Darboux Transformation and Symbolic Computation on Multi-Soliton and Periodic Solutions for Multi-Component Nonlinear Schr¨odinger Equations in an Isotropic Medium

2009 ◽  
Vol 64 (5-6) ◽  
pp. 300-308 ◽  
Author(s):  
Hai-Qiang Zhang ◽  
Tao Xu ◽  
Juan Li ◽  
Li-Li Li ◽  
Cheng Zhang ◽  
...  
Keyword(s):  
Periodic Solutions ◽  
Iterative Algorithm ◽  
Integrable System ◽  
Symbolic Computation ◽  
Darboux Transformation ◽  
Isotropic Medium ◽  
Soliton Solutions ◽  
Practical Interest ◽  
Lax Pair ◽  
Block Matrices

Abstract The Darboux transformation is applied to a multi-component nonlinear Schr¨odinger system, which governs the propagation of polarized optical waves in an isotropic medium. Based on the Lax pair associated with this integrable system, the formula for the n-times iterative Darboux transformation is constructed in the form of block matrices. The purely algebraic iterative algorithm is carried out via symbolic computation, and two different kinds of solutions of practical interest, i. e., bright multi-soliton solutions and periodic solutions, are also presented according to the zero and nonzero backgrounds.

Darboux transformation and new soliton solutions for the (2+1)-dimensional complex modified Korteweg–de Vries and Maxwell–Bloch equations

2018 ◽  
Vol 32 (23) ◽  
pp. 1850277 ◽  
Author(s):  
Rui Guo ◽  
Run Zhou ◽  
Xiao-Mei Zhang ◽  
Yu-Zhen Chai
Keyword(s):  
Symbolic Computation ◽  
Darboux Transformation ◽  
Soliton Solutions ◽  
Lax Pair ◽  
Dynamic Features ◽  
Bloch Equations ◽  
De Vries ◽  
Maxwell Bloch Equations

In this paper, we focus on the (2[Formula: see text]+[Formula: see text]1)-dimensional complex modified Korteweg–de Vries and Maxwell–Bloch (CMKdV-MB) equations. According to the relevant Lax pair, the n-fold Darboux transformation (DT) is constructed. In addition, via the DT and symbolic computation, two different kinds of soliton solutions are derived from trivial seed solutions. Meanwhile, the dynamic features of soliton solutions are graphically analyzed.

GAUGE TRANSFORMATION AND SOLITON SOLUTIONS FOR THE WHITHAM–BROER–KAUP SYSTEM IN THE SHALLOW WATER

2012 ◽  
Vol 26 (25) ◽  
pp. 1250164 ◽  
Author(s):  
WEN-RUI SHAN ◽  
YAN ZHAN ◽  
BO TIAN
Keyword(s):  
Shallow Water ◽  
Gauge Transformation ◽  
Symbolic Computation ◽  
Darboux Transformation ◽  
Soliton Solutions ◽  
Lax Pair ◽  
Long Waves ◽  
Akns System

In the shallow-water studies, the Whitham–Broer–Kaup (WBK) system can be used to describe the propagation of the long waves. In this paper, based on the Lax pair of the WBK system, we derive the gauge transformation from the WBK system to the Ablowitz–Kaup–Newell–Segur (AKNS) system with the help of symbolic computation. Applying the Darboux transformation of the AKNS system, we obtain some soliton solutions of the WBK system. Those results might be useful in the investigations on the propagation of solitons in such situation as shallow water.

A REDUCTION mKdV METHOD WITH SYMBOLIC COMPUTATION TO CONSTRUCT NEW DOUBLY-PERIODIC SOLUTIONS FOR NONLINEAR WAVE EQUATIONS

2003 ◽  
Vol 14 (05) ◽  
pp. 661-672 ◽  
Author(s):  
ZHENYA YAN
Keyword(s):  
Periodic Solutions ◽  
Nonlinear Wave ◽  
Symbolic Computation ◽  
Wave Equations ◽  
Kdv Equation ◽  
Soliton Solutions ◽  
Transformation Method ◽  
Mkdv Equation ◽  
Nonlinear Wave Equations ◽  
Cubic Nonlinear Schrödinger Equation

Firstly twenty-four types of doubly-periodic solutions of the reduction mKdV equation are given. Secondly based on the reduction mKdV equation and its solutions, a systemic transformation method (called the reduction mKdV method) is developed to construct new doubly-periodic solutions of nonlinear equations. Thirdly with the aid of symbolic computation, we choose the KdV equation, the coupled variant Boussinesq equation and the cubic nonlinear Schrödinger equation to illustrate our method. As a result many types of solutions are obtained. These show that this method is simple and powerful to obtain more exact solutions including doubly-periodic solutions, soliton solutions and singly-periodic solutions to a wide class of nonlinear wave equations. Finally we further extended the method to a general form.

scholarly journals A New Discrete Integrable System Derived from a Generalized Ablowitz-Ladik Hierarchy and Its Darboux Transformation

2012 ◽  
Vol 2012 ◽  
pp. 1-19 ◽  
Author(s):  
Xianbin Wu ◽  
Weiguo Rui ◽  
Xiaochun Hong
Keyword(s):  
Exact Solutions ◽  
Integrable System ◽  
Darboux Transformation ◽  
Dynamic Characteristics ◽  
Discrete System ◽  
Soliton Solutions ◽  
Interesting Phenomenon ◽  
Discrete Integrable System ◽  
New System

We find an interesting phenomenon that the discrete system appearing in a reference can be reduced to the old integrable system given by Merola, Ragnisco, and Tu in another reference. Differing from the works appearing in the above two references, a new discrete integrable system is obtained by the generalized Ablowitz-Ladik hierarchy; the Darboux transformation of this new discrete integrable system is established further. As applications of this Darboux transformation, different kinds of exact solutions of this new system are explicitly given. Investigatingthe properties of these exact solutions, we find that these exact solutions are not pure soliton solutions, but their dynamic characteristics are very interesting.

Uniformly Constructing a Series of Nonlinear Wave and Coefficient Functions’ Soliton Solutions and Double Periodic Solutions for the (2 + 1)-Dimensional Broer-Kaup-Kupershmidt Equation

2005 ◽  
Vol 60 (3) ◽  
pp. 127-138
Author(s):  
Yong Chen ◽  
Qi Wang ◽  
Yanghuai Lang
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Periodic Solutions ◽  
Nonlinear Wave ◽  
Symbolic Computation ◽  
Nonlinear Partial Differential Equations ◽  
Soliton Solutions ◽  
Algebraic Method ◽  
Extended Tanh Method ◽  
Tanh Method

By using a new more general ansatz with the aid of symbolic computation, we extended the unified algebraic method proposed by Fan [Computer Phys. Commun. 153, 17 (2003)] and the improved extended tanh method by Yomba [Chaos, Solitons and Fractals 20, 1135 (2004)] to uniformly construct a series of soliton-like solutions and double-like periodic solutions for nonlinear partial differential equations. The efficiency of the method is demonstrated on the (2+1)-dimensional Broer-Kaup- Kupershmidt equation.

SOLITON PROPAGATION IN NONUNIFORM OPTICAL FIBERS

2003 ◽  
Vol 12 (03) ◽  
pp. 341-348 ◽  
Author(s):  
YAN XIAO ◽  
ZHIYONG XU ◽  
LU LI ◽  
ZHONGHAO LI ◽  
GUOSHENG ZHOU
Keyword(s):  
Optical Fibers ◽  
Soliton Solution ◽  
Darboux Transformation ◽  
Soliton Solutions ◽  
Lax Pair ◽  
Transmission System ◽  
Soliton Propagation ◽  
Soliton Transmission System ◽  
Soliton Transmission

In this paper, we construct the Lax pair for a soliton transmission system in nonuniform optical fibers and give N-soliton solution using the Darboux transformation. The explicit one-soliton and two-soliton solutions are presented. Further, we discuss the interaction scenario between two neighboring solitons and the effect of the inhomogeneities of the fiber (z0) on the interaction between two neighboring solitons.

N-fold Darboux transformation of a six-field integrable lattice system

2021 ◽  
pp. 2150248
Author(s):  
Yanan Qin
Keyword(s):  
Conservation Laws ◽  
Gauge Transformation ◽  
Darboux Transformation ◽  
Soliton Solutions ◽  
Lax Pair ◽  
Lattice System ◽  
Local Conservation ◽  
Integrable Lattice ◽  
Coupled Equation ◽  
Math Phys

In this paper, we studied a semidiscrete coupled equation, which is integrable in the sense of admitting Lax representations. Proposed first by Vakhnenko in 2006, local conservation laws and one-fold Darboux transformation were presented with different forms, respectively, in O. O. Vakhnenko, J. Phys. Soc. Jpn. 84, 014003 (2015); O. O. Vakhnenko, J. Math. Phys. 56, 033505 (2015); O. O. Vakhnenko, J. Math. Phys. 56, 033505 (2015). On the basis of these results, we principally construct [Formula: see text]-fold Darboux transformation by means of researching gauge transformation of its Lax pair, and work out its explicit multisolutions. Given a set of seed solutions and appropriate parameters, we can calculate two-soliton solutions and plot their figures when [Formula: see text].

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