Dynamics of bright and dark multi-soliton solutions for two higher-order Toda lattice equations for nonlinear waves

Xiao-Yong Wen,Ling Xu,Nan Liu

doi:10.1186/s13662-018-1748-y

Xiao-Yong Wen, Ling Xu + Show 1 more

Open Access

https://doi.org/10.1186/s13662-018-1748-y

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Abstract

Discrete N-fold Darboux transformation (DT) is used to derive new bright and dark multi-soliton solutions of two higher-order Toda lattice equations. Propagation and elastic interaction structures of such soliton solutions are shown graphically. The details of their evolutions are studied via numerical simulations. Numerical results show the accuracy of our numerical scheme and the stable evolutions of such bright and dark multi-solitons without a noise. To compare the numerical evolution results with the classical Toda lattice equation, we also investigate the dynamical behaviors of the multi-soliton solutions for Toda lattice equation via numerical simulations, and we find that the multi-soliton solutions of Toda lattice equation have better stability and are more robust against a big noise than its two corresponding higher-order equations. The same small noise has different effect on the evolutions of the multi-soliton solutions for three different equations in the same hierarchy. The possible reason is that the higher-order nonlinear terms of the higher-order equation cause the instability of the wave propagation. The discrete generalized (n,N-n)-fold DTs are constructed and used to derive some discrete rational solutions of three equations, and a few mathematical features for such rational solutions are also discussed. Results might be helpful for understanding the propagation of nonlinear waves in soliton theory.

Highlights

The nonlinear lattice equations (NLEs) [1,2,3,4], treated as the spatially discrete analogues of the nonlinear partial differential equations (NPDEs) [5,6,7,8,9], have received certain attention [10,11,12,13,14]
Dynamical behaviors of solitons in the continuous and discrete cases are described by the NPDEs and NLEs [10], respectively
It is worth noting that the corresponding (2N)-soliton solutions for both Eqs. (5) and (6) can reduce to the (2N – 1)-soliton solutions by choosing spectral parameter λ

Summary

Introduction

The nonlinear lattice equations (NLEs) [1,2,3,4], treated as the spatially discrete analogues of the nonlinear partial differential equations (NPDEs) [5,6,7,8,9], have received certain attention [10,11,12,13,14]. Figures 4(a1)–(c1) present the overtaking collision interactions among three unidirectional anti-bell-shape dark solitons with different amplitudes for solution un in (45), the minimums of amplitude for the three dark solitons are 1.5, 2, and 3 units under the background level 1, respectively. Figures 4(a2)–(c2) reveal the overtaking collision interactions among three unidirectional bell-shape bright solitons with different amplitudes for solution un in (45), the maximums of amplitude for the three bright solitons are 2, 3, and 5.25 units, respectively, above the background level 1. (III) When N = 3, according to Theorem 2, based on the generalized (1, 2)-fold DT, we have the third-order rational solution of Eq (1) as follows: un = un + dn(2) – dn(2+), vn vn + c(n2) 1 – b(n2). Further research is needed whether or not Eq (1) has some nonsingular rational solutions

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Conclusion