbased on a Fourier split-step method. In order to resolve the singularity of the kernel properly,which is of crucial importance when studying collapse phenomena, we use a method outlinedin appendix D. In the following, we will also investigate the impact of an additional localrepulsive (defocusing) term on the collapse dynamics. In particular, we will be interestedwhether the presence of the repulsive local interaction can arrest the collapse. To this end, letus consider a NNLS equation of the following form:

i∂tψ(x, t) = −�ψ(x, t) −[∫

1

|x − y|2 |ψ(y, t)|2 dy − δ|ψ(x, t)|2

]ψ(x, t), (23)

where the parameter δ in front of the local term is either 0 or 1. On physical grounds, theadditional local repulsive term occurs in BECs as a result of s-wave scattering [5].

4.1. Approximate variational and exact numerical solutions for the ground state

To find approximate localized ground state solutions to the NNLS equation we use the followingansatz:

ψ(x, t) := A0 exp(

− r2

2σ 2

)exp(iEt) (24)

and vary the corresponding Lagrangian with respect to width σ and amplitude A0. Here, werestrict our considerations to dimension n = 3, so that x = (x1, x2, x3). The results are shownin figures 1(a) and (b), which depict the so-called existence curves for the solutions, i.e., therelations between mass M := ‖ψ‖22, energy E and width σ .

The solid lines in figure 1(a) represent families of solutions obtained by variational analysiswith (black line) and without (blue line) repulsive local interaction (δ = 0), and the dashedgreen horizontal line together with the blue dots show the same for the numerically exactsoliton solution of equation (23). We expect from this plot that the ground state may indeedcollapse without repulsion (δ = 0), since for fixed mass M the width σ can be arbitrarily small.Typical length-, time- and amplitude-scales Rc, Tc and ψc for an exact solution behave like

Collapse in the nonlocal nonlinear Schrödinger equation 1995

Figure 1. Existence curves of the ground state soliton ((a) and (b)) obtained by variational analysisand numerical computation for nonlocal attractive 1/r2 response (blue solid and dashed greenline, respectively) and with competing local repulsive interaction (black solid and dotted blue line,respectively). (c) Time evolution of the amplitude of the numerically exact ground state. If the massof the ground state is increased, the solution collapses (shown here for

√1.01 and 1.01 × ψGS),

and if the mass is decreased, the solution spreads upon propagation (shown here for 0.99 and√0.99 × ψGS). The inset shows a zoom into the early propagation stage.

1/Tc ∼ 1/R2c ∼ ψ2c Rc. Hence, the invariance of the mass M with respect to energy E canbe expected from M ∼ ψ2c R3c ∼ const. The same argument holds for arbitrary dimension n,where we have 1/R2c ∼ ψ2c Rnc /R2c . In contrast, when including the additional local repulsiveinteraction competing with the nonlocal attractive one, we expect the absence of collapse fromfigure 1(a), since the mass M of the ground state is increasing with decreasing width σ .

Before presenting rigorous numerical simulations of the time evolution of the abovesolutions, let us evaluate the expression for the critical collapse time tcr derived in section 3.3.Plugging the Gaussian approximate variational ground state as well as the numerically exactsoliton as self-similar profile into equation (22), the critical time tcr diverges, and we expectthat collapse happens at infinite times. However, when multiplying the profiles by some factorlarger than one, tcr becomes finite, and we expect collapse at time tcr.

These predictions are confirmed by our numerics for the exact ground state, as can be seenin figure 1(c). The ground state is self-trapped and evolves with small change in amplitude(invisible in the figure) due to numerical imperfections. If we multiply the exact ground state bya factor smaller than one (e.g.

√0.99), it spreads with time, whereas if we multiply it by a factor

larger than one (e.g.√

1.01), it collapses. The critical times tcr obtained from equation (22)are in excellent agreement with the simulations. In contrast, using the approximate variationalground state as initial condition, we see that it collapses (figure 2(a)) after rather short time.Thus, the estimate for the critical time tcr = ∞ is incorrect. This is related to the factthat the approximate variational Gaussian profile has a larger mass than the exact soliton(M = 3 > MGS). When we multiply the variational ground state by a factor larger than one(see figure 2(b)), we see collapse earlier than predicted by equation (22). Collapse-times tcr aresystematically overestimated. Finally, collapse can be prevented by adding a local repulsiveterm to the nonlinear potential in the original NNLS equation (see figure 2(c)).

The time evolution of the initially Gaussian variational ground state in figure 2(a) canprovide us with some deeper insight into the collapse dynamics. From our previous attemptsto predict the critical collapse time tcr, we may suspect that the exact soliton itself is the self-similar profile. In order to corroborate this hypothesis, figures 3(a)–(c) show some snapshotsof the radial profiles during time evolution corresponding to figure 2(a). Already after evolvingover a short time, the wave function clearly deviates from a Gaussian profile, and resemblesmore the numerically exact solution (see black curves in figure 2(b)). Further propagation

1996 F Maucher et al

Figure 2. The evolution of the amplitude for the initial variational ground state is shown in (a). Theblue circles correspond to snap shots of the evolution shown in figure 3. In (b), the evolution of thevariational ground state multiplied by 2 (left curve) and 1.5 (right curve) is plotted and comparedwith the estimates for the critical time tcr . In (c), the possibility to prevent collapse using a localrepulsive term is illustrated: the variational ground state multiplied by a factor of 5 remains muchsmaller than the collapsing ones without repulsion shown in (b).

Figure 3. Snapshots of the time evolution of the initially Gaussian variational ground state offigure 2(a) at (a) t = 0, (b) t = 1.95 and (c) t = 2.25. The black curves show numerically exactsolitons with the same amplitude. Clearly, some part of the initially Gaussian wave function isradiated away, and the actually collapsing part converges towards the exact soliton.

shows that the collapsing solution converges to the exact ground state. A similar behaviouris known for the classical two-dimensional local nonlinear Schrödinger equation, where thefamous Townes soliton naturally emerges in the collapse process (see, e.g. [51, 52]). Moreover,we checked that an initially ellipsoidal wave function with slightly too large mass will collapseas a radially symmetric formation, which again resembles the numerically exact solution (notshown).

4.2. Evolution scenarios for toroidal states

Finally, we want to have a look at higher order solitons and their collapse behaviour. Forthis purpose, we consider toroidal states, i.e. the three-dimensional generalization of a vortex.Toroidal states have an azimuthally symmetric density together with an azimuthal phase ramp,which leads to a phase singularity of integer topological charge m in the origin

ψ(r, φ, z, t) = ρ(r, z) exp(imφ) exp(iEt). (25)In order to find numerically exact toroidal solitons (or vortices), we use imaginary timeevolution starting from tori computed from a variational ansatz. As for the ground statesoliton in section 4.1, we find that adding 1% to the mass M leads to collapse of the toroidal

Collapse in the nonlocal nonlinear Schrödinger equation 1997

Figure 4. Time evolution scenarios for the single charged (m = 1) toroidal soliton. (a) Timeevolution of the amplitude. When reducing the mass by 1%, the solution spreads, whereas itcollapses for masses M � 7.98. (b)–(f ) The vortex may decay into two ground state solitonsdue to an azimuthal instability, which then collapse individually. (g) The azimuthal instability canbe triggered by adding random noise (here 1% in amplitude), so that instead of spreading (bluesolid line) a torus with M = 7.90 decays and collapses (black dashed line). The corresponding3D evolution of the modulus square of the wave function in the (x1, x2) plane (x3 = 0) isdepicted in (b)–(f ).

soliton, whereas subtracting 1% leads to delocalization of the wavefunction, which cannot self-trap anymore (see figure 4(a)). However, the toroidal state is azimuthally unstable. Addinga small amplitude noise is enough to trigger the instability and due to the interplay betweenazimuthal phase ramp and amplitude modulation along the ring the whole structure starts torotate [53]. Upon further propagation, the ring breaks into two humps (ground state solitons)that eventually collapse individually (see figures 4(b)–(f )). This secondary collapse is possiblebecause the mass of the torus fulfils MTorus > 2 × MGS. As for the ground state, we verifiedthat collapse can be prevented by additional local repulsive interaction.

5. Conclusions

The phenomenon of collapse in the nonlocal nonlinear Schrödinger equation has beeninvestigated in arbitrary dimensions. We showed that nonsingular nonlocal kernels do notsupport collapsing solutions. More generally, it has been shown that for singular kernels∼1/rα , the necessary condition for collapse to occur is α � 2. Various collapse scenarios forthe 1/r2 kernel have been studied numerically in the physically relevant case of three transversedimensions (n = 3). The critical time (or distance) when collapse occurs has been estimatedusing self-similar solutions. Apart from the ground state solution, collapse of a toroidal statewith the absence and the presence of local repulsive interaction has also been studied in thisparticular system. Generally, it appears that the initial torus is azimuthally unstable. Finally,we showed that independently of the initial state, collapse can be arrested by adding a localrepulsive contribution to the attractive 1/r2 kernel.

Acknowledgments

The authors (SS) would like to thank SK Turitsyn for fruitful discussions and the anonymousreferees for valuable comments which helped to improve the paper. Numerical simulations

1998 F Maucher et al

were performed using HPC resources of the Australian Partnership for Advanced Computing(APAC) and at Rechenzentrum Garching. This research was supported by the AustralianResearch Council.

Appendix A. Notations

We use common functional analysis definitions. The symbol Lp(Rn) denotes the space ofmeasurable functions for which the Lebesgue measure

‖ψ‖p :=(∫

|ψ |p dx)1/p

(A.1)

exists. A function is in L∞(Rn), if there is a constant C > 0, such that |ψ | is smaller thanC for almost every x ∈ Rn. The symbol H 1(Rn) (Sobolev space) is the space of measurablefunctions for which the norm, defined as

‖ψ‖H 1 := ‖∇ψ‖2 + ‖ψ‖2 (A.2)is smaller than ∞. Here, ∫ denotes ∫ ∞−∞.

The Fourier transform f̃ of a function f is given by

f̃ (k) =∫

e−2iπ(k,x)f (x) dx (A.3)

with x, k ∈ Rn, and (·, ·) denotes the scalar product.Apart from that, |x| =

√x21 + x

22 + · · · = r is the usual Euclidean distance.

Appendix B. Hardy–Littlewood–Sobolev inequality

Let p, l > 1 and 0 < α < n with 1/p + α/n + 1/l = 2. Let f ∈ Lp(Rn) and h ∈ Ll(Rn).Then there exists a sharp constant C(n, α, p) independent of f and h, such that∣∣∣∣

∫ ∫f (x)h(y)

|x − y|α dx dy∣∣∣∣ � C(n, α, p)‖f ‖p‖h‖l . (B.1)

The explicit functional dependence of the C(n, α, p) is known (see, e.g., [41]). For ourpurposes, the special case p = l = 2n/(2n − α) is important, where the sharp constant isgiven by

C(n, α, p) = C(n, α) = πα/2 �(n/2 − α/2)�(n − α/2)

[�(n/2)

�(n)

]−1+α/n. (B.2)

In this case there is equality in (B.1) if and only if h = Bf andf (x) = A (γ 2 + |x − a|2)−(2n−α)/2 (B.3)

for some complex constants A and B, 0 = γ real and a ∈ Rn [41].

Appendix C. Gagliardo–Nirenberg–Sobolev inequality

Let f ∈ H 1(Rn) and 0 < σ < 2/(n − 2). Then there exists a sharp constant K(σ, n)independent of f , such that∫

|f (x)|2σ+2 dx � K(σ, n)(∫

|∇f (x)|2 dx) σn

2(∫

|f (x)|2 dx) 2+σ(2−n)

2

. (C.1)

Collapse in the nonlocal nonlinear Schrödinger equation 1999

The explicit functional dependence of the K(σ, n) is known (see, e.g., [1]), and reads

K(σ, n) = (σ + 1)2(2 + 2σ − σn)−1+σn/2

(σn)σn/2

1

‖R‖2σ2, (C.2)

where R is the positive solution of �R − R + R2σ+1 = 0.

Appendix D. Numerical implementation of the singularity of 1/r2

Since we study collapse phenomena, it is crucial to define the properly in a numerical sensekernel 1/r2 and, in particular, its value at r = 0 . To this end we used the following procedure(see, e.g. [54]). Firstly, we decompose the singular kernel into a short range, singular at theorigin, and a nonsingular, long-range contributions.

1

r2= e

−r2/w20

r2+

1 − e−r2/w20r2

. (D.1)

The long-range part can be used directly as it is in a real space, whereas the short-range part

( e−r2/w20

r2) will be treated in the Fourier domain. In our numerics, the parameter w0 was chosen

to be w0 = 7�x/2π , with �x the step size of the spatial grid. In the Fourier domain, theproduct of two functions becomes a convolution, that is in particular also well defined forcertain singular functions such as 1/rα with α < n, if the other function is ‘well behaved’.Hence, in the Fourier domain the short ranged contribution can be easily calculated, and thelimit |k| → 0 is then well defined.

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1. Introduction2. Model3. Analytical estimates for possible evolution scenarios3.1. Lyapunoff methods3.2. Virial identities3.3. Self-similar solutions for the 1/r2 kernel

4. (3+1)-dimensional numerical investigation of collapse dynamics and collapse arrest for the 1/r2 kernel4.1. Approximate variational and exact numerical solutions for the ground state 4.2. Evolution scenarios for toroidal states

5. Conclusions AcknowledgmentsAppendix A. NotationsAppendix B. Hardy--Littlewood--Sobolev inequalityAppendix C. Gagliardo--Nirenberg--Sobolev inequalityAppendix D. Numerical implementation of the singularity of 1/r2 References