Darboux transformation and exact solutions for a (2 + 1)-dimensional integrable system of nonlinear evolution equations

2020 ◽  
Vol 34 (34) ◽  
pp. 2050392
Author(s):  
Zhen Chuan Zhou ◽  
Xiao Ming Zhu
Keyword(s):  
Exact Solutions ◽  
Spectral Problem ◽  
Integrable System ◽  
Darboux Transformation ◽  
Evolution Equations ◽  
Nonlinear Evolution ◽  
Lax Pair ◽  
Nonlinear Evolution Equations ◽  
Recursion Operator

In this paper, starting from a spectral problem, we construct a [Formula: see text]-dimensional integrable system of nonlinear evolution equations. Based on the Lax pair, the recursion operator and Darboux transformation for the whole hierarchy were constructed. As an application, some exact solutions for the hierarchy are obtained by using the Darboux transformation.

Darboux transformation approach for two new coupled nonlinear evolution equations

2019 ◽  
Vol 34 (01) ◽  
pp. 2050004
Author(s):  
Dan Zhao ◽  
Zhaqilao
Keyword(s):  
Darboux Transformation ◽  
Evolution Equations ◽  
Burgers Equation ◽  
Nonlinear Evolution ◽  
Kdv Equation ◽  
Lax Pair ◽  
Nonlinear Evolution Equations ◽  
Coupled Equations ◽  
Covariant Property ◽  
Coupled Burgers Equation

A new coupled Burgers equation and a new coupled KdV equation which are associated with [Formula: see text] matrix spectial problem are investigated for complete integrability and covariant property. For integrability, Lax pair and conservation laws of the two new coupled equations with four potentials are established. For covariant property, Darboux transformation (DT) is used to construct explicit solutions of the two new coupled equations.

scholarly journals Equations with Peakon Solutions in the Negative Order Camassa-Holm Hierarchy

2018 ◽  
Vol 2018 ◽  
pp. 1-11
Author(s):  
Fengfeng Dong ◽  
Lingjun Zhou
Keyword(s):  
Spectral Problem ◽  
Inverse Scattering ◽  
Continued Fractions ◽  
Evolution Equations ◽  
Nonlinear Evolution ◽  
Lax Pair ◽  
Nonlinear Evolution Equations ◽  
Negative Order

The negative order Camassa-Holm (CH) hierarchy consists of nonlinear evolution equations associated with the CH spectral problem. In this paper, we show that all the negative order CH equations admit peakon solutions; the Lax pair of the N-order CH equation given by the hierarchy is compatible with its peakon solutions. Special peakon-antipeakon solutions for equations of orders -3 and -4 are obtained. Indeed, for N≤-2, the peakons of N-order CH equation can be constructed explicitly by the inverse scattering approach using Stieltjes continued fractions. The properties of peakons for N-order CH equation when N is odd are much different from the CH peakons; we present the case N=-3 as an example.

scholarly journals New Jacobi Elliptic Function Solutions for the Zakharov Equations

2012 ◽  
Vol 2012 ◽  
pp. 1-16 ◽  
Author(s):  
Yun-Mei Zhao
Keyword(s):  
Exact Solutions ◽  
Elliptic Function ◽  
Evolution Equations ◽  
Nonlinear Evolution ◽  
Expansion Method ◽  
Mathematical Physics ◽  
Nonlinear Evolution Equations ◽  
Jacobi Elliptic Function

A generalized(G′/G)-expansion method is proposed to seek the exact solutions of nonlinear evolution equations. Being concise and straightforward, this method is applied to the Zakharov equations. As a result, some new Jacobi elliptic function solutions of the Zakharov equations are obtained. This method can also be applied to other nonlinear evolution equations in mathematical physics.

scholarly journals Some Methods about Finding the Exact Solutions of Nonlinear Modified BBM Equation

2021 ◽  
Vol 2021 ◽  
pp. 1-12
Author(s):  
Fan Niu ◽  
Jianming Qi ◽  
Zhiyong Zhou
Keyword(s):  
Exact Solutions ◽  
Rational Function ◽  
Elliptic Function ◽  
Evolution Equations ◽  
Nonlinear Evolution ◽  
Mathematical Physics ◽  
Nonlinear Evolution Equations ◽  
Complex Method ◽  
First Time ◽  
Bbm Equation

Finding exact solutions of nonlinear equations plays an important role in nonlinear science, especially in engineering and mathematical physics. In this paper, we employed the complex method to get eight exact solutions of the modified BBM equation for the first time, including two elliptic function solutions, two simply periodic solutions, and four rational function solutions. We used the exp − ϕ z -expansion methods to get fourteen forms of solutions of the modified BBM equation. We also used the sine-cosine method to obtain eight styles’ exact solutions of the modified BBM equation. Only the complex method can obtain elliptic function solutions. We believe that the complex method presented in this paper can be more effectively applied to seek solutions of other nonlinear evolution equations.

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