Darboux transformation and soliton solutions of the (2+1)-dimensional Schwarz–Korteweg–de Vries equation

2020 ◽  
Vol 34 (25) ◽  
pp. 2050270
Author(s):  
Xuemei Li ◽  
Mingxiao Zhang
Keyword(s):  
Darboux Transformation ◽  
Trivial Solution ◽  
Vries Equation ◽  
Soliton Solutions ◽  
Lax Pairs ◽  
De Vries ◽  
Asymptotic Analyses

In this paper, we deduce the (2+1)-dimensional Schwarz–Korteweg–de Vries equation from two (1+1)-dimensional equations. Based on the resulting Lax pairs, we present its [Formula: see text]-fold Darboux transformation. From a trivial solution, we get [Formula: see text]-soliton and [Formula: see text]-soliton solutions of the (2+1)-dimensional Schwarz–Korteweg–de Vries equation. The asymptotic analyses of the [Formula: see text]-soliton, [Formula: see text]-soliton and [Formula: see text]-soliton solutions are presented theoretically and graphically.

Decomposition, Darboux transformation and soliton solutions for the (2 + 1)-dimensional integrable nonlocal “breaking soliton” equation

2020 ◽  
Vol 34 (24) ◽  
pp. 2050251
Author(s):  
Xiaoming Zhu ◽  
Kelei Tian
Keyword(s):  
Darboux Transformation ◽  
Schrodinger Equation ◽  
Vries Equation ◽  
Soliton Solutions ◽  
Soliton Equation ◽  
Interaction Mechanisms ◽  
Dark Solitons ◽  
De Vries ◽  
Nonlocal Nonlinear Schrödinger Equation ◽  
Nonlocal Nonlinear

In this paper, we investigate an integrable nonlocal “breaking soliton” equation, which can be decomposed into the nonlocal nonlinear Schrödinger equation and the nonlocal complex modified Korteweg–de Vries equation. As an application, with the use of this decomposition and Darboux transformation, the dark solitons, antidark solitons, rational dark solitons and rational antidark solitons of the considered equation are given explicitly. In particular, the interaction mechanisms of these solutions are discussed and illustrated through some figures.

Analytical study of exact solutions of the nonlinear Korteweg–de Vries equation with space–time fractional derivatives

2018 ◽  
Vol 32 (02) ◽  
pp. 1850012 ◽  
Author(s):  
Jiangen Liu ◽  
Yufeng Zhang
Keyword(s):  
Exact Solutions ◽  
Fractional Derivatives ◽  
Analytical Study ◽  
Vries Equation ◽  
Soliton Solutions ◽  
Traveling Wave Solutions ◽  
Expansion Method ◽  
Space Time ◽  
Strong Basis ◽  
De Vries

This paper gives an analytical study of dynamic behavior of the exact solutions of nonlinear Korteweg–de Vries equation with space–time local fractional derivatives. By using the improved [Formula: see text]-expansion method, the explicit traveling wave solutions including periodic solutions, dark soliton solutions, soliton solutions and soliton-like solutions, are obtained for the first time. They can better help us further understand the physical phenomena and provide a strong basis. Meanwhile, some solutions are presented through 3D-graphs.

Noncommutative coupled complex modified Korteweg–de Vries equation: Darboux and binary Darboux transformations

2019 ◽  
Vol 34 (07n08) ◽  
pp. 1950054
Author(s):  
H. Wajahat A. Riaz
Keyword(s):  
Evolution Equations ◽  
Nonlinear Evolution ◽  
Vries Equation ◽  
Soliton Solutions ◽  
Higher Order ◽  
Nonlinear Evolution Equations ◽  
Order Effects ◽  
Darboux Transformations ◽  
De Vries ◽  
Higher Order Effects

Higher-order nonlinear evolution equations are important for describing the wave propagation of second- and higher-order number of fields in optical fiber systems with higher-order effects. One of these equations is the coupled complex modified Korteweg–de Vries (ccmKdV) equation. In this paper, we study noncommutative (nc) generalization of ccmKdV equation. We present Darboux and binary Darboux transformations (DTs) for the nc-ccmKdV equation and then construct its Quasi-Grammian solutions. Further, single and double-hump soliton solutions of first- and second-order are given in commutative settings.

scholarly journals Investigation of Interaction Solutions for Modified Korteweg-de Vries Equation by Consistent Riccati Expansion Method

2019 ◽  
Vol 2019 ◽  
pp. 1-8
Author(s):  
Jin-Fu Liang ◽  
Xun Wang
Keyword(s):  
Vries Equation ◽  
Soliton Solutions ◽  
Wave Interaction ◽  
Expansion Method ◽  
Periodic Wave ◽  
Mkdv Equation ◽  
Jacobi Elliptic Function ◽  
Interaction Solutions ◽  
De Vries ◽  
Consistent Riccati Expansion

A consistent Riccati expansion (CRE) method is proposed for obtaining interaction solutions to the modified Korteweg-de Vries (mKdV) equation. Using the CRE method, it is shown that interaction solutions such as the soliton-tangent (or soliton-cotangent) wave cannot be constructed for the mKdV equation. More importantly, exact soliton-cnoidal periodic wave interaction solutions are presented. While soliton-cnoidal interaction solutions were found to degenerate to special resonant soliton solutions for the values of modulus (n) closer to one (upper bound of modulus) in the Jacobi elliptic function, a normal kink-shaped soliton was observed for values of n closer to zero (lower bound).

Solitons for a forced generalized variable-coefficient Korteweg–de Vries equation for the atmospheric blocking phenomenon

2016 ◽  
Vol 30 (35) ◽  
pp. 1650318 ◽  
Author(s):  
Jun Chai ◽  
Bo Tian ◽  
Xi-Yang Xie ◽  
Han-Peng Chai
Keyword(s):  
Variable Coefficient ◽  
Vries Equation ◽  
Soliton Solutions ◽  
Atmospheric Blocking ◽  
Graphic Analysis ◽  
Bilinear Bäcklund Transformation ◽  
Different Types ◽  
De Vries ◽  
Soliton Amplitude ◽  
The One

Investigation is given to a forced generalized variable-coefficient Korteweg–de Vries equation for the atmospheric blocking phenomenon. Applying the double-logarithmic and rational transformations, respectively, under certain variable-coefficient constraints, we get two different types of bilinear forms: (a) Based on the first type, the bilinear Bäcklund transformation (BT) is derived, the [Formula: see text]-soliton solutions in the Wronskian form are constructed, and the [Formula: see text]- and [Formula: see text]-soliton solutions are proved to satisfy the bilinear BT; (b) Based on the second type, via the Hirota method, the one- and two-soliton solutions are obtained. Those two types of solutions are different. Graphic analysis on the two types shows that the soliton velocity depends on [Formula: see text], [Formula: see text], [Formula: see text] and [Formula: see text], the soliton amplitude is merely related to [Formula: see text], and the background depends on [Formula: see text] and [Formula: see text], where [Formula: see text], [Formula: see text], [Formula: see text] and [Formula: see text] are the dissipative, dispersive, nonuniform and line-damping coefficients, respectively, and [Formula: see text] is the external-force term. We present some types of interactions between the two solitons, including the head-on and overtaking interactions, interactions between the velocity- and amplitude-unvarying two solitons, between the velocity-varying while amplitude-unvarying two solitons and between the velocity- and amplitude-varying two solitons, as well as the interactions occurring on the constant and varying backgrounds.

scholarly journals Soliton Solutions for a Nonisospectral Semi-Discrete Ablowitz–Kaup–Newell–Segur Equation

Mathematics ◽  
2020 ◽  
Vol 8 (11) ◽  
pp. 1889 ◽  
Author(s):  
Song-Lin Zhao
Keyword(s):  
Asymptotic Analysis ◽  
Vries Equation ◽  
Soliton Solutions ◽  
Hirota’S Method ◽  
De Vries ◽  
Hirota's Method ◽  
Multisoliton Solutions

In this paper, we study a nonisospectral semi-discrete Ablowitz–Kaup–Newell–Segur equation. Multisoliton solutions for this equation are given by Hirota’s method. Dynamics of some soliton solutions are analyzed and illustrated by asymptotic analysis. Multisoliton solutions and dynamics to a nonisospectral semi-discrete modified Korteweg-de Vries equation are also discussed.

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