Modified non-linear Schrödinger models, {\cal C}{\cal P}_s{\cal T}_d symmetry, dark solitons and infinite towers of anomalous charges
aa r X i v : . [ h e p - t h ] F e b Modified non-linear Schr¨odinger models, CP s T d symmetry, dark solitons andinfinite towers of anomalous charges H. Blas ( a ) , M. Cerna Magui˜na ( b ) and L.F. dos Santos ( c )( a ) Instituto de F´ısicaUniversidade Federal de Mato GrossoAv. Fernando Correa, N ( b ) Departamento de Matem´aticaUniversidad Nacional Santiago Ant´unez de MayoloCampus Shancay´an, Av. Centenario 200, Huaraz - Per´u ( c ) Centro Federado de Educa¸c˜ao Tecnologica-CEFET-RJCampus Angra dos Reis, Rua do Areal, 522, Angra dos Reis- RJ -Brazil
Abstract
Some modified (defocusing) non-linear Schr¨odinger models (MNLS) possess infinite towers of anoma-lous conservation laws with asymptotically conserved charges. The so-called anomalies of the quasi-conservation laws vanish upon space-time integration for a special CP s T d symmetric field configurations.We verify numerically the degree of modifications of the charges around the dark-soliton interaction re-gions by computing numerically some representative anomalies related to lowest order quasi-conservationlaws of the non-integrable cubic-quintic NLS model as a modified (defocusing) NLS model. This modifi-cation depends on the parameter ǫ , such that the standard NLS is recovered for ǫ = 0. Here we presentthe numerical simulations for small values of | ǫ | , and show that the collision of two dark solitons areelastic. The NLS-type equations are quite ubiquitous in several areas of non-linear science. Introduction
Some non-linear field theory models with important physical applications and solitary wave solutions arenot integrable. Recently, some deformations of integrable models, such as sine-Gordon, Korteweg-de Vriesand non-linear Schr¨odinger models [1][2][3][6], which exhibit soliton-type properties, have been put forward.Quasi-integrability properties of the deformations of the integrable models have recently been examined inthe frameworks of the anomalous zero-curvature formulations [1][4][6] and the deformations of the Riccati-type pseudo-potential approach [2][3][5]. Recently, it has been considered the properties of the modified(focusing) non-linear Schr¨odinger model with bright solitons [5]. Here we tackle the problem of constructing,analytically and numerically, new towers of anomalous charges for the modified (defocusing) non-linearSchr¨odinger model with dark solitons; so extending the results of [4] by providing novel infinite towers ofquasi-conservation laws. The both type of models (focusing and defocusing) differ in the relevant signs (+ / − )of their coupling constants and the boundary conditions (b.c.) imposed on their soliton solutions. So, in thefocusing (defocusing) case one has bright (dark) solitons with vanishing (non-vanishing) b.c.’s. Consider the modified non-linear Schr¨odinger models (MNLS) i∂ t ψ ( x, t ) + ∂ x ψ ( x, t ) − [ δV ( | ψ | ) δ | ψ | ] ψ ( x, t ) = 0 , (2.1)where ψ ∈ C and V : R + → R is the deformed potential.Let us consider a special space-time reflection around a fixed point ( x ∆ , t ∆ ) as a symmetry of soliton-typesolutions of the model e P : ( e x, e t ) → ( − e x, − e t ); e x = x − x ∆ , e t = t − t ∆ . (2.2)The transformation e P defines a shifted parity P s for the spatial variable x and a delayed time reversal T d for the time variable t . It is assumed that the ψ solution of the deformed NLS model (2.1) possesses thefollowing property under the transformation (2.2) e P ≡ P s T d , e P ( ψ ) = e iδ ¯ ψ, e P ( ¯ ψ ) = e − iδ ψ, ¯ ψ ≡ ψ ⋆ , δ = constant. (2.3)In [5] it has been provided a method to construct an infinite number of towers of quasi-conservation laws.Here we consider the lowest order and the first three towers of quasi-conservation laws and discuss them inthe context of the defocusing NLS with dark soliton solutions. The first order charge and its generalization becomes ddt Q ( t ) = Z dx ˆ α , ˆ α ≡ F (1) ( I ) ∂ x [ ∂ x ¯ ψ∂ x ψ ] (2.4) Q ( t ) = Z ∞−∞ dx [ iF ( I ) ¯ ψ∂ x ψ − ψ∂ x ¯ ψ ¯ ψψ ] , I ≡ | ψ | , (2.5) Formally, one can assume deg ( ψ ± ) = deg ( ¯ ψ ± ) = ± , deg ( ∂ x ) = deg ( ∂ t ) = 1. F ( n ) ( I ) ≡ d n dI n F ( I ). For F = 1 one has ˆ α = 0 in (2.4) and the relevant charge Q top turns out to bethe topological charge of the dark soliton’s phase. For arbitrary F and the special solutions satisfying theparity property (2.2)-(2.3) one has Z e t − e t dt Z e x − e x dx ˆ α = 0 , for e t → ∞ , e x → ∞ . (2.6)Therefore, integrating in t on the b.h.s.’s of (2.4) one can get Q ( e t ) = Q ( − e t ) , e t → ∞ . (2.7)The special function F ( I ) = e − I has been used in [7] to study the first integrals in the study of soliton-gasand integrable turbulence.A tower of infinite number of quasi-conservation laws can be constructed on top of a given lowest orderexact (quasi-)conservation law. Next, we will present the first few of them. First tower
One can construct a tower of quasi-conserved charges on top of the exact conserved charge Q = R dx ( ¯ ψψ ). So, one has ddt Q n = Z dx ˆ β n ; ˆ β n ≡ − n ∂ x [( ¯ ψψ ) n − ] i ( ¯ ψ∂ x ψ − ψ∂ x ¯ ψ ) , n = 2 , , ... (2.8) Q n = 12 n Z dx ( ¯ ψψ ) n . (2.9)For the field ψ satisfying (2.2)-(2.3) the anomaly density ˆ β n possesses an odd parity for any n . Therefore,one must have the vanishing of the space-time integral of the anomaly ˆ β n and then, the asymptoticallyconserved charges satisfy Q n ( e t ) = Q n ( − e t ) , e t → ∞ , n = 2 , , ... (2.10) Second tower
The next tower of quasi-conserved charges is constructed on top of the exact conserved charge e Q ( t ) = i R dx ( ¯ ψ∂ x ψ − ψ∂ x ¯ ψ ). So, one has ddt e Q n = Z dx ˆ γ n ; ˆ γ n ≡ − ( i ) n n ∂ x [( ¯ ψ∂ x ψ − ψ∂ x ¯ ψ ) n − ] × (2.11) h ∂ x ψ∂ x ¯ ψ − ψ∂ x ¯ ψ − ¯ ψ∂ x ψ + 2 V (1) | ψ | − V i , e Q n ( t ) = Z dx ( i ) n n ( ¯ ψ∂ x ψ − ψ∂ x ¯ ψ ) n , n = 2 , , .... (2.12)Similarly, one has the asymptotically conserved charges e Q n ( e t ) = e Q n ( − e t ) , e t → ∞ , n = 2 , , ... (2.13) Third tower ddt K n = Z dx ˆ δ n , n = 1 , , ... (2.14) K n = Z dx [ ∂ x ¯ ψ∂ x ψ ] n . (2.15)The general form of the anomalies ˆ δ n are provided in [5]. Below we will consider the case n = 1ˆ δ ≡ i [( ¯ ψ∂ x ψ ) − ( ψ∂ x ¯ ψ ) ] V (2) ( I ) , V (2) ≡ d dI V ( I ) . (2.16)Regarding the standard (defocusing) NLS model one can argue that all the anomalies will vanish uponintegration in space-time provided that the N-dark solitons satisfy (2.2)-(2.3). Consequently, their associatedcharges will be asymptotically conserved even for the standard NLS model. In fact, these type of solutionshave been constructed in [5] for N-bright solitons. We already have those results for standard NLS darksolitons and they will appear elsewhere. So, those examples show an analytical, and not only numerical,demonstration of the vanishing of the space-time integrals of the anomalies associated to the infinite towersof infinitely many quasi-conservation laws in soliton theory. We consider the non-integrable cubic-quintic NLS (CQNLS) model i ∂ψ ( x, t ) ∂t + ∂ ψ ( x, t ) ∂x − (cid:16) η | ψ ( x, t ) | − ǫ | ψ ( x, t ) | (cid:17) ψ ( x, t ) = 0 , (3.1)where η > , ǫ ∈ R . The model (3.1) possesses a solitary wave solution of the form ψ ( x, t ) = Φ( z )exp[ i Θ( z )+ iwt ] , z = x − vt (see [4]) Φ ± ( z ) = ξ + rξ tanh [ k ± ( z − z )]1 + r tanh [ k ± ( z − z )] (3.2)Θ ± ( z ) = ∓ arctan hs r ξ ξ tanh [ k ± ( z − z )] i (3.3)where r ≡ | ψ | − ξ ξ − | ψ | , k ± ≡ r | ǫ | p ( ± )( ξ − | ψ | )( | ψ | − ξ ) , (3.4) ξ = B − √ B − v ǫ ǫ , ξ = B + √ B − v ǫ ǫ , B ≡ η − ǫ | ψ | . (3.5)The notations Φ ± and Θ ± correspond to ǫ > ǫ <
0, respectively. So, we will take two one-dark solitarywaves located some distance apart as the initial condition for our numerical simulations of two-dark solitoncollisions. Below, we numerically compute the space and space-time integrals of the anomaly densities ˆ α ,ˆ β , ˆ γ and ˆ δ , appearing in (2.4), (2.8), (2.11) and (2.16), respectively, for two type of two-soliton collisionsof the CQNLS model (3.1). 3igure 1: (color online) Collision of two dark solitons of the CQNLS model (3.1) for ǫ = +0 . , | ψ | = 6 , η =2 .
5. The initial solitons ( t i =green line) travel with velocities v ≈ − . √ v ≈ √ t c = blue line) in their closest approximation and thentransmit to each other. The dark solitons after collision are plotted as a red line ( t f ).Figure 2: (color online) Reflection of two dark solitons of the cubic-quintic NLS model (3.1) plotted for ǫ = − . , | ψ | = 6 , η = 2 .
5. The initial solitons ( t i =green line) travel in opposite direction with velocity | v | ≈ . √
2. They partially overlap ( t c = blue line) in their closest approximation and then reflect to eachother. The dark solitons after collision are plotted as a red line ( t f ).4igure 3: (color online) The left Fig. shows the plot R + e x − e x ˆ α dx vs t and the right one shows the plot R + e t − e t dt R + e x − e x dx ˆ α vs t for the anomaly ˆ α in (2.4) and for the 2-soliton collision of Fig. 1. We numerically compute the space and space-time integrals of the anomaly densities ˆ α , ˆ β , ˆ γ and ˆ δ , asshown in Figs. 3, 4, 5 and 6, respectively, for the transmission of two-dark solitons of the CQNLS model(3.1) as plotted in Fig. 1. 5igure 4: (color online) Top Fig. shows the profile at initial (green), collision (blue) and final (red) times ofthe anomaly density ˆ β in (2.8) for the 2-soliton collision of Fig. 1. In the bottom the left Fig. shows theplot R + e x − e x ˆ β dx vs t and the right one shows the plot R + e t − e t dt R + e x − e x dx ˆ β vs t .Figure 5: (color online) The left Fig. shows the plot R + e x − e x ˆ γ dx vs t and the right one shows the plot R + e t − e t dt R + e x − e x dx ˆ γ vs t of the anomaly density ˆ γ in (2.11) for the 2-soliton collision of Fig. 1.6igure 6: (color online) Top Fig. shows the profile at initial (green), collision (blue) and final (red) timesof the anomaly density ˆ δ in (2.16) for the 2-soliton collision of Fig. 1. The bottom left shows the plot R + e x − e x ˆ δ dx vs t and the right one shows the plot R + e t − e t dt R + e x − e x dx ˆ δ vs t .7igure 7: (color online) The left Fig. shows the plot R + e x − e x ˆ α dx vs t and the right one shows the plot R + e t − e t dt R + e x − e x dx ˆ α vs t , for the anomaly ˆ α in (2.4) computed for the soliton reflection in Fig. 2.Figure 8: (color online) The left Fig. shows the plot R + e x − e x ˆ β dx vs t and the right one shows the plot R + e t − e t dt R + e x − e x dx ˆ β vs t for the anomaly ˆ β in (2.8) computed for the soliton reflection in Fig. 2. Next, we numerically simulate the space and space-time integrals of the anomaly densities ˆ α , ˆ β , ˆ γ andˆ δ , as shown in Figs. 7, 8, 9 and 10, respectively, for the reflection of two-dark solitons of the CQNLS model(3.1) as plotted in Fig. 2.Some comments are in order here. First, in our numerical simulations of the 2-dark soliton collisions of theCQNLS model (3.1) we have not observed appreciable emission of radiation during the collisions; so, it can beargued that the linear superposition of well separated two solitary waves of the CQNLS model is an adequateinitial condition. Second, we have shown the vanishing of the space-time integrals of the anomaly densitiesˆ α , ˆ β , ˆ γ and ˆ δ , appearing in (2.4), (2.8), (2.11) and (2.16), respectively, within numerical accuracy. Third,we have performed extensive numerical simulations for a wide range of values in the parameter space; i.e.the deformation parameter | ǫ | < η ≈ .
5, several amplitudes and relative velocitiesfor 2-soliton collisions, obtaining the vanishing of those anomalies, within numerical accuracy.Sometimes the vanishing of the anomaly, within numerical accuracy, already happens for the spaceintegration alone, e.g. as in the Figs. 3 and 7. In fact, in the Fig. 7 one has R + e x − e x dx ˆ α ≈ − . This factcan be explained by some symmetry considerations of the anomalies [4] written in a new parametrization of8igure 9: (color online) The left Fig. shows the plot R + e x − e x ˆ γ dx vs t and the right one shows the plot R + e t − e t dt R + e x − e x dx ˆ γ vs t for the anomaly ˆ γ in (2.11) computed for the soliton reflection in Fig. 2.Figure 10: (color online) The left Fig. shows the plot R + e x − e x ˆ δ dx vs t and the right one shows the plot R + e t − e t dt R + e x − e x dx ˆ δ vs t for the anomaly ˆ δ in (2.16) computed for the soliton reflection in Fig. 2.9he field ψ . So, let us write the anomaly density of eq. (2.4) ˆ α asˆ α = 2 F (1) ( I ) ∂ x [ ( ∂ x I ) I + 12 I ( ∂ x ϕ ) ] , ψ ≡ √ Ie iϕ/ . (3.6)Notice that this anomaly density is an odd function under the space reflection x → − x , provided that I → I and ϕ → ϕ . In the Fig 2. for the collision of two dark solitons one has the plot of the modulus | ψ | = √ I forthree successive times which shows this type of symmetry for each time.On the other hand, the vanishing R + e x − e x dx ˆ α ≈ − in Fig. 3 might happen for some other reasons thanthe existence of some symmetry arguments as above, since this fact can not be visualized qualitatively inthe collision of two dark solitons in Fig. 1. In fact, the exact analytic 2-soliton solutions of the modifiedNLS model (3.1) are not known, so it is not possible to show this type of symmetries for the explicit fieldconfigurations. However, in the case of the standard NLS model the analytic N-soliton solutions are availableand the relevant space-time symmetries can be examined for the solutions and the various anomalies [4][6].The true understanding of the vanishing of the space-time integral of the anomalies and the relevanceof them for the dynamics of the collision of the solitons of the modified NLS model are under currentinvestigations. The only explanation, so far, for the vanishing of the integrated anomalies, is the symmetryargument as presented above.Remarkably, infinite number of anomalies and the related quasi-conserved charges are also present inthe standard NLS model [5]. So, an exact conserved charge of certain order can be constructed as a linearcombination of some quasi-conserved charges of the same order, and when a linear combination of theirrelated anomalies vanish, even before the space-time integration of them are performed. Quasi-integrability properties of the CQNLS model have been examined by providing novel anomalouscharges related to infinite towers of quasi-conservation laws. The anomaly densities exhibit odd paritiesunder the special space-time symmetry (2.2)-(2.3) of the field configurations.Through numerical simulations of 2 − dark soliton collisions we have checked the quasi-conservation prop-erties of the lowest order charges of the CQNLS model defined in (2.4), (2.8), (2.11) and (2.14), respectively.So, we computed the space and space-time integrals of their associated anomaly densities ˆ α , ˆ β , ˆ γ and ˆ δ ,for two types of two-soliton collisions of the CQNLS model (3.1) as plotted in the Figs. 1 and 2, respectively.In our numerical simulations presented in the Figs 3-6 and 7-10 we have observed that the space-time inte-grals of the set of anomalies ˆ α , ˆ β , ˆ γ and ˆ δ vanish within numerical accuracy. So, one can conclude thatfor 2-dark solitons the relevant charges are asymptotically conserved and their collisions are elastic withinnumerical accuracy, for a wide range of values of the set { η, ǫ } and a variety of amplitudes, velocities andrelative initial phases. Since the modified NLS equations are quite ubiquitous, our results may find potentialapplications in several areas of non-linear science. 10 Acknowledgments
HB thanks FC-UNI (Lima-Per´u) and FC-UNASAM (Huaraz-Per´u) for hospitality during the initial stageof the work. MC thanks the Peruvian agency Concytec for partial financial support. LFdS thanks CEFETCelso Sukow da Fonseca-Rio de Janeiro-Brazil for kind support. The authors thank A. C. R. do Bonfim,H. F. Callisaya, C. A. Aguirre, J. P. R. Campos, R. Q. Bellido, J.M.J. Monsalve and A. Vilela for usefuldiscussions.
References [1] L.A. Ferreira and Wojtek J. Zakrzewski (2011)
JHEP05 .L.A. Ferreira, G. Luchini and Wojtek J. Zakrzewski (2012)
JHEP09 .F. ter Braak, L. A. Ferreira and W. J. Zakrzewski (2019)
NPB
Nucl. Phys. B :114852-114905.[3] H. Blas, R. Ochoa and D. Suarez (2020) Quasi-integrable KdV models, towers of infinite number ofanomalous charges and soliton collisions
JHEP03 :1-48.[4] H.Blas, and M. Zambrano (2016) Quasi-integrability in the modified defocusing non-linear Schr¨odingermodel and dark solitons
JHEP03 :1-47.[5] H. Blas, M. Cerna and L.F. dos Santos (2020) Modified non-linear Schr¨odinger models, CPT invariantN-bright solitons and infinite towers of anomalous charges, arXiv:2007.13910 [hep-th] [6] H. Blas, A.C.R. do Bonfim and A.M. Vilela (2017) Quasi-integrable non-linear Schr¨odinger models,infinite towers of exactly conserved charges and bright solitons
JHEP05 :1-28.[7] G. Roberti, G. El, S. Randoux and P. Suret (2019)
PRE
PRE98