A Riccati-Bernoulli sub-ODE method for nonlinear partial differential equations and its application

Advances in Difference Equations

Xiao-Feng Yang¹  Yi Wei²  Zi-Chen Deng⁴ 

[1] Department of Applied Mathematics, Northwestern Polytechnical University, Xi’School of Mechanics, Civil Engineering and Architecture, Northwestern Polytechnical University, Xi’State Key Laboratory of Structural Analysis of Industrial Equipment, Dalian University of Technology, Dalian, P.R. China;an, P.R. China

关键词: Riccati-Bernoulli sub-ODE method;    Bäcklund transformation;    traveling wave solution;    solitary wave solution;    peaked wave solution;    35Q55;    35Q80;    35G25;   

DOI  :  10.1186/s13662-015-0452-4

学科分类：数学（综合）

来源: SpringerOpen

【 摘 要 】

The Riccati-Bernoulli sub-ODE method is firstly proposed to construct exact traveling wave solutions, solitary wave solutions, and peaked wave solutions for nonlinear partial differential equations. A Bäcklund transformation of the Riccati-Bernoulli equation is given. By using a traveling wave transformation and the Riccati-Bernoulli equation, nonlinear partial differential equations can be converted into a set of algebraic equations. Exact solutions of nonlinear partial differential equations can be obtained by solving a set of algebraic equations. By applying the Riccati-Bernoulli sub-ODE method to the Eckhaus equation, the nonlinear fractional Klein-Gordon equation, the generalized Ostrovsky equation, and the generalized Zakharov-Kuznetsov-Burgers equation, traveling solutions, solitary wave solutions, and peaked wave solutions are obtained directly. Applying a Bäcklund transformation of the Riccati-Bernoulli equation, an infinite sequence of solutions of the above equations is obtained. The proposed method provides a powerful and simple mathematical tool for solving some nonlinear partial differential equations in mathematical physics.

【 授权许可】

CC BY