Bright Solitary Waves on a Torus: Existence, Stability and Dynamics for the Nonlinear Schrödinger Model
BBright Solitary Waves on a Torus: Existence, Stability and Dynamics for theNonlinear Schr¨odinger Model
J. D’Ambroise, P.G. Kevrekidis, and P. Schmelcher
3, 4 Department of Mathematics, Computer & Information Science,State University of New York (SUNY) College at Old Westbury, Westbury, NY, 11568, USA Department of Mathematics and Statistics, University of Massachusetts, Amherst, MA, 01003, USA Center for Optical Quantum Technologies, Department of Physics,University of Hamburg, Luruper Chaussee 149, 22761 Hamburg Germany The Hamburg Centre for Ultrafast Imaging, University of Hamburg,Luruper Chaussee 149, 22761 Hamburg, Germany
Motivated by recent developments in the realm of matter waves, we explore the potential ofcreating solitary waves on the surface of a torus. This is an intriguing perspective due to the role ofcurvature in the shape and dynamics of the coherent structures. We find different families of brightsolitary waves for attractive nonlinearities including ones localized in both angular directions, aswell as waves localized in one direction and homogeneous in the other. The waves localized in bothangular directions have also been partitioned into two types: those whose magnitude decays to zeroand those who do not. The stability properties of the waves are examined and one family is foundto be spectrally stable while most are spectrally unstable, a feature that we comment on. Finally,the nature of the ensuing nonlinear dynamics is touched upon.
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I. INTRODUCTION
The atomic physics platform of Bose-Einstein conden-sates (BECs) has served over the past two decades asan excellent testbed for ideas stemming from nonlinearwaves and their interplay with geometric and topologicalnotions among many others [1–3]. In particular, a widevariety of experiments has been performed to exploreideas related to bright coherent structures in effectivelyself-focusing and attractively interacting BECs [4–7] andrelated to repulsively interacting BECs with their effec-tive self-defocusing nonlinearity [8–16]. Important struc-tures that were investigated include the gap solitons [17]in optical lattices, multi-component solitons [18], vorticesand multi-vortex configurations [19, 20], as well as soli-tonic vortices and vortex rings [21].Typically, these structures are placed in simple con-fining potentials with the most canonical example beingthat of the parabolic trap [1–3]. However, this doesn’tpreclude the possibility that not only periodic [22], butalso more elaborate non-harmonic potentials are accessi-ble experimentally, as e.g. described in [23]. In fact, cer-tain experimental techniques enable the formation of ar-bitrary trapping geometries [24] rendering atomic BECsa particularly well suited platform for exploring the in-terplay of nonlinearity and geometry. Nevertheless, otherareas too, including nonlinear optics, provide case exam-ples of different types of confinement through the manip-ulation of the optical medium’s refractive index [25].A particular motivation of the present work consistsof a recent theoretical proposal towards exploring opti-cal lattice settings with a torus topology in the contextof BECs, with the atoms being confined to the surfaceof a torus [26]. This represents a striking example of ageometrically and topologically “nontrivial” trap which bears great promise for novel nonlinear wave effects. In-deed there is a plethora of curved and confining surfacesof different geometry and topology that might be real-izable and accessible experimentally using modern lightmanipulation techniques and tools. In parallel to that,there is an active interest in the field of pattern formationtowards studying nonlinear partial differential equationson curved soft substrates; a prime example of that isthe study of defect formation and localization througha generalized Swift-Hohenberg theory in elastic surfacecrystals [27, 28]. Motivated by an interweaving of thesethemes, our aim in the present work is to explore theexistence, stability and dynamics of solitary waves in anonlinear Schr¨odinger (NLS) model with attractive inter-particle interactions on the surface of a torus. This isa prototypical continuum Hamiltonian case example asopposed to the lattice one of [26] and the dissipative dy-namics of [27, 28] for the exploration of such dynamics.Some of our main findings involve the identification oflocalized solutions in both angular directions of the torus.Such solutions come in two families, one on the inside andone on the outside of the torus. Also solutions dependingon only one of the angular variables (independent of theother) are equally considered and their families have beenidentified. Among the doubly localized solutions, we findtwo varieties, one that is more solitonic in nature and onethat is more reminiscent of a periodic solution that doesnot decay to the proximity of zero within its spatial vari-ation. This last family, when centered on the outside ofthe torus is the only that is found to be spectrally stablesufficiently close to the small amplitude, i.e. near linearlimit. The dynamics of localized solutions often lead tocollapsing spots on the torus, while those depending ona single angle may lead to multiple such collapsing spots.Our presentation is structured as follows. In section II a r X i v : . [ n li n . PS ] J un we provide the model, the benchmarks used for the linearproblem and the theoretical setup of the nonlinear prob-lem. Then, in section III, we will examine the differenttypes of nonlinear solutions, localized and homogeneousones, that we have identified in this geometric settinginvolving the Laplacian operator on the 2d torus. Thecontinuation of the different branches of solutions versusthe eigenvalue parameter µ which is the frequency of thesolution, is given and the stability of these states is con-sidered. When the solutions are identified as unstabletheir dynamics will be followed. Importantly, however,we also identify stable solutions sufficiently close to thelinear limit of the problem. Lastly, we will summarizeour findings in section IV and present a few of the manydirections that are opening up for this promising researchtheme. II. MODEL AND THEORETICAL SETUP
The torus with major radius R and minor radius r is centered at the origin and has parameterization in R given by x = ( R + r cos( θ )) cos( φ ), y = ( R + r cos( θ )) sin( φ ), and z = r sin( θ ). The toroidal angle isdenoted by φ and poloidal angle θ . For a diagram of thetorus and relevant parameters see Figure 1. Consider the2D NLS equation on the surface of the torus with radii R > r as follows iψ t = −
12 ∆ ψ + V ( θ, φ ) ψ − σ | ψ | ψ (1)where σ = 1 is the focusing and σ = − r ∂ θθ − sin θr ( R + r cos θ ) ∂ θ + 1( R + r cos θ ) ∂ φφ . (2)In what follows, we define α = r/R ; moreover, as a start-ing point we will hereafter set V ( θ, φ ) = 0 to examine thepotential of the torus for intrinsically localized states. Anatural benchmark that we have performed en route tothe consideration of the nonlinear problem has been thestudy of the linear spectrum of the underlying Laplacianoperator as identified, e.g., in [29]. This spectral analysishas been done as a function of the parameter α charac-terizing the nature of the torus as regards the size of theminor radius r over the major radius R . A key featurein this case is that the relevant spectrum is discrete andconsists of isolated eigenvalues due to the presence of afinite interval for the values of θ and φ (both running inthe [0 , π ] interval).As is customary in the NLS setting, we consider sta-tionary solutions by setting ψ = e − iµt u and obtaining theboundary value problem naturally with periodic bound-ary conditions −
12 ∆ u + V ( θ, φ ) u − σ | u | u − µu = 0 . (3) Once the solutions (among them we are interested in theones that bear some form of localization) to Eq. (3) havebeen identified, their spectral stability can be monitoredby setting ψ = ( u + δ (cid:104) a ( θ, φ ) e νt + b ( θ, φ ) ∗ e ν ∗ t (cid:105) ) e − iµt . (4)From this, we obtain to order δ the linear system (cid:20) M M − M ∗ − M ∗ (cid:21) (cid:20) ab (cid:21) = − iν (cid:20) ab (cid:21) (5)where M = ∆ − V + µ + 2 σ | u | and M = σu . FromEq. (4) it follows that max(Re( ν )) > θ ) = 0. Nextwe separate categories of solutions based on whether thebulk of the solutions is centered near θ ≈ θ ≈ π on the insideof the torus (Types -in). Note also that the operatorin (2) remains translationally invariant in φ and so cor-respondingly we find some solutions which appear as asolid stripe wrapping around the torus in the toroidal φ direction.The classification of solutions below is also based onwhich directions the solutions are localized. Using New-ton’s method to solve the stationary equation (3), onecan select an initial guess bearing the desired localiza-tion properties of an intended solution in order to obtainconvergence of the iterative process. We focus on solu-tions which have a single concentration of mass. All ofthe solutions which we have identified for the nonlinearproblem with σ = 1 in the context of the present studylie on a spectrum between the category types outlinedbelow, and have been obtained via continuation in thefrequency parameter µ .A Type I solution has localization properties similar toan initial guess in the form of u = A sech (cid:16)(cid:112) | µ | [ B ( θ − θ ) + C ( φ − φ ) ] (cid:17) (6)for constants A , B , C ∈ R . For sufficiently large | µ | the solutions are localized both in the toroidal and in thepoloidal direction. As | µ | transitions to smaller valuesthe solution footprint grows and comes to wrap aroundthe torus in both directions.Such solutions appear as a two-dimensional sech-shapein both the θ and φ directions on the surface of the torus.There are two primary subtypes. We will call a Type Isolution Type I-in if its bulk is on the interior of the torus;such solutions can be obtained from an initial guess ofthe form (6) with φ , θ ≈ π . Type I-out solutions bearmass predominantly on the exterior of the torus and theycan be obtained from an initial guess (6) with φ ≈ θ ≈
0. One of these configurations will correspond to alocal energy minimum, while the other will correspond toa nonlinear Hamiltonian energy maximum though whichone is which also depends on the value of α .Type II solutions have localization properties similarto either an initial guess of the form u = sech( A ( φ − φ ))(1 + B cos( θ − θ )) (7)for constants A , B ∈ R , or the same format but with θ, φ switched. These solutions are localized in the onedirection and wrap around the torus (but without ap-proaching zero) in the other direction. There are onceagain two sub-types. Type II-in solutions bear most oftheir mass on the interior of the torus and can be ob-tained from initial guess (7) with θ , φ ≈ π , and TypeII-out solutions are principally localized on the exterior ofthe torus and can be obtained from initial guess (7) with θ ≈ φ ≈ π . Recall, however, that type I and type IIsolutions can equivalently be localized at any other valueof φ , given the translational invariance of the Laplacianoperator on φ .Finally, Type III solutions have localization propertiessimilar to an initial guess of the form u = sech( A ( θ − θ )) (8)for A ∈ R a constant. Such resulting solutions appearas a localized shape in the poloidal θ direction and a solidstripe, i.e. it does not depend on and is thus uniform inthe toroidal φ direction. The Type III-in solutions areapproximately centered at θ ≈ π and the Type III-outat θ ≈ P = (cid:90) π (cid:90) π | ψ | dS (9)where the surface element on the torus is dS = | ˆ φ × ˆ θ | dθdφ . Here hats denote the unit vectors in the differ-ent (toroidal and poloidal) directions. Thus, | ˆ φ × ˆ θ | = R + r cos( θ ). Figures 2-7 show examples of stationary so-lutions with their magnitude shown in color on the sur-face of the corresponding torus. The power (used as abifurcation parameter for the solution branch) and themaximum real part of the eigenvalues, identifying thespectral stability of the solution, are shown as a functionof µ . Figure 8 shows examples of stationary solutionswith the magnitude of solutions shown in terms of θ ver-sus φ in the two-dimensional flat rectangle [0 , π ] × [0 , π ].Figures 9-11 show the dynamics of unstable solutions.These results will be discussed in more detail in the nextsection.It is relevant to note that additional wave solutions ofEq. (3) exist and they can be found using an initial guessof e ilφ for l = 1 , , . . . . Such solutions are unstable with the fundamental (non-vortical) states of l = 0 being moredynamically robust than those with l (cid:54) = 0 for the caseswe have considered. All examples observed for l (cid:54) = 0 fea-ture blowup similar to the fate of the localized solutionsdescribed in the next section. III. NUMERICAL RESULTS
We now turn to the presentation of our numerical find-ings. We will consider different values of α progressingfrom smaller to larger ones. In Figures 2 and 3 twobranches of solutions are shown for α = 0 .
15. One branch(solid line in Figure 2 (a)-(b)) has Type II-in solutionsthat are shown in Figure 2 (c)-(f). These solutions aresech-shaped in the toroidal direction and wrap aroundin the poloidal direction (without getting close to zero),with the bulk of the solution located on the interior ofthe torus. For larger values of | µ | the solutions have asmaller footprint on the torus, i.e. they become morelocalized as we expect in this setting of stronger non-linearity. For smaller values of | µ | on this branch thesolution footprint is broader on the torus. This branchhas increasing power until µ ≈ − . µ -values decrease (increase in absolute value). The otherbranch (dashed line) has Type II-out solutions that areshown on the torus in Figure 3 (a)-(d). These solutionsare similar functionally but with the bulk of the solutionlocated on the exterior of the torus. This branch has in-creasing and then decreasing power as a function of µ .This change in the power’s monotonicity is caused by achange in the number of real eigenvalues, i.e., a change instability. There are six real eigenvalues for higher valuesof | µ | . Four are zero, and two have symmetric nonzeroreal values accounting for the instability of the branch.Around µ ≈ − .
75 the count of real eigenvalues changesto four (all are zero) and the solutions spectrally stabilizeas the previously mentioned two nonzero real eigenvaluesnow merge onto the imaginary axis.In Figures 4-5 four branches of solutions are demon-strated with α = 0 .
5. One branch (solid line in Figure4 (a)-(b)) has Type I-in solutions which are sech shapedin both directions. Here too, lower | µ | leads to a widerfootprint, while higher | µ | has smaller footprint on thesurface of the torus. The latter also leads to higher in-tensity in this more highly nonlinear regime. The Type Isolutions are shown in Figure 4 (c)-(f). Another branch(dashed line) has Type I-out solutions which are localizedin both directions for higher | µ | values, and when theyare continued in the µ parameter to lower | µ | values theywiden and eventually wrap around in the poloidal direc-tion to approach Type II-out solutions. Other branches(dotted and dash-dotted line) are found here for α = 0 . α > ∼ .
45 too. These branches
FIG. 1: Sketch of a torus with outer radius R and inner radius r < R . The arrows point in the direction of increasing toroidalangle φ and increasing poloidal angle θ . typically carry larger mass, as the solitary structure isquasi-one-dimensional being uniform along the toroidaldirection. The Types II and III examples are shown inFigure 5 (a)-(d).In Figures 6-7 again four branches are found to existfor α = 0 .
75. One branch (solid line in Figure 6 (a)-(b)) has Type I-in solutions which are sech-shaped inboth directions for large | µ | . As | µ | decreases, the solu-tion widens and wraps around in the toroidal direction.This effectively implies that Type I solutions morph (forthis value of α ) into Type II solutions, upon the relevantvariation of µ . This is to be contrasted with the α = 0 . | µ | decreases.For the α = 0 .
75 case (fatter torus) the broadening of thesolutions occurs only in the toroidal φ direction. An-other branch (dashed line) is similar to the dashed linefor the α = 0 . α = 0 . α > ∼ .
45. It is relevant to note in passing that all ofthese solutions for the larger values of α have been foundto be unstable.In Figure 8 the magnitude of some solutions is shownin a flat rendering on the rectangle [0 , π ] × [0 , π ] withrespect to θ and φ to give a “planar” sense of the wave-forms. The top row (a)-(b) shows typical Type II solu-tions which wrap around in the toroidal direction (with-out returning to the proximity of zero) and are localizedin the poloidal direction. The middle row (c)-(d) shows astandard example of a Type I solutions localized in bothdirections for sufficiently large values of | µ | . Upon para- FIG. 2: These graphs correspond to α = 0 .
15 and R = 1 . r = αR = 0 .
24. In (a) the power P ( µ ) of solutionsis shown as a function of µ . In (b) the maximum real partof the eigenvalues ν is shown as a function of µ . The mag-nitude of stationary solutions | u | is shown according to thecolorbar on the surface of the torus. A solution with µ = − µ gives the other Type II-insolution with µ = − .
75 in subfigures (e) and (f) (two views,front and back) that is localized in the toroidal direction andwraps around in the poloidal direction. As the branch con-tinues and the power curve turns upwards to greater powervalues the solutions continue to broaden in the toroidal di-rection. The power and eigenvalues for these solutions andothers obtained by continuation in µ are shown in the toppanels with a solid line. Examples of solutions in Figure 3correspond to the other branch shown with a dashed line.The latter branch is spectrally stable if µ is sufficiently large(and its absolute value sufficiently small). metric continuation in µ the solution footprint expandsin both directions so that for small enough values of | µ | solutions wrap around in both toroidal and poloidal di-rections. The bottom row (c)-(f) shows some basic exam-ples of Type III solutions which represent a solid stripein the toroidal direction.According to the panels (b) in Figures 2, 4, 6 whichsummarize the stability features of the obtained solu-tions, all the branches identified as having stable solu-tions for a range of frequencies are of Type II-out, i.e.,they are localized in the toroidal direction but do not ap-proach zero along the poloidal direction, and they havemost of their mass on the outside of the torus. Notice FIG. 3: These graphs on the torus are similar to those seenin Figure 2. The solutions here correspond to the power andeigenvalue curves from Figure 2 with a dashed line. The so-lution with µ = − µ from the othersolution of Type II-out in subfigures (c) and (d) (front andback views) which has value µ = − .
5. The solution with µ = − that this happens only for an interval of µ , indicatingthat this stability is a byproduct of the interplay betweenthe nonlinearity and geometry. For the two-dimensionalplane in the absence of curvature, recall that solitarywave two dimensional solutions are prone to collapse asis discussed in detail e.g. in [30, 31]. On the other hand,the geometry breaks the translational invariance of themodel, as well as the scale invariance thereof, leading tothe existence (for sufficiently strong nonlinearity) of realeigenvalue pairs. It is only for this (II-out) branch of so-lutions and for a range of frequencies in the vicinity ofthe small amplitude limit that the interplay of curvatureand nonlinearity permits spectral stability. Such stablesolutions exist for a wider range of µ -values if α is small(on a thinner torus), and they exist for a smaller rangeof µ -values for α large (on a fatter torus). Plots of stablesolutions are shown in plots (c) and (d) in Figures 3, 5,7 for α = 0 . , . , .
75 respectively. By comparing thesestable solutions one can see that the amount of mass ofthese stable solutions which wraps around in the poloidaldirection starts to diminish as α increases.It is relevant to note that from the stability perspec-tive, solutions of Type I and Type II possess two pairs ofeigenvalues at the origin, reflecting the invariance withrespect to phase and the translational invariance alongthe toroidal φ direction. On the other hand, this freedomto locate the solution arbitrarily in φ is lost in the Type-III solutions leading to a single pair of eigenvalues at theorigin of the spectral plane. In the case of the stable FIG. 4: As in Figure 2 but for the values α = 0 . R =1 .
6, and r = αR = 0 .
8. The solid lines in subfigures (a)and (b) correspond to Type I-in solutions which look as asech shape in both directions. Examples on this branch areshown in (e) and (f) µ = − . µ = −
5. The dashed lines in subfigures (a) and(b) correspond to Type I-out solutions which are sech shapeon the exterior of the torus and after continuation to lowervalues of | µ | transition into Type II-out solutions which wraparound in the poloidal direction. In (c) an example is shownon the dashed branch for µ = − Type II-out solutions reported above, e.g., for α = 0 . FIG. 5: These graphs on the torus are similar to those pro-vided in Figure 4. The solutions subfigures (a) and (b) areof Type III-in and correspond to the dotted line branch fromFigure 4. For these Type III-in solutions the stripe visible onthe interior of the torus is solid around the hole of the torusdue to the independence of the solution on φ . The solutionshown here in (c) and (d) (front and back views) is a Type II-out example with µ = − . lutions bearing an additional phase factor of e ilφ . An ex-ample of this type for the waveform of Type III is shownin Figure 12. From the left panels illustrating the spec-tral planes ( ν r , ν i ) of eigenvalues ν = ν r + iν i , we caninfer that solutions with l = 1 are more unstable (via alarger number of unstable modes and associated growthrates) than the l = 0 ones. This has been typical in thecases we have examined herein, hence we do not focus onthese vortical waveforms further. IV. CONCLUSIONS AND OUTLOOK
In the present work, we have set the stage for con-sidering nonlinear Schr¨odinger equations and related dis-persive wave models on the torus. While in the math-ematical community, this is synonymous to a periodicboundary condition system, as opposed to an infinite do-main, here the details of the geometry critically affectthe linear (Laplacian) operator adding a natural param-eter to this system, namely the ratio of minor over majoraxis of the torus. Furthermore, the explict presence ofthe poloidal variable and the finite size periodicity of thetoroidal one induce drastic differences from the transla-tionally invariant two-dimensional case that is familiar inthe NLS realm [30, 31]. Here, we encounter a situationwhere (partially) translational as well as scale invarianceare broken by the presence of curvature leading to a po-tential spectral destabilization of the resulting solutions.We have identified different types of states including ones
FIG. 6: These graphs are similar to Figure 2 but here withvalues α = 0 . R = 1 .
6, and r = αR = 1 .
2. Since thepower becomes large for some branches in (a), this graph isdisplayed with a log scale on the y-axis. The solid lines in(a) and (b) correspond to localized in both directions TypeI-in solutions for higher | µ | values which upon continuation tolower | µ | values give Type II-in solutions as they widen in thetoroidal direction. Examples on this solid line branch are seenin subfigures (e) and (f) for µ = − . µ = −
10. The dashed lines in (a) and(b) correspond to Type I-out solutions for higher values of | µ | which transition into Type II-out solutions for lower valuesof | µ | . In (c) an example is shown on this dashed branch for µ = − that are localized in both spatial directions, localized inone and wrapping around (yet staying far from zero) inthe other, as well as localized in one and homogeneousin the other. Among these, we found that only the sec-ond type can have a range of frequencies (near the linearlimit) where the solutions are spectrally stable. For toohigh frequencies and for all branches, nonlinearity takesover and leads to collapse instabilities. However, suffi-ciently close to the small amplitude limit, the interplayof curvature and nonlinearity may dynamically lead toperiodic oscillations around a spectrally stable state, asshown in our dynamics above.Naturally, we feel that this first stab at this class ofproblems opens numerous new directions to consider. Avery canonical one among them is to explore the possi-bility of self-defocusing nonlinearities. On the one hand,these are canonical for numerous atomic gases such as Rb or N a [1, 2] while on the other hand, they presentthe potential for fundamentally distinct structures in-cluding ones bearing vorticity in the two-dimensional
FIG. 7: These graphs on the torus are similar to those seenin Figure 6. The solutions in (a) and (b) are of Type III-inand correspond to the dotted line branch from Figure 6. Forthese Type III-in solutions the stripe visible on the interiorof the torus is solid around the hole of the torus given thehomogeneous nature of the solution in the corresponding ( φ )variable. The solution here in subfigures (e) and (f) (front andback views) is a Type II-out example with µ = − . realm. This vorticity could be in the form of local-ized vorticity (point vortices), or in that of vorticity fil-aments, such as rings wrapping potentially around ei-ther the toroidal or the poloidal direction. Exploringsuch states and their stability has been a major themein atomic BECs [3] and its interplay with geometry inthis setting would present novel challenges and potentialoutcomes. Yet another relevant possibility could be toremain on the focusing interaction realm but “negate”the possibility of collapse by introducing photorefractivenonlinearities as, e.g., in [32]. In this case, the model atsufficiently high intensity returns to its linear form andtherefore collapse type instabilities no longer occur. Itwould be especially relevant in this latter setting to ex-plore which among the different types of unstable solu-tions identified herein become stabilized (or possibly viceversa). These studies will be deferred to future work. Acknowledgements.
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FIG. 8: The graphs demonstrate some of the different shapesof solutions, represented flat with respect to θ, φ ∈ [0 , π ].Plot (a) here is the same solution as (f) in Figure 2. Subfigure(b) here is the same solution as (d) on Figure 3; since this isan out-type solution the flat axis here is centered at the peakof the solution. Subfigures (c) and (d) here are the samesolutions in (f) and (c) in Figure 4 respectively. Subfigures(e) and (f) here are Type III-in and Type III-out respectivelyfor α = 0 .
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FIG. 12: Plot (a) shows the eigenvalues of the solution for l = 0 which is shown on the torus in (b). Plot (c) similarlyfor l = 1 in (d). Both examples correspond to α = 0 .
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