Soliton-like behaviour in non-integrable systems
Raghavendra Nimiwal, Urbashi Satpathi, Vishal Vasan, Manas Kulkarni
SSoliton-like behaviour in non-integrable systems
Raghavendra Nimiwal, Urbashi Satpathi, Vishal Vasan, ManasKulkarni
International centre for theoretical sciences, Tata Institute of Fundamental Research,Bangalore - 560089, IndiaE-mail: [email protected], [email protected],[email protected], [email protected]
Abstract.
We present a general scheme for constructing robust excitations (soliton-like) in non-integrable multicomponent systems. By robust, we mean localisedexcitations that propagate with almost constant velocity and which interact cleanlywith little to no radiation. We achieve this via a reduction of these complexsystems to more familiar effective chiral field-theories using perturbation techniquesand the Fredholm alternative. As a specific platform, we consider the generalisedmulticomponent Nonlinear Schr¨odinger Equations (MNLS) with arbitrary interactioncoefficients. This non-integrable system reduces to uncoupled Korteweg-de Vries(KdV) equations, one for each sound speed of the system. This reduction then enablesus to exploit the multi-soliton solutions of the KdV equation which in turn leadsto the construction of soliton-like profiles for the original non-integrable system. Wedemonstrate that this powerful technique leads to the coherent evolution of excitationswith minimal radiative loss in arbitrary non-integrable systems. These constructedcoherent objects for non-integrable systems bear remarkably close resemblance to truesolitons of integrable models. Although we use the ubiquitous MNLS system as aplatform, our findings are a major step forward towards constructing excitations ingeneric continuum non-integrable systems.
1. Introduction and summary of results
Nonlinear dynamics has been a subject of great interest both from a theoretical [1,2] andan experimental [3–6] perspective. Such far-from-equilibrium dynamics can be dividedinto two broad classes: (i) integrable [7,8] and (ii) non-integrable (generic) [9,10] models.If one restricts oneself to conservative systems, then the two hallmarks of far-from-equilibrium dynamics are nonlinearity and dispersion [11, 12]. Nonlinearity is dominantwhen the dynamics is far from its equilibrium state. For instance, if one starts withinitial conditions which are a significant perturbation on top of a background state,we expect nonlinearity to play a role. Such dynamics cannot be merely described bylinearisation which will yield a wave equation. Dispersive terms on the other hand aredominant when the profiles have large higher-order spatial derivatives. a r X i v : . [ n li n . PS ] J a n oliton-like behaviour in non-integrable systems oliton-like behaviour in non-integrable systems i ∂ψ k ∂t = − ∂ ψ k ∂x + N (cid:88) j =1 F kj ( | ψ j | ) ψ k (1)where ψ k ( x, t ) is a complex field (for the k -th species) in one space and time dimensionand the mass is set to 1. Here, N denotes the number of components or species(i.e., k = 1 , ...N ). Therefore, Eq. 1 represents N coupled complex partial differentialequations (PDEs). This equation is a conservative system and for a suitable choice ofPoisson bracket, it is in fact Hamiltonian. The elements of the interaction matrix F kj ( | ψ j | ) are functions of | ψ j | . The matrix structure we consider is given by F kj ( | ψ j | ) = α kj | ψ j | + δ kj G j | ψ j | where the coefficient α kj defines both the self and crosscouplings of cubic order and the coefficient G j specifies only self coupling of quintic type.This particular choice of F kj is typical of several physical systems (for e.g. cold atomicgases [25–30] and nonlinear optics [33–35]). In ultra-cold gases, the complex-valuedfunction ψ k ( x, t ) represents the condensate wavefunction, the absolute-value squared ofwhich is the density of atoms while its argument is the phase. On the other hand, innonlinear optics, the complex field represents the pulse and the absolute-value squaredis the intensity of light. In nonlinear optics, the role of time (in Eq. 1) is played by anadditional spatial z direction [32].Aside from certain exceptional choices for F jk [44–46], Eq. 1 is in general not integrable. Therefore for most choices of the couplings, one does not expect solitonsolutions. One may still wonder if it is indeed possible to design special initial conditions ψ k ( x,
0) ( k = 1 , ...N ), which give rise to solutions of Eq. 1, that are soliton-like in thesense that they bear as much resemblance as possible, to properties of true solitonsof integrable models. In this paper, we ask whether there are localised solutions toEq. 1 that propagate at almost constant speed? If so, do these localised solutionsinteract with one another with minimal scattering/radiation? As mentioned earlier,one suspects arbitrary initial conditions are subject to complex nonlinear dynamics,typically leading to breaking of profiles due to nonlinear and dispersive effects. Thusto answer such questions is not entirely trivial. In this work, we answer both questionsin the affirmative and give a general recipe to find such initial conditions, for arbitrarycoupling coefficients and background densities. In particular • We obtain special soliton-like initial conditions for the original non-integrableMNLS (Eq. 1) and show that the resulting time evolution bears a remarkableresemblance (Fig. 1 and Fig. 3) to that of solitons of integrable models. oliton-like behaviour in non-integrable systems • We achieve the above by employing known multi-soliton ( M − soliton) solutions [47]for the reduced chiral KdV equation (Eq. 18). These M − soliton solutions are usedto construct special initial conditions for the original MNLS. As specific examples,we present analytical and numerical results for the two-soliton case ( M = 2) of thethree component MNLS ( N = 3) but our method is valid for arbitrary M and N . • We demonstrate that even small perturbations of the optimal initial conditionsinvariably lead to radiation and subsequent failure of the robust evolution. Thisindicates that crude attempts to guess optimal initial conditions are doomed to fail(top panel of Fig. 2). • We manifestly establish that the initial condition so constructed exhibits dissimilarevolution under the original MNLS and the linearised version of MNLS. Thisestablishes the fact that the dynamics is undoubtedly nonlinear (bottom panelof Fig. 2) and the robust evolution we find is far from trivial. • We present a reduction to an effective uncoupled-system of KdV equations forthe MNLS. Even though the KdV equations are uncoupled, the coefficients of eachKdV encodes information of all the components in terms of base states (backgroundfields) and coupling coefficients. • The MNLS-to-KdV reduction is based on a detailed spectral analysis of the linearversion of MNLS and in the current work we address issues that were left open froman earlier work [48]. For example, we establish explicit expressions for eigenvectorswhich gives an analytical handle on deriving the coefficients of the effective chiralfield theory. As a consequence we obtain a map from coupling coefficients andbackground densities of MNLS to parameters for the initial condition required togenerate robustly propagating and stably interacting soliton-like solutions. • Although most of the results we present are for the case when the sound speeds ofthe underlying linear problem are distinct, we end the paper with a few statementsregarding the case of repeated speeds.The paper is organised as follows. In Sec. 2 we recast MNLS Eq. 1 into suitablehydrodynamic variables (density and velocity fields). We then recap relevant resultsfrom an earlier work [48] and present new results on the spectral analysis. In Sec. 3 wewrite down the effective chiral field-theory which is given by 2 N (in general) decoupled KdV equations ( N in each chiral sector) whose coefficients encode the informationof the various components. M − soliton solutions are discussed and the subsequentconstruction of optimal initial conditions is then explained. Sec. 4 is dedicated tonumerical simulations. Comparisons between original (fully nonlinear) dynamics andthe effective dynamics are made. We then make some important remarks regarding theinteraction of these coherent solutions with one another. In Sec. 5 we briefly commenton the case of repeated sound speeds. More precisely, we give necessary and sufficientconditions to ensure distinct sound speeds thereby describing the parameter-space ofvalidity for the results of Sec. 3. When the sound speeds are not distinct, one remarkably oliton-like behaviour in non-integrable systems
2. Spectral Analysis
We rewrite Eq. 1 in terms of density and velocity fields using a Madelungtransformation [49] ψ k ( x, t ) = (cid:112) ρ k ( x, t ) e i (cid:82) x v k ( x (cid:48) ,t ) dx (cid:48) to obtain ∂ρ k ∂t = − ∂∂x (cid:0) ρ k v k (cid:1) , ∂v k ∂t = − ∂∂x (cid:20) N (cid:88) j =1 α kj ρ j + v k G k ρ k − ∂ x √ ρ k √ ρ k (cid:21) . (2)We then linearise the above problem about a base state ( ρ k ,
0) with non-zerobackground densities and zero background velocity. We define the set of perturbedfields ( δρ k , δv k ) such that ρ k = ρ k + (cid:15) δρ k ( (cid:15)x, (cid:15)t ) and v k = (cid:15) δv k ( (cid:15)x, (cid:15)t ) where (cid:15) is asmall parameter. We substitute the form of the perturbation into Eq. 2 and drop termsof O ( (cid:15) ) and higher to obtain the linear equation ∂ t (cid:32) δρδv (cid:33) = − ∂ x A (cid:32) δρδv (cid:33) where A = (cid:32) N × N ρ ˜ α N × N (cid:33) . (3)Note ρ is an N × N diagonal matrix with strictly positive elements ρ k (the backgrounddensities) and δρ , δv are the N × α is given by [48]˜ α ij = (cid:40) ˜ g i , i = j,h, i (cid:54) = j. where ˜ g i = g i + 2 G i ρ i (4)with g i being the diagonal elements of the α matrix. All elements of the ˜ α matrix(Eq. 4) are strictly positive. The spectral analysis of the 2 N × N matrix A (Eq. 3)can be accomplished by a spectral analysis of the N × N matrix ˜ αρ [48]. Indeed, underthe assumption that ˜ α is symmetric positive definite (for ˜ α as defined above this is truewhen h < min ˜ g i [48]), all eigenvalues of ˜ αρ are real and positive which we denote by λ .Additionally, the matrix ˜ αρ is diagonalisable. Equivalently the algebraic multiplicity(degeneracy of eigenvalue) and the geometric multiplicity (dimension of the span ofall eigenvectors associated with the degenerate eigenvalue) of every eigenvalue of ˜ αρ are equal. And lastly, to each eigenvector of ˜ αρ there corresponds an independenteigenvector of A . Hence A too is diagonalisable and explicit expressions for itseigenvectors can be stated in terms of eigenvectors of ˜ αρ . We will discuss the case whenan eigenvalue of ˜ αρ has multiplicity m = 1, i.e. the case of a simple eigenvalue. Thecase of repeated eigenvalues (which corresponds to the case of repeated or non-uniquesound speeds) and necessary and sufficient conditions for that to happen is discussedlater (see Sec. 5). oliton-like behaviour in non-integrable systems αρ (and hence for A ) when λ is a simple eigenvalue ( m = 1). We assume we know the value of this eigenvalueand then construct the associated eigenvector. If λ is a simple eigenvalue of ρ ˜ α witheigenvector q then ρ ˜ αq = λ q ⇒ ( ρ ˜ α − λ ) q = 0. Thus to determine the eigenvector of ρ ˜ α it suffices to determine the null-vector of ρ ˜ α − λ .It so happens that one can readily write down eigenvectors for all simple eigenvaluesif one can write down the determinant of the matrix ρ ˜ α − λ . To explain this point wetake an important detour. Suppose B is a 3 × B = a a a b b b c c c = a T b T c T (5)with zero determinant and hence there exists a non-trivial null-vector. Suppose zerois a simple eigenvalue of B and hence the null-space is spanned by a single vector(up to scaling). Using the standard cross product in 3 − dimensions, the vector b × c isorthogonal to vectors b and c . On the other hand, the determinant of the singular matrix B is given by det B = a ( b c − b c ) − a ( b c − c b ) + a ( b c − c b ) = a · ( b × c ) = 0,and hence b × c is orthogonal to a, b and c . Since b × c is orthogonal to every row of B ,we have determined a null-vector for the matrix B . This implies we have determinedthe eigenvector for the null-space of B . If B = A − λ where λ is a simple eigenvalue of A , then we have determined the eigenvector of A associated with eigenvalue λ . Recall a · ( b × c ) = c · ( a × b ) = b · ( c × a ). As a consequence, b × c , a × b and c × a are all equallyvalid expressions for the null-vector of B . However as zero is a simple eigenvalue of B ,the null-space is spanned by a single vector and the seemingly different expressions arein fact linearly dependent: they are equal up to scaling.The idea explained above easily generalises into higher dimensions. Indeed thedeterminant of the same 3 × B may also be written as a i b j c k ε ijk where ε ijk is the fully anti-symmetric alternating tensor which equals +1 when i, j, k is an evenpermutation of 1 , ,
3; equals − i, j, k is an odd permutation of 1 , , b j c k ε ijk . This particular representation extends to larger matrices in a straightforwardmanner. Suppose B were an N × N matrix with b ( i ) , i = 1 , , . . . , N , the rows of B .Then det B = b (1) i · · · b ( N ) i N ε i ··· i N where ε i ··· i N is the fully anti-symmetric tensor in N symbols. If det B = 0 then we readily have an expression for the null-vector which isgiven by b (2) i · · · b ( N ) i N ε i ··· i N . The main takeaway from the preceeding discussion is thatthe algebraic expression for the determinant of a matrix, when suitably interpreted,contains the elements of the null-vector (equivalently, the eigenvector). One only has to‘recognise’ the determinant as a dot-product between any row of the matrix with someother vector. The other vector in question, is the eigenvector. Note this statement doesnot assume any special structure for the matrix.We now return to the problem at hand. Note that ρ ˜ α − λ = hρX N ⇒ det ( ρ ˜ α − λ ) = (det X N ) (cid:81) Ni =1 ( ρ i h ) where the elements of the N × N matrix X N oliton-like behaviour in non-integrable systems X N ) ij = (cid:40) γ i , i = j, , i (cid:54) = j, γ i = ρ i ˜ g i − λ ρ i h . (6)Recall, in our previous work [48], we showed that det X N satisfied a recursion relation.Let us define the notation [ γ · · · γ N ] ≡ det X N . Then the recursion relation is[ γ · · · γ N ] = γ N [ γ · · · γ N − ] − N − (cid:88) k =1 [ γ · · · γ k − γ k +1 · · · γ N − ] . (7)Here [ γ · · · γ N − ] is the determinant of the ( N − × ( N −
1) matrix X N − with γ i , i = 1 , . . . , N − X N − but with 1 for k -th element along the diagonal. This impliesdet( ρ ˜ α − λ ) = ˜ γ N [˜ γ · · · ˜ γ N − ] ρ − ρ N h N − (cid:88) k =1 [˜ γ · · · ˜ γ k − ρ k h ˜ γ k +1 · · · ˜ γ N − ] ρ (8)where[˜ γ · · · ˜ γ N − ] ρ = det ˜ X N − , ( ˜ X N − ) ij = (cid:40) ˜ γ i , i = j,hρ i , i (cid:54) = j, and ˜ γ i = ρ i ˜ g i − λ . (9)The subscript ρ in Eq. 8 and Eq. 9 indicates that the concerned matrices contain hρ i as off-diagonal elements. The elements within the square brackets denote thediagonal elements of ˜ X N − . It is easy to note that the N − dimensional vector (cid:0) ρ N h, ρ N h · · · ρ N h, ( ρ N g N − λ ) (cid:1) T is the last row of ( ρ ˜ α − λ ). Moreover we seethat Eq. 8 expresses the determinant of ˜ X N as a dot-product of this vector (a row of˜ X N ) with some other vector. We write out this vector as p ≡ − [ hρ ˜ γ · · · ˜ γ N − ] ρ − [˜ γ hρ ˜ γ · · · ˜ γ N − ] ρ ... − [˜ γ · · · ˜ γ N − hρ N − ] ρ [˜ γ · · · ˜ γ N − ] ρ . (10)Hence Eq. 10 is an explicit representation for the simple eigenvector associated witheigenvalue λ for the matrix ρ ˜ α . From our previous results [48], we have explicit formulaefor elements of the eigenvector. Indeed[˜ γ · · · ˜ γ N − ] ρ = [ γ · · · γ N − ] N − (cid:89) i =1 ( hρ i ) (11)[˜ γ · · · ˜ γ k − ρ k h ˜ γ k +1 · · · ˜ γ N − ] ρ = [ γ · · · γ k − γ k +1 · · · γ N − ] N − (cid:89) i =1 ( hρ i ) (12) oliton-like behaviour in non-integrable systems γ · · · γ N ] = Sym( γ i , N ) + N (cid:88) k =2 ( − k − ( k − γ i , N − k ) , γ i = ρ i ˜ g i − λ hρ i . (13)At this point we present some illustrative examples. Let λ be a known simple eigenvalueof the three component ( N = 3) case. Then the matrix ρ ˜ α − λ = ˜ γ hρ hρ hρ ˜ γ hρ hρ hρ ˜ γ , ˜ γ i = ρ i ˜ g i − λ (14)has vanishing determinant and − [ hρ , ˜ γ ] ρ − [˜ γ , hρ ] ρ [˜ γ , ˜ γ ] ρ = − ˜ γ hρ + ρ ρ h − ˜ γ hρ + ρ ρ h ˜ γ ˜ γ − ρ ρ h (15)is the associated eigenvector of ρ ˜ α . Indeed one can compute the product of the matrix( ρ ˜ α − λ ) with the vector given above to obtain the trivial vector if and only ifdet( ρ ˜ α − λ ) = 0.Likewise for N = 4 we explicitly have − hρ ˜ γ ˜ γ + ( ρ ˜ γ + ρ ˜ γ ) ρ h − ρ ρ ρ h − ˜ γ hρ ˜ γ + ( ρ ˜ γ + ρ ˜ γ ) ρ h − ρ ρ ρ h − ˜ γ ˜ γ hρ + ( ρ ˜ γ + ρ ˜ γ ) ρ h − ρ ρ ρ h ˜ γ ˜ γ ˜ γ − ( ρ ρ ˜ γ + ρ ρ ˜ γ + ρ ρ ˜ γ ) h + 2 ρ ρ ρ h (16)and the general case (for any N ) is given by Eq. 10 which allows us to explicitly write aneigenvector p of ρ ˜ α with simple eigenvalue λ . We conclude by noting that [ p ± λρ − p ] T is an eigenvector of A (Eq. 3) with eigenvalue ± λ and q ≡ ρ − p is then the eigenvectorof ˜ αρ .Before we move to the next section which deals with chiral theory, we introducesome matrices which are of use there. Let Q be the N × N matrix of eigenvectorsof ˜ αρ (i.e., Q represents the matrix with columns q i where q i is the i − th eigenvectorof ˜ αρ ). Let Λ be the N × N diagonal matrix containing positive eigenvalues ( λ ) andlet L = Q T ρQ . We then have the following 2 N × N matrix V whose columns areeigenvectors of A (Eq. 3) V = (cid:32) ρQ Λ − − ρQ Λ − Q Q (cid:33) , ( V − ) T = 12 (cid:32) QL − Λ − QL − Λ ρQL − ρQL − (cid:33) . (17)A major step forward in comparison to the earlier work [48] is that we now cananalytically compute Q and therefore V (Eq. 17). The relevant input is the couplingcoefficients and background densities, once the eigenvalue (sound speed) has beendetermined. oliton-like behaviour in non-integrable systems
3. Effective chiral theory and optimal initial conditions
This section is dedicated to writing down the effective chiral theory (Sec. 3.1) and thendiscussing our prescription to generate optimal initial conditions (Sec. 3.2).
Recall the original MNLS Eq. 1 is a system of nonlinear coupled PDEs of N complexfunctions which can be equivalently written as a system of 2 N PDEs for real fields(Eq. 2). Here the central question is whether we can make a chiral reduction [48] to N chiral equations ( N in each chiral sector). It turns out that we can indeed reducethe system into a total of 2 N uncoupled KdV equations (when all eigenvalues of A aresimple, i.e. distinct sound speeds) ∂ τ f j + K NL f j ∂ ξ f j + K DS ∂ ξ f j = 0 , j = 1 , ...N (for each chiral sector) . (18)We now describe the ingredients of Eq. 18. Every KdV equation in Eq. 18 is for aparticular eigenvalue λ j (speed of sound; for the subsequent discussion we fix a specific j value). Using the small parameter (cid:15) , the new (stretched) space-time variables ( ξ and τ ) used in Eq. 18 are given in terms of the old space-time variables ( x and t ) by theexpressions ξ = (cid:15)x − (cid:15)λ j t and τ = (cid:15) t . In Eq. 18, f j ( ξ, τ ) are chiral fields in stretchedvariables and are related to the original fields by (cid:32) (cid:126)ρ(cid:126)v (cid:33) = (cid:32) (cid:126)ρ (cid:33) + (cid:15) f j ( ξ, τ ) V j + O ( (cid:15) ) , V j = (cid:32) ρq j /λ j q j (cid:33) . (19)Here (cid:126)ρ and (cid:126)v denote the set of density fields { ρ k ( x, t ) } and velocity fields { v k ( x, t ) } .Furthermore (cid:126)ρ is the set of background densities { ρ k } and as before ρ denotes the N × N diagonal matrix with ρ k along the diagonal. V j denotes the j − th column of the2 N × N matrix V (Eq. 17). Looking at Eq. 17, it is easy to note that the j − th columnof V is given by the elements ρ i q ( i ) j /λ j for i = 1 , , ..N and q ( i ) j for i = N + 1 , N + 2 , .. N where q ( i ) j is the i − th element of the N × q j . Note q j is the eigenvector of˜ αρ corresponding to eigenvalue λ j and can be readily extracted from Eq. 10 (along withthe fact that q j ≡ ρ − p j ). It remains to discuss the nonlinear and dispersive coefficients, K NL and K DS respectively. For this purpose, we introduce a 2 N × N givenby [48] N = f j ∂ ξ f j N A + ∂ ξ f j N B = f j ∂ ξ f j ρ ( q j ◦ q j ) λ j ( q j ◦ q j ) + 2 λ j G ◦ ( ρq j ) ◦ ( ρq j ) + ∂ ξ f j − q j λ j (20)where “ ◦ ” in Eq. 20 denotes the element-wise multiplication of N × N is a 2 N × f j ∂ ξ f j ρ i (cid:0) q ( i ) j (cid:1) λ j for oliton-like behaviour in non-integrable systems i = 1 , , ..N and f j ∂ ξ f j (cid:18)(cid:0) q ( i ) j (cid:1) + 2 λ j G i (cid:0) ρ i q ( i ) j (cid:1) (cid:19) − ∂ ξ f j q ( i ) j λ j for i = N + 1 , N + 2 , .. N .The KdV equation (Eq. 18) can be equivalently written as ∂ τ f j + (cid:10) ( V − ) Tj | N (cid:11) = 0where ( V − ) Tj is the j − th column of the 2 N × N matrix ( V − ) T given in Eq. 17. Moreprecisely, ( V − ) Tj = 12 (cid:32) q j λ j /l j ρq j /l j (cid:33) (21)where l j are the diagonal elements of the matrix L in Eq. 17 (in fact L is a diagonalmatrix [48]). It is easy to see that l j = (cid:104) q j | ρq j (cid:105) . We therefore finally get the coefficients K NL and K DS K NL = (cid:10) ( V − ) Tj | N A (cid:11) = 32 l j N (cid:88) i =1 ρ i (cid:0) q ( i ) j (cid:1) + 1 λ j l j N (cid:88) i =1 G i (cid:0) ρ i q ( i ) j (cid:1) (22) K DS = (cid:10) ( V − ) Tj | N B (cid:11) = − λ j l j N (cid:88) i =1 ρ i (cid:0) q ( i ) j (cid:1) . (23)And finally we emphasize even though the KdV equations (Eq. 18) are uncoupled, thecoefficients K NL and K DS for each KdV equation depends on the background fields andcoupling coefficients of all the components. ψ k The M − soliton solutions of the integrable Eq. 18 are given by [47] f j ( ξ, τ ) = 12 K DS K NL ∂ ξ log (cid:18) (cid:88) { σ a ,σ b }∈{ , } e W [ σ a ,σ b ] (cid:19) (24)with W [ σ a , σ b ] = M (cid:88) a =1 σ a η a ( ξ, τ ) + M (cid:88) a,b =1 a
0) which can then be employed as initial conditions for Eq. 1. oliton-like behaviour in non-integrable systems t = 0.0 t = 100.0 t = 200.0
40 30 20 10 0 101.501.521.541.561.581.60 t = 300.0
40 30 20 10 0 10 t = 400.0
40 30 20 10 0 10 t = 500.0 NumericalAnalytical x − λ t D e n s i t y x − λ t
40 30 20 10 0 10 t D e n s i t y
40 30 20 10 0 100100200300400500 1.5001.5151.5301.5451.5601.5751.5901.605 x − λ t T i m e Figure 1. (colour on-line) (Top) Comparison of the numerical evolution of density ofthe original dynamics (Eq. 1) and the chiral dynamics (Eq. 18). The arrows indicatesthe direction of the two soliton peaks. (Bottom) Time evolution of the two soliton-likepeaks with the original dynamics (Eq. 1). Bottom left shows a three dimensional plot(along with the peak positions marked in red) and the bottom right shows the contourplot of the same figure. The contour plot shows how the soliton-like profiles essentiallypass through each other. For all the plots, we choose the following values for theparameters: g = 1 , g = 1 . , g = 1 . , h = 0 . , (cid:15) = 0 . , ρ = 1 . , ρ = 1 . , ρ =1 . , G = 0 . , G = 0 . , G = 0 . dx = 0 . dt = 5 . × − . The solitonparameters are chosen as k = 1 . , k = 1 . , η = − . , η = 8 .
0. The plots showtime evolution of the first component of the three component system ( N = 3) evolvedusing the finite difference scheme given by Eq. 27. We emphasize again, aside from the soliton parameters which are free variables, thephysically relevant information needed to construct these initial conditions is just theset of background densities and coupling coefficients. In the next section, we willdiscuss numerical experiments that were performed following the above procedure andthe remarkable consequence of soliton-like interaction. oliton-like behaviour in non-integrable systems
4. Numerical Simulations
This section is devoted to numerical simulations and has three subsections. In Sec. 4.1we compare the time evolution between original dynamics (Eq. 1) and the effective chiraldynamics (Eq. 18) for the optimally chosen initial condition. In particular, without lossof generality, we will discuss the two-soliton case ( M = 2) for the three componentMNLS ( N = 3). Remarkable soliton-like properties are exhibited. In Sec. 4.2 wedemonstrate that (i) ad hoc guesses for optimal initial condition are doomed to fail and(ii) the dynamics of our optimal initial condition (obtained by the protocol discussedin previous section) are definitely nonlinear. In Sec. 4.3 we present an interesting caseof bidirectional motion, which strictly speaking goes beyond the formalism discussed inthis paper, but nevertheless displays unexpected soliton-like behaviour.Before we discuss our numerical results, we briefly explain the method involved [50].Eq. 1 is discretised using a central difference scheme in space and time, and its finitedifference representation is given by i ψ m,t +∆ tk − ψ m,t − ∆ tk t = 12 ψ m +1 ,tk − ψ m,tk + ψ m − ,tk ∆ x + N (cid:88) j =1 F kj ( | ψ m,tj | ) ψ m,tk (27)where ψ m,tk is the value of ψ for k th component at grid-point m and at time-step t .The method is linearly (dropping the nonlinear terms) stable for ∆ t/ (∆ x ) ≤ / (cid:0) O (∆ t ) + O (∆ x ) (cid:1) . No-flux boundarycondition ∂ψ k /∂x = 0 is used at the boundary points and the domain is large enough toensure that there are no boundary effects. The fourth-order Runge-Kutta scheme wasused for the first time step and Eq. 27 was used for the subsequent time steps. Similarly,the hydrodynamic version (Eq. 2) was also numerically solved by finite difference. Here we compare the time evolution of the original dynamics and the reduced chiraldynamics. As an example we explicitly discuss the three component ( N = 3) MNLS caseadmitting a two-soliton ( M = 2) solution for the chiral equation. Using the effectivechiral theory, MNLS reduces to six distinct KdV equations, each corresponding to adistinct eigenvalue λ j ( j = 1 , , j = 4 , , λ (i.e., j = 1). A two-soliton initial condition solution f ( ξ, τ = 0) is obtainedusing Eq. 24 (by substituting M = 2). We then get the initial condition for density { ρ ( x, , ρ ( x, , ρ ( x, } and velocity, { v ( x, , v ( x, , v ( x, } using Eq. 19. Oncewe have this, we employ the Madelung transformation to get { ψ ( x, , ψ ( x, , ψ ( x, } oliton-like behaviour in non-integrable systems ψ k ( x,
0) = (cid:115) ρ k + (cid:15) ρ k q ( k )1 λ f ( (cid:15)x, e i(cid:15) q ( k )1 (cid:82) x f ( (cid:15)x, dx for k = 1 , , ξ = (cid:15)x − (cid:15)λ t and τ = (cid:15) t ) f ( ξ, τ ) = 12 K DS K NL ∂ ξ log (cid:20) e η + e η + (cid:18) k − k k + k (cid:19) e η + η (cid:21) . (29)The three equations (Eq. 28) are evolved according to Eq. 1 by direct numerical simu-lation. It is interesting to note that the indefinite integral in Eq. 28 can be explicitlyevaluated by the very nature of f ( (cid:15)x,
0) given by Eq. 29 and is direct consequence ofHirota’s bilinear method [47]. We refer to this solution as numerical . On the otherhand Eq. 24 in combination with Eq. 19 represents an analytical expression for the fields ψ j ( x, t ) that are approximate solutions to Eq. 1. We refer to this expression as analytical .In the top panel of Fig. 1 we show a comparison of the density of the first componentfollowing the chiral dynamics (Eq. 18) and the original dynamics (Eq. 1). The timeevolution is shown in the travelling frame of reference (where the chosen eigenvalue λ is the speed of the moving frame). In this frame, the two peaks (of different heights andwidths) move in opposite directions (as depicted by the arrows). As time progresses,these peaks merge and then separate. In other words, they pass through each other.This is expected of a true KdV multi-soliton however we find that this is almost thecase even with the solution of the original non-integrable MNLS (Eq. 1), i.e. we findvery good agreement between the two dynamics (see top panel of Fig. 1 which showsthe various time snapshots). A three-dimensional plot showing these coherent objectspassing through each other is shown in the bottom left of Fig. 1. To get a better insightinto the world-line of the soliton-like objects we also show how the position of the peaksmove in time (bottom right of Fig. 1). Therefore in Fig. 1, we have successfully shownthat we can engineer optimal initial conditions which when evolved according to Eq. 1,behave almost as solitons although Eq. 1 is not integrable. From the discussions in the previous Sec. 4.1, two important questions naturally arise.The first question is whether one needs to go through the systematic procedure discussedin the paper to engineer optimal initial conditions. In other words, one might wonderwhat would happen if we choose initial conditions which seem reasonable based onphysical insights. The second question that arises is how nonlinear the dynamicspresented in Fig. 1 actually is. Another way of posing this question is whether thesecoherent excitations are indeed non-trivial or are they merely pulses evolving accordingto a linearised version of Eq. 1 or equivalently a linearised version of Eq. 2. oliton-like behaviour in non-integrable systems
20 0 20 401.1851.1901.1951.200 t = 0.0
20 0 20 40 t = 140.0 NumericalAnalyticalPerturbed
10 0 10-0.04-0.03-0.02-0.010.00
VelocityPerturbed Velocity x − λ t D e n s i t y
40 20 0 20 401.1851.1901.1951.200 t = 0.0
40 20 0 20 40 t = 140.0
NumericalAnalyticalLinear x − λ t D e n s i t y Figure 2. (colour on-line) (Top) Comparison of the time evolution of the initial profilefrom the effective description and a slightly perturbed initial profile. The perturbedinitial profile is made by using a Gaussian of same height and a slightly wider basefor the velocity profile (see inset). The plots above are for two component ( N = 2)system and the velocity components are given by v = − . × exp [ − x / (2 × . )]and v = − . × exp [ − x / (2 × . )]. The L − infinity norm of the difference comesout to be 0 . . g = 1 , g = 1 . , h = 0 . , (cid:15) =0 . , ρ = 1 . , ρ = 1 . , G = 0 . , G = 0 . , dx = 0 . dt = 5 . × − .The soliton parameters are chosen as k = 1 . , η = 0 . To answer the first question, we consider a rather extreme scenario. We demonstratethat not only ad hoc initial conditions (ad hoc but still based on physical insights andrespecting some basic properties) are doomed to fail but also even minor deviations fromthe optimal initial conditions (derived by our systematic procedure) lead to significantradiative effects.As an example, in top panel of Fig. 2, we have chosen a two component ( N = 2)MNLS with a one-soliton ( M = 1) initial condition. The time evolution of the densityof the first component is shown. The initial density profile is the same as the oneobtained by our procedure. However the velocity profile is slightly perturbed. Indeedthis velocity perturbation results in the density evolution significantly differing from thesolution for the optimal initial condition, in the sense that, we see far more radiation inthe former (see top right panel of Fig. 2 at a particular late time snapshot). In additionto radiation, the size (depth, width) and location of the peak is also different. Thishighlights the importance of our systematic analysis in engineering robust propagating oliton-like behaviour in non-integrable systems k = 1 , , ...N )linearised version of Eq. 2 ∂ t δρ k + ρ k ∂ x δv k = 0 , ∂ t δv k + ∂ x (cid:32) N (cid:88) j =1 α kj δρ j + 2 G k ρ k δρ k (cid:33) + 14 ρ k ∂ δρ k ∂x = 0 . (30)The bottom panel of Fig. 2 shows a clear difference between the true nonlinearand the linear dynamics (when we evolve our carefully engineered initial condition inboth cases). We see that the linear dynamics predicts the incorrect peak position and isplagued by significant dispersive effects. In fact, in the nonlinear dynamics, it is preciselythe intricate balance between nonlinearity and dispersion that results in robust soliton-like behaviour even in a non-integrable model. Till now we only discussed situations when we fix a particular moving frame. In otherwords, we pick a simple eigenvalue λ j and thereby obtain a single specific KdV equation(Eq. 18). Said another way, we focus on perturbations propagating with a single soundspeed. In this section, we prepare an initial condition that is completely outside theparadigm of our paper and the scheme discussed. Here, we prepare the initial profile,for our original dynamics (Eq. 1), derived from the soliton solutions of two different chiral equations. Note, since KdV is an equation with only one time derivative, itcannot model the interaction of disturbances propagating in opposite directions. Moreimportantly there is no a priori reason to suspect the asymptotic analysis, which gaveus approximate solutions and optimal initial conditions, will continue to hold true wheninitial conditions are prepared using different chiral sectors.Let us consider the three component ( N = 3) MNLS with a one-soliton ( M = 1).The opposite speeds (simple eigenvalues) are λ (right moving) and λ (left moving).This means that we have two single-soliton solutions (see Eq. 24) f ( ξ, t ) and f ( ξ, t ).We combine these two solutions to prepare an initial condition for the original MNLS(Eq. 1) (cid:32) (cid:126)ρ ( x, (cid:126)v ( x, (cid:33) = (cid:32) (cid:126)ρ (cid:33) + (cid:15) f ( ξ, τ ) V + (cid:15) f ( ξ, τ ) V [Bidirectional Initial Condition](31)where V and V in Eq. 31 can be obtained using Eq. 19. We re-emphasize that preparinga bidirectional initial profile as shown in Eq. 31 does not fall under the paradigm of oureffective chiral reduction. Nevertheless we do this to test the limits of our constructions.To our surprise the two opposite moving excitations behave as if they are solitons (seeFig. 3) passing through one another with minimal radiative loss. This is a completely oliton-like behaviour in non-integrable systems t = 0.0 t = 8.0 t = 16.0
50 0 501.0941.0951.0961.0971.0981.0991.100 t = 24.0
50 0 50 t = 32.0
50 0 50 t = 40.0 NumericalAnalytical x D e n s i t y x
80 60 40 20 0 20 40 60 80 t D e n s i t y
80 60 40 20 0 20 40 60 800510152025303540 1.09301.09451.09601.09751.09901.1005 x T i m e Figure 3. (colour on-line) Time evolution of an artificial initial condition createdby joining two opposite travelling well separated peaks (Eq. 31). (Top) Comparisonof the numerical evolution of density of the original dynamics (Eq. 1) with theartificial initial condition and the linear superposition of two oppositely travellingKdV solitons (i.e., solitons from j = 1 and j = 6 in Eq. 18). The arrows indicatesthe direction of the two soliton peaks. (Bottom) Time evolution of the two solitonpeaks with the original dynamics. Bottom left shows a three dimensional plot (alongwith the peak positions marked in red) and bottom right shows the contour plot ofthe same figure. For the plots, we chose the following values for the parameters: g = 1 , g = 1 . , g = 1 . , h = 0 . , (cid:15) = 0 . , ρ = 1 . , ρ = 1 . , ρ = 1 . , G =0 . , G = 0 . , G = 0 . dx = 0 . dt = 5 . × − . The soliton parametersare chosen as k = 0 . , η = − . λ (left moving) and k = 1 . , η = 15 . λ (right moving). The plots showtime evolution of the first component of the three component system ( N = 3) evolvedusing the finite difference scheme given by Eq. 27. oliton-like behaviour in non-integrable systems
5. Repeated sound speeds
Thus far in this paper we limited the discussion to the case when all eigenvaluesof A are simple. As a result we obtained a single KdV equation (Eq. 18) once aspecific eigenvalue λ j was chosen. This eigenvalue (and the coupling coefficients andbackground densities) allowed us to construct the optimal initial conditions for robustpropagation. We then compared the resulting approximate time dynamics with thatdictated by original equation (Eq. 1). The alternative and, in our opinion, fascinatingphenomenon where A (Eq. 3) admits repeated eigenvalues will be part of a subsequentwork. Here we simply note the resulting chiral equations take the form of a system ofcoupled KdV-type equations and present a non-trivial generalisation of the results in thepresent manuscript. In this section, we will lay down the foundation for the subsequentinvestigation. We first put forward the following important theorem. Theorem 5.1
Suppose λ is an eigenvalue of ρ ˜ α . For m ≥ , the multiplicity of λ is m if and only if the eigenvalue takes the form λ = ρ i (˜ g i − h ) for m + 1 pairs ( ρ i , ˜ g i ) . We do not present the proof of this theorem here since it is not particularlyilluminating. We remark in passing that this peculiar property of ρ ˜ α is entirely aconsequence of the specific form for the cross-species coupling assumed. The mainimport of the theorem is that any time a repeated eigenvalue occurs, the eigenvaluemust be of the form specified. This then implies all coefficients of the associated reducedKdV equations are fully defined only in terms of the coupling coefficients and backgrounddensities.Indeed there are additional consequences of the aforementioned condition on λ .Since there are m + 1 ( ρ i , ˜ g i ) pairs involved in the theorem, without loss of generality,let us number them i = 1 , , . . . , m +1. This is permissible since each species is coupled inexactly the same manner to every other species. Only the self-interaction ˜ g i distinguishesspecies. Hence we have λ = ρ (˜ g − h ) = . . . = ρ m +1 (˜ g m +1 − h ) . (32)Note that the above expression is true for a particular h . Thus eliminating h from anytwo ( ρ i , ˜ g i ) pairs we obtain multiple expressions for this value h = h ij = ρ i ˜ g i − ρ j ˜ g j ρ i − ρ j , i (cid:54) = j, i, j ∈ { , , . . . , m + 1 } . (33)By definition h ij = h ji . Next observe that by equating any two h ij we obtain a constraintbetween various ˜ g i , ρ i . One might naively expect a large number of such constraintshowever many of these are redundant in the sense that h ij = h jk ⇐⇒ h ik = h kj ⇐⇒ h ij = h ik . (34)Indeed all three of the above equalities (Eq. 34) reduce to precisely one constraint ρ i ˜ g i ( ρ j − ρ k ) + ρ j ˜ g j ( ρ k − ρ i ) + ρ k ˜ g k ( ρ i − ρ j ) = 0 . (35) oliton-like behaviour in non-integrable systems h ij which imply all others in Eq. 33 is h = h , h = h , . . . h m − m = h m m +1 . (36)The above Eq. 36 is equivalent to the following ( m −
1) constraint equations ρ ˜ g ( ρ − ρ ) + ρ ˜ g ( ρ − ρ ) + ρ ˜ g ( ρ − ρ ) = 0(37) ρ ˜ g ( ρ − ρ ) + ρ ˜ g ( ρ − ρ ) + ρ ˜ g ( ρ − ρ ) = 0(38)... ρ m − ˜ g m − ( ρ m − ρ m +1 ) + ρ m ˜ g m ( ρ m +1 − ρ m − ) + ρ m +1 ˜ g m +1 ( ρ m − − ρ m ) = 0 . (39)The above system of equations represents a necessary condition for repeated eigenvalues(repeated sound speeds) to exist in our system. This condition is given in terms of thecoupling coefficients and background densities. Irrespective of the size of the system N ,if a repeated eigenvalue exists then some subset of m + 1 pairs ( ρ i , ˜ g i ) satisfy the above( m −
1) equations. Thus if no subset of the ( ρ i , ˜ g i ) satisfy the above relations, then thesystem consists of only simple eigenvalues and the results from the earlier part of thecurrent paper are operative.Eqns. 37 - 39 are also a sufficient condition for repeated eigenvalues. Indeed if oneselects the coupling constants and background densities to satisfy the above constraint,then a little algebra shows one can define h such that Eq. 33 holds. If this h is positive,then one has indeed found a value of the cross-coupling constant to guarantee a repeatedeigenvalue of ρ ˜ α . Furthermore, from Eq. 32, this repeated eigenvalue (of multiplicity m ) is given by λ = ρ i (˜ g i − h ) where one may use any of the i = 1 , . . . , m + 1.Since there are ( m −
1) equations in 2( m + 1) variables, one can always find asolution. Evidently there is a 2( m + 1) − ( m −
1) = m + 3 dimensional real manifold suchthat each point on the manifold is a potential system to guarantee repeated eigenvalues.Thus there are in fact many ways to construct a system with repeated eigenvalues.Solving Eqns. 37 - 39 is also straightforward. For instance one can pick any ρ i > g i . The vectorof ˜ g i , i = 1 , , . . . m + 1 belong to the null space of matrix. One can show this matrixgenerically has a two-dimensional null-space (when all ρ i are unequal).For each such choice of ( ρ i , ˜ g i ), there is a unique h that gives rise to the repeatedeigenvalue. If h is varied even slightly from this critical value (while keeping ( ρ i , ˜ g i )fixed) then the repeated eigenvalue splits up into unequal eigenvalues. Nevertheless theprescription described in this section gives the experimentalist the precise value of thecross-component coupling h and that will guarantee repeated sound speeds in a verytransparent manner. This also closes a gap in the analysis of our previous work [48]where we stated multiple eigenvalues were possible for specific values of h but were notable to provide a full description of that scenario. oliton-like behaviour in non-integrable systems
6. Conclusions and Outlook
In this paper, we addressed a general question of the possibility of constructing initialconditions for generic non-integrable models, such that they bear as much resemblance aspossible to solitons of integrable models. We successfully found such localised excitationsthat move at almost constant speed and barely show scattering / radiation effects. Ourconstruction is systematic and its success was demonstrated in Fig. 1. We also discussedtwo natural questions - (i) Can one make crude attempts to design initial conditionsby circumventing our systematic prescription? and (ii) How truly nonlinear is ourdynamics? We provided convincing evidence that (i) Crude attempts to design initialconditions by evading our procedure is doomed to fail (top panel of Fig. 2) and (ii) Ourdesigned initial conditions undergo truly nonlinear evolution (bottom panel of Fig. 2).We also presented an interesting finding on bidirectional evolution that is composed ofboth chiral sectors (Fig. 3). As this falls outside the paradigm of our formalism wepresent this as an interesting observation but no theoretical explanation is offered. It isindeed remarkable that one can find excitations moving in opposite directions (for a non-integrable model such as Eq. 1) that have strong resemblance with solitons of integrablemodels. Needless to mention, the findings in this paper can serve as a guiding principle toengineer initial conditions (localised excitations) in experiments (such as cold atoms ornonlinear optics) that can subsequently display robust soliton-like evolution. Indeed ourexpressions may also serve as suitable initial guesses for numerical routines employingthe Ansatz-based approach to obtain travelling-wave excitations. The expressions andthe resulting dynamics also strongly suggest there do in fact exist stably propagatinglocalised solutions to systems of coupled PDEs such as Eq. 1.In the future, we plan to investigate the case of degeneracy (repeated eigenvalues),the foundation for which has already been laid out in this paper (Sec. 5). This isnaturally expected to result in coupled KdV equations which might in turn give rise topossibility of finding new integrable chiral field-theories (non-trivial generalisations toEq. 18). It is also paramount to mention that although, we used MNLS (Eq. 1) as aplatform, our formalism can be exploited to understand nonlinear dynamics in severalother models, such as classical spin chains [51–55] which too is a subject of futureinvestigation.
Acknowledgements
We thank Ziad Musslimani, Andrea Trombettoni, Chiara D’Errico, Nicolas Pavloff,Bernard Deconinck, Konstantinos Makris and Swetlana Swarup for useful discussions.MK would like to acknowledge support from the project 6004-1 of the Indo-FrenchCentre for the Promotion of Advanced Research (IFCPAR), Ramanujan Fellowship(SB/S2/RJN-114/2016), SERB Early Career Research Award (ECR/2018/002085) andSERB Matrics Grant (MTR/2019/001101) from the Science and Engineering ResearchBoard (SERB), Department of Science and Technology, Government of India. VV oliton-like behaviour in non-integrable systems
References [1] Jackson E A 1989
Perspectives of Nonlinear Dynamics: Volume 1 vol 1 (CUP Archive)[2] Morsch O and Oberthaler M 2006
Reviews of Modern Physics Science
Science
Physical Review Letters Physical Review Letters
Integrable models vol 30 (World scientific)[8] Babelon O, Bernard D and Talon M 2003
Introduction to classical integrable systems (CambridgeUniversity Press)[9] Thompson J M T and Stewart H B 2002
Nonlinear dynamics and chaos (John Wiley & Sons)[10] Tabor M 1989
Chaos and integrability in nonlinear dynamics: an introduction (Wiley)[11] Kulkarni M and Abanov A G 2012
Physical Review A Nonlinear dispersive equations: local and global analysis
106 (American MathematicalSoc.)[13] Drazin P G and Johnson R S 1989
Solitons: an introduction vol 2 (Cambridge university press)[14] Ablowitz M J and Segur H 1981
Solitons and the inverse scattering transform (SIAM)[15] Doedel E J 1981
Congr. Numer Numerical continuation methods: an introduction vol 13 (SpringerScience & Business Media)[17] Ablowitz M J, Ablowitz M, Prinari B and Trubatch A 2004
Discrete and continuous nonlinearSchr¨odinger systems vol 302 (Cambridge University Press)[18] Ablowitz M and Ladik J 1976
Studies in Applied Mathematics The discrete nonlinear Schr¨odinger equation: mathematical analysis,numerical computations and physical perspectives vol 232 (Springer Science & Business Media)[20] Ablowitz M J and Musslimani Z H 2013
Physical Review Letters
Nonlinearity Studies in Applied Mathematics
Nonlinearity with Disorder (Springer) pp 67–84[24] Dalfovo F, Giorgini S, Pitaevskii L P and Stringari S 1999
Reviews of Modern Physics et al. arXiv:1509.06844 [26] Inouye S, Andrews M, Stenger J, Miesner H J, Stamper-Kurn D and Ketterle W 1998 Nature
Physical Review A Physical Review A Applied Mathematical Modelling Journal of the Physical Society of Japan Optical solitons: from fibers to photonic crystals (Academicpress) oliton-like behaviour in non-integrable systems [32] Agrawal G P 2000 Nonlinear fiber optics Nonlinear Science at the Dawn of the 21st Century (Springer) pp 195–211[33] Pushkarov D and Tanev S 1996
Optics Communications
Physical Review E OpticsCommunications et al.
Solitons in molecular systems (Springer)[38] Chin C, Grimm R, Julienne P and Tiesinga E 2010
Reviews of Modern Physics PhysicalReview A Physical Review Letters et al.
Science
Physical Review A et al. Physical Review A R3381[44] Manakov S V 1974
Soviet Physics-JETP Journal of Nonlinear Science arXiv:1512.06754 [47] Hirota R 2004 The Direct Method in Soliton Theory (Cambridge University Press, page 54)[48] Swarup S, Vasan V and Kulkarni M 2020
Journal of Physics A Zeitschrift f¨ur Physik Journal of Computational Physics
203 – 230 ISSN 0021-9991[51] Lakshmanan M 2011
Philosophical Transactions of the Royal Society A: Mathematical, Physicaland Engineering Sciences
Physica A: Statistical Mechanics and itsApplications Physical Review B Journal ofStatistical Physics
Journal of Mathematical Physics33