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.

Darboux transformation and the exact solutions of the (2+1)-dimensional nonlocal complex modified Korteweg-de Vries and Maxwell–Bloch equations

2020 ◽  
Vol 34 (22) ◽  
pp. 2050230
Author(s):  
Na-Na Li ◽  
Hui-Qin Hao ◽  
Rui Guo
Keyword(s):  
Exact Solutions ◽  
Darboux Transformation ◽  
Soliton Solutions ◽  
Lax Pair ◽  
Darboux Transformations ◽  
Bloch Equations ◽  
Dynamic Behaviors ◽  
De Vries ◽  
Maxwell Bloch Equations

In this paper, we consider the (2[Formula: see text]+[Formula: see text]1)-dimensional nonlocal complex modified Korteweg-de Vries and Maxwell–Bloch (cmKdV-MB) equations. According to the relevant Lax pair presented, we construct one- and two-fold Darboux transformations (DT). The exact solutions are derived from the trivial seeds by DT and the dynamic behaviors of soliton solutions are analyzed by individual pictures.

scholarly journals Single and Multi-Soliton Solutions for a Spectrally Deformed Set of Maxwell-Bloch Equations

Symmetry ◽  
2019 ◽  
Vol 11 (3) ◽  
pp. 435
Author(s):  
Mehmet Baran
Keyword(s):  
Nonlinear Optics ◽  
Darboux Transformation ◽  
Soliton Solutions ◽  
Bloch Equations ◽  
Spectral Deformation ◽  
Maxwell Bloch Equations

A specific spectral deformation of the Maxwell-Bloch equations of nonlinear optics is investigated. The Darboux transformation formalism is adapted to this spectrally deformed system to construct its single and multi-soliton solutions. The Effects of spectral deformation on soliton behaviour is studied.

N-soliton solutions for the Maxwell–Bloch equations via the Riemann–Hilbert approach

2021 ◽  
pp. 2150356
Author(s):  
Yan Li ◽  
Jian Li ◽  
Ruiqi Wang
Keyword(s):  
Spectral Analysis ◽  
Exact Solutions ◽  
Soliton Solutions ◽  
Lax Pair ◽  
Bloch Equations ◽  
Jump Matrix ◽  
Maxwell Bloch Equations

We mainly study [Formula: see text]-soliton solutions for the Maxwell–Bloch equations via the Riemann–Hilbert (RH) approach in this paper. The relevant RH problem has been constructed by performing spectral analysis of Lax pair. Then the jump matrix of the Maxwell–Bloch equations has been obtained. Finally, we gain the exact solutions of the Maxwell–Bloch equations by solving the special RH problem with reflectionless case.

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.

Lax Pair and Darboux Transformation for a Variable-Coefficient Fifth-Order Korteweg-de Vries Equation with Symbolic Computation

2008 ◽  
Vol 49 (4) ◽  
pp. 833-838 ◽  
Author(s):  
Zhang Ya-Xing ◽  
Zhang Hai-Qiang ◽  
Li Juan ◽  
Xu Tao ◽  
Zhang Chun-Yi ◽  
...  
Keyword(s):  
Symbolic Computation ◽  
Darboux Transformation ◽  
Variable Coefficient ◽  
Vries Equation ◽  
Lax Pair ◽  
De Vries ◽  
Fifth Order

N-SOLITON SOLUTIONS, AUTO-BÄCKLUND TRANSFORMATIONS AND LAX PAIR FOR A NONISOSPECTRAL AND VARIABLE-COEFFICIENT KORTEWEG-DE VRIES EQUATION VIA SYMBOLIC COMPUTATION

2009 ◽  
Vol 23 (10) ◽  
pp. 2383-2393 ◽  
Author(s):  
LI-LI LI ◽  
BO TIAN ◽  
CHUN-YI ZHANG ◽  
HAI-QIANG ZHANG ◽  
JUAN LI ◽  
...  
Keyword(s):  
Symbolic Computation ◽  
Bilinear Form ◽  
Variable Coefficient ◽  
Vries Equation ◽  
Soliton Solutions ◽  
Lax Pair ◽  
Bäcklund Transformations ◽  
Nonlinear Superposition ◽  
Backlund Transformations ◽  
De Vries

In this paper, a nonisospectral and variable-coefficient Korteweg-de Vries equation is investigated based on the ideas of the variable-coefficient balancing-act method and Hirota method. Via symbolic computation, we obtain the analytic N-soliton solutions, variable-coefficient bilinear form, auto-Bäcklund transformations (in both the bilinear form and Lax pair form), Lax pair and nonlinear superposition formula for such an equation in explicit form. Moreover, some figures are plotted to analyze the effects of the variable coefficients on the stabilities and propagation characteristics of the solitonic waves.

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.

Soliton solutions, conservation laws and modulation instability for the discrete coupled modified Korteweg–de Vries equations

2018 ◽  
Vol 32 (14) ◽  
pp. 1850152 ◽  
Author(s):  
Rui Guo ◽  
Jiang-Yan Song ◽  
Hong-Tao Zhang ◽  
Feng-Hua Qi
Keyword(s):  
Conservation Laws ◽  
Darboux Transformation ◽  
Modulation Instability ◽  
Soliton Solutions ◽  
Lax Pair ◽  
Discrete Soliton ◽  
Transformation Discrète ◽  
De Vries ◽  
Discrete Darboux Transformation

In this paper, the discrete coupled modified Korteweg–de Vries equations are systematically investigated. Based on the Lax pair, N-fold discrete Darboux transformation, discrete soliton solutions, conservation laws and modulation instability are analyzed and presented.

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