Chess-Board-Like Spatio-Temporal Interference Patterns and Their Excitation

Chong Liu1,2,3, Zhan-Ying Yang1,3, Wen-Li Yang1,3,4, and Nail Akhmediev2

1School of Physics, Northwest University, Xi’an 710069, China2Optical Sciences Group, Research School of Physics and Engineering,The Australian National University, Canberra, ACT 2600, Australia

3Shaanxi Key Laboratory for Theoretical Physics Frontiers, Xi’an 710069, China and4Institute of Modern Physics, Northwest University, Xi’an 710069, China

We discover new type of interference patterns generated in the focusing nonlinear Schrodingerequation (NLSE) with localised periodic initial conditions. At special conditions, found in thepresent work, these patterns exhibit novel chess-board-like spatio-temporal structures which canbe observed as the outcome of collision of two breathers. The infinitely extended chess-board-likepatterns correspond to the continuous spectrum bands of the NLSE theory. More complicatedpatterns can be observed when the initial condition contains several localised periodic swells. Thesepatterns can be observed in a variety of physical situations ranging from optics and hydrodynamicsto Bose-Einstein condensates and plasma.

I. INTRODUCTION

The concept of solitons is well known and it was provento be useful in studies of tsunami waves, pulses in op-tical fibres and matter waves in Bose-Einstein conden-sate. Large part of theoretical studies on solitons is basedon the Korteweg-de Vries and the nonlinear Schrodingerequations (NLSE). The latter describes multiplicity ofphenomena in various branches of physics and can beconsidered as an archetypical model of nonlinear wavepropagation. In addition to solitons, the NLSE has also afamily of breather solutions. In contrast to ordinary soli-tons, breather solutions are not localised. They are locat-ed on top of a plane wave. Breathers have been studiedin hydromechanics [1, 2], nonlinear optics [3, 4], optome-chanics [5, 6], plasma [7], metamaterials [8], and Bose-Einstein condensates [9]. They play an important role indescribing variety of physical phenomena such as mod-ulation instability (MI) [10–13], supercontinuum gener-ation [14], turbulence [15–19], Fermi-Pasta-Ulam recur-rence [20–24], Talbot effect [25], and rogue wave events[26]. Recently, significant progress has been made on theexperimental observation of fundamental breathers [27–31] and their nonlinear superpositions [32–34], both innonlinear optics and hydrodynamics. Just as solitons,breathers may collide with each other creating complexinterference patterns. They reveal high amplitude peakswhen the breathers are synchronised [26, 35, 36], similarto the case of the soliton synchronisation [37, 38].

One way of creating breathers in nonlinear physics isthrough MI. The latter is the exponential growth of pe-riodic perturbations of a plane wave in unstable medi-a [39]. It is known as Benjamin-Fair instability [40] inthe case of water waves or Bespalov-Talanov instabili-ty [41] in the case of optical beams. The NLSE has anexact solution that describes both the process of the ini-tial growth and the following full scale recurrent evolu-tion back to the plane wave. These solutions have beenfirst given in [42, 43] and presently known as ‘Akhmedievbreathers’ (AB). A single breather produces localised pe-

riodic fringe pattern. Fringes are also produced when twosolitons collide (soliton interference pattern) [44–47]. Onthe contrary, collision of two breathers can produce newspatio-temporal patterns which are doubly periodic. Thisis an important difference between solitons and breathersthat deserves special attention.

Before entering the details of this phenomenon let usrecall that MI can be produced using localised periodicperturbations. Analytic solutions for this phenomenonare known as super-regular breathers [13, 34, 48]. Thelatter is the result of collision of two ABs propagating atsmall angles of opposite sign to the direction of evolution.It has been found [13] that modulation in this case alsogrows producing periodic fringes just like in the case ofordinary MI. Modulation remains localised although thesize of localisation increases in evolution. However, thishappens only if the period of modulation remains withinthe instability interval. If the period is chosen outside ofthis interval, the evolution pattern changes drastically.In the present work, we found that the pattern trans-forms itself into a chess-board-like structure. Moreover,expanding this pattern leads to a continuous spectrumof eigenvalues involved in the NLSE theory. The latteris a nontrivial result that may significantly influence ourvision of the role of breathers in nonlinear wave evolution.

II. ANALYTICAL RESULTS

We start with the focusing dimensionless NLSE [49]:

i∂ψ

∂z+

1

2

∂2ψ

∂t2+ |ψ|2ψ = 0, (1)

where ψ(z, t) is the wave envelope, z is the propagationvariable, and t is the transverse variable. This equationgoverns the nonlinear wave evolution in various media. Inparticular, it describes gravity waves in deep-water con-ditions [1] and light waves in optical fibers [3]. In eachcase, the function ψ(z, t) describes the envelope of themodulated waves and its absolute value carries informa-

2

tion about either wave elevation above the average watersurface or the intensity of optical waves.

Let us consider the simplest case of symmetric collisionof two breathers with the same amplitude and oppositegroup and phase velocities. This exact solution can beconstructed using Darboux transformation with a pair ofcomplex eigenvalues λj with equal real parts and equalimaginary parts of opposite sign [42]. The solution can begeneralised for arbitrary number of breathers [10]. Theeigenvalues can be parameterised using the Joukowsky(Zhukovsky) transform [13]. Namely, λj = i(ξ + 1/ξ)/2,with ξ = re±iα, i.e., λj = (iν ∓ µ)/2, with ν = ε+ cosα,µ = ε− sinα, ε± = r ± 1/r. Here r (≥ 1) and α are theradius and the angle on the plane of polar coordinatesthat determine the value of λj . This transformation pro-vides certain convenience in presentation of the conditionof chess-board-like interference patterns.

The two-breather solution can be written in the form:

ψ(z, t) =

[1− G(z, t) + iH(z, t)

D(z, t)

]exp(iz), (2)

where G, H, and D are real functions of two variables

G = νµ{

4νµ [cosh(κ1 − κ2) + cos(φ1 + φ2)]

+νA2∆1 − µA1∆2

}, (3)

H = νµ

[2

(r2 − 1

r2

)cos 2α sinh(κ1 − κ2) + γB1Ξ1

−2

(r2 +

1

r2

)sin 2α sin(φ1 + φ2) + δB1Ξ2

], (4)

D = −1

2γ2B1 cos(φ1 − φ2)− 2 sin2 2α cos(φ1 + φ2)

−1

2ν2A2 sinhκ1 sinhκ2 − 2νµ (µ∆2 + ν∆1)

+

(1

2ν2B2 + ε2+µ

2

)coshκ1 coshκ2, (5)

with ∆1 = sinφ1 sinhκ2 − sinφ2 sinhκ1, ∆2 =cosφ1 coshκ2 + cosφ2 coshκ1, Ξ1 = cosφ1 sinhκ2 −cosφ2 sinhκ1, Ξ2 = sinφ1 coshκ2 + sinφ2 coshκ1, γ =ε− cosα, δ = ε+ sinα, Aj = ε2+ + 2 cos 2α ± 2, Bj =r2 + 1/r2 ± 2 cos 2α. Here, κ1 = γ(t − Vgrz), κ2 =γ(t+Vgrz), φ1 = δ(t−Vphz)+θ1, φ2 = −δ(t+Vphz)+θ2,Vgr = −(δν/γ+µ)/2, Vph = (γν/δ−µ)/2. Here, Vgr andVph are the group and phase velocities while θj are arbi-trary phases.

The solution depends on free parameters r, α, andphases θj . An example of intensity pattern producedby the above solution is shown in Fig. 1(a). This figureshows the collision of two breathers with chess-board-likedouble-periodic interference pattern in the region of col-lision. This pattern is qualitatively different from thesoliton collision areas studied earlier. In the latter case,the interference pattern consists of parallel stripes, i.e. itis single periodic. Single periodic patterns have been al-so observed theoretically and experimentally in breathercollision areas [13, 34]. In contrast, breather collisions

FIG. 1: (a) Chess-board-like interference pattern formed bythe two-breather collision calculated using Eq. (2) with theparameters r = 3, α = π/2.16, θ1 = θ2 = π/2. (b) Magnifiedcentral part of the pattern in (a). (c) The complex plane oftransformed eigenvalues ξ = re±iα of two colliding breathers.Particular points or curves on this plane correspond to thefollowing cases: “AB” (Akhmediev breather, r = 1, the unitcircle), “SRB” (Super-regular breather, r → 1, θ1 + θ2 = π),“PRW” (Peregrine rogue wave, r = 1, α = 0), and “KMB”(Kuznetsov-Ma breather, r > 1, α = 0). “CBL” (Chess-board-like) pattern appears for all points within the triangularareas |α| → π/2 limited by the red lines outside of the bluecircle r > 1. The pair of green circles corresponds to thepattern shown in (a). (d) Variation of Nt, Nz, interferenceperiods (Dt, Dz), and maximum amplitudes (|ψ|max), as rchanges in the range of α ∈ [π/2.2, π/2.1]. The red dashedlines correspond to α = π/2.16.

can produce more complicated structures [50–52]. Thepattern presented in Figs. 1(a,b) demonstrates clearlyone of these complications. Namely, Figure 1(a) showssymmetric collision of two breathers propagating at finiteangle to each other. For each breather, the widths in tand z directions are given by

∆t ∼ 1/ |γ| , ∆z ∼ 1/ |γVgr| . (6)

The periods of breathers along t and z axes are

Dt = 2π/ |δ| , Dz = 2π/ |δVph| , (7)

respectively. These values can be used to calculate thenumber of fringes Nt and Nz within the interference pat-tern in each direction: ∆t ∼ NtDt, ∆z ∼ NzDz. Forclear double-periodic interference pattern, both integersNt and Nz should be sufficiently large. From Eqs. (6)and (7), it follows that

Nt ∼ |tanα|(r + 1/r

r − 1/r

), Nz ∼ |cot 2α|

(r2 − 1/r2

r2 + 1/r2

).

(8)For an arbitrary r > 1, the double-periodic interferencepattern appears when |α| → π/2 (or |tanα| , |cot 2α| →∞) to ensure that Nt, Nz are big enough.

3

Once α is fixed (|α| → π/2), the two limiting cases,(i) r → 1, and (ii) r → ∞, can be dropped. Indeed,when r → 1, thus Nt → ∞, Nz → 0, the interferencepattern exhibits small amplitude oscillations only in t.This can be seen from Fig. 1(d). On the other hand,when r → ∞, meaning Nt ∼ |tanα|, Nz ∼ |cot 2α|, theinterference pattern is confined to an infinitesimal regionwith very large amplitude. Thus, we consider interfer-ence patterns for modest values of r. An example of theinterference pattern with α = π/2.16 for r = 3 shownin Figs. 1(a) and 1(b) exhibits double periodicity with-in the collision region. Its temporal and spatial periodsare Dt, Dz, respectively. An interesting feature of thepattern is that there is a periodic π phase shift at thehalf-periods kDz/2 (where k is an odd number) alongthe z axis. This phase shift results in the chess-board-like structure. The complex plane in Fig. 1(c) shows theregion of existence of these patterns. Namely, they arelocated in the symmetric grey-coloured triangular areaon this plot. Location of eigenvalues for other types ofNLSE solutions is also shown on this plot.

From experimental point of view and for numericalmodelling, an important question is what type of ini-tial conditions may create the chess-board-like patterns.In previous studies [33, 53, 54], the breather collisionswere triggered by the input field with multiple complexexponentials. Here, we extract the initial condition fromthe exact solution (2) at z = 0, i.e., ψ(z = 0, t). Thephase parameters θ1, θ2 play a key role in forming theinitial state. If θ1 + θ2 = 2kπ (where k = 0,±1,±2...),the function H = 0. Then the initial state is purely real.If θ1 + θ2 = (2k + 1)π, then G = 0. Consequently, themodulation part of the initial condition is purely imag-inary. General case of complex initial conditions is toocomplicated and does not produce anything new. Con-sidering |α| → π/2, or cos2 α→ 0, we analytically obtainthe approximate expressions for these two types of initialconditions:

ψ(0, t)i ≈ 1− i ρi sechγt cos δ, (9)

ψ(0, t)r ≈ 1− ρr sechγt cos(δ + kπ), (10)

where ρi = 2B1 cosα/ε−, ρr = 2(A1−4ν) cosα/(4 cosα−ε+), δ = δt + θ with θ = (θ1 − θ2)/2. The approximateformulae (9)-(10) can be used as initial conditions forgenerating the interference patterns with high accuracy.Both ψ(0, t)i and ψ(0, t)r consist of a localised functionsechγt and the modulation cos δt responsible for the tem-poral period Dt of interference patterns. An example ofinitial conditions and the resulting pattern are shown inFig. 2. There is an excellent agreement between the pat-tern obtained from the exact solution (2) and the onesimulated numerically using the initial condition (9).

Clearly, the function ψ(z, t = 0) has the spatial period2π/|δVph|. The two periods of spatio-temporal interfer-ence pattern are given by Dt, Dz. Once r and α are fixed,the interference patterns have these periods in t and z nomatter what is the shape of the swell. This is becauseθj have no effect on the discrete symmetric spectra λj .

FIG. 2: (a) Amplitude and phase profiles of approximate ini-tial conditions (9) (red dashed line) and exact solution (2) atz = 0 (black solid line). (b) Pattern started from the approx-imate condition in (a). Parameters are the same as in Fig.1.

FIG. 3: Chess-board-like interference patterns from the ini-tial condition ψ(0, t) = 1 − iρisech(t/b1) cos δt with differentwidths (a) b1 = 3bs, (b) b1 = 30bs where bs = 1/|γ|. Otherparameters are the same as in Fig. 2.

They influence only the maximal amplitudes and loca-tion of interference maxima. When |α| → π/2, implyingthat | sinα| → 1, the modulation frequency satisfies thecondition |δ| = (r + 1/r)| sinα| ≥ 2. We can also see,from Fig. 1(d) that Dt = 2π/|δ| ≤ π. This means thatthe chess-board-like pattern occurs in a high-frequencyregion out of the MI gain curve (|δ| < 2).

III. NUMERICAL SIMULATIONS

Let us take a closer look at Eqs. (9)-(10) by a more gen-eral case: ψ(0, t) = 1−σρ1sech(t/b1) cos δ1t, where σ = ior 1, δ1 is the frequency, ρ1 and b1 are the amplitude andwidth, respectively. To excite interference patterns inFig. 2, σρ1 can be either purely real or purely imaginary,while the frequency must satisfy |δ1| ≥ 2. This frequencyis outside of the MI gain curve. For simplicity, we takeδ1 = δ. The critical width of localisation which allowsto excite the double-periodic pattern is b1 = bs = 1/|γ|.When b1 ≥ bs, numerical simulations clearly demonstratethe double periodicity. When b1 →∞, the periodic pat-tern occupies the whole half-plane (z, t). Instead, whenb1 < bs, the interference pattern weakens and disappearscompletely at small b1.

According to (9)-(10), for any b1 (≥ bs), purely imag-inary amplitude is fixed by ρ1 = ρi; the maximal realamplitude is ρ1 = ρr. The generated interference pat-terns in each case have the periods Dt, Dz defined by

4

Eq. (7). When b1 →∞, the pattern has the same max-imum amplitude 1 − ρr, no matter whether it is real orimaginary. In either case, the high-frequency interfer-ence patterns can be excited when b1 ≥ bs. The largerρ1 is, the bigger is the maximum amplitude of interfer-ence patterns and the smaller is Dz. Figure 3 shows anexample of excitation with two different widths b1. Ascan be seen from comparison of two panels (a) and (b),when b1 increases, the size of the interference pattern getslarger while the periods Dt, Dz remain the same. Whenb1 → ∞, the interference pattern spreads indefinitely tooccupy the whole space-time plane.

Initial conditions can be analysed using the eigenval-ues of the NLSE theory [18, 55, 56]. An example of suchanalysis is given in Fig. 4(a). The spectrum that cor-responds to the excitation of breathers is discrete. Thespectrum becomes continuous in the limit of infinitely pe-riodic solution. The latter case is similar to the double-periodic A- and B-type Akhmediev solutions of the NLSE[43, 57, 58]. These solutions have been studied experi-mentally in [22, 24]. They can be considered as a pertur-bation of ABs which shift solution from the heteroclinicseparatrix trajectory in an infinite-dimensional phase s-pace to a periodic one. These A- and B-type solutionsare located on different sides of the separatrix which isthe AB [43]. This transformation produces significantqualitative changes in the intensity pattern. Fringe pat-tern is transformed into the chess-board-like structure.Moreover, discrete eigenvalue spectra of breathers aretransformed into continuous spectra of double-periodicsolutions. In order to show this, we calculated numeri-cally the spectrum of the A-type solution. It is shown inFig. 4(b). The spectrum is indeed continuous with dis-creteness in the lower part solely due to the discretenessof numerical scheme.

The initial condition for generation of A- and B-typesolutions can be expressed in terms of Jacobi elliptic func-tions. Their fundamental periods can be calculated ex-actly [59]. Clearly, the A-type solution describes periodicstructures with the transverse period located in the in-terval Dt ∈ (0,

√2π) whereas B-type solutions exist in

the region Dt ∈ (√

2π,∞). The chess-board-like interfer-ence patterns can be excited when the period is locatedin the range Dt ∈ (0, π). This means that the A-type so-lutions with the periods within the range Dt ∈ (0, π) candescribe the infinitely extended of interference pattern(b1 →∞) created by two-breather collision. Calculationof eigenvalues for the two cases confirms this. The twospectra nearly overlap as can be seen from Fig. 4(b).

More complicated patterns can be produced when us-ing muti-swell initial conditions:

ψ(0, t) = 1− σn∑j=1

Lj(t− tj) cos (δjt+ Θj), (11)

where σ = i or 1, tj is the time shift, δj is the frequency,Θj is the phase shift and Lj(t − tj) is smooth localisedfunctions. This can be sech-type, Gaussian, Lorentzian

or another similar function. In what follows, we used thesech-type function, Lj(t−tj) = ρjsech[(t−tj)/bj ]. Whenn = 1, Eq. (11) generates the two-breather interferencepattern shown above. When n ≥ 2, the pattern corre-sponds to multiple collisions. Indeed, if the swells arewell separated by the choice of tj , Eq. (11) provides agood approximation for modelling 2n-breather collision.

Figure 5 shows two examples of excitation that startedfrom three localised swells with the same envelope (n =3) with equal bj (≥ bs). Each collision in Fig. 5(a) withbj = bs is similar to that in Fig. 2(b). The total numberof collisions is n(n+1)/2. The periods in t and z are wellpredicted by the analytical expressions for Dt and Dz.For larger values of bj , the width of each swell becomeslarger and the areas of collision also increase in size. Oneexample with bj = 2bs is shown in Fig. 5(b). Overallgeometry of collision is the same as in Fig. 5(a) but thenumber of high-amplitude maxima is higher.

FIG. 4: (a) Location of discrete and continuous eigenvaluesλ on the upper complex half plane calculated for the initialcondition ψ(0, t) = 1 − iρisech(t/b1) cos δt with b1 = bs (redtriangles) and b1 → ∞ (blue diamonds) for r = 1, ..., 6, andα = π/2.16. The latter case produces six different contin-uous spectra. (b) Eigenvalues λ calculated for the A-typeAkhmediev solution with κ = 0.11 (red) and a periodic initialcondition ψ(0, t) = 1 − i2.7 cos 3.3t (blue). The two spectrain (b) are continuous and practically overlap.

FIG. 5: Multiple (n = 3) excitations of chess-board-like inter-ference patterns started from the initial condition ψ(0, t) =1 − i

∑3j=1 ρisech[(t − tj)/bj ] cos δt with different widths (a)

bj = bs, (b) bj = 2bs. Here tj = −15, 0, 15. Other parametersare the same as in Fig. 2.

5

IV. CONCLUSION

In conclusion, we introduced new chess-board-likespatio-temporal interference patterns generated by lo-calised periodic initial conditions applied to the NLSE.These patterns appear when the initial periodicity is out-side of the MI range. We shown that they correspond totwo-breather collision with specific double-periodic inter-ference patterns. When the pattern extends to infinity,the spectrum of eigenvalues is transformed from discreteto continuous. Multiple collision patterns can be excitedwhen the initial conditions contain several swells. Con-sidering the universality of the NLSE, these collision-likedouble-periodic interference patterns can be observed invariety of physical systems ranging from optics and hy-drodynamics to Bose-Einstein condensates and plasma.

ACKNOWLEDGEMENTS

This work has been done when C.L. visited Optical Sci-ences Group, ANU. This work is supported by Nation-al Natural Science Foundation of China (NSFC) (Nos.11705145, 11434013, and 11425522), the Major Basic Re-search Program of Natural Science of Shaanxi Province(No. 2017KCT-12), Natural Science Basic Research Planin Shaanxi Province of China (Grant No. 2018JQ1003),Australian Research Council (ARC) (Discovery ProjectsDP140100265 and DP150102057), and the VolkswagenStiftung. N.A. is a recipient of the Alexander von Hum-boldt Award.

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[59] The initial periods of A- and B-type Akhmediev solutionsare Dt = 4

√κ2Km=

√1−κ2

, Dt = 4√1+κ

Km=

√1−κ1+κ

, where

Km =∫ π/20

dθ/√

1−m sin2 θ, and κ ∈ (0, 1). The freeparameter for the families of these solutions is κ and it isassumed that the background averaged amplitude is one.From here, we find that the A-type solution describesperiodic structures with the transverse period located inthe interval Dt ∈ (0,

√2π) whereas B-type solutions exist

in the region Dt ∈ (√

2π,∞).