EXACT SOLUTIONS FOR SEVERAL HIGHER-ORDER NONLINEAR WAVE EQUATIONS

2015 ◽  
Vol 91 (2) ◽  
pp. 143-156
Author(s):  
Yuan-Xi Xie
Keyword(s):  
Exact Solutions ◽  
Nonlinear Wave ◽  
Wave Equations ◽  
Higher Order ◽  
Nonlinear Wave Equations

scholarly journals Complex PT -symmetric nonlinear Schrödinger equation and Burgers equation

2013 ◽  
Vol 371 (1989) ◽  
pp. 20120059 ◽  
Author(s):  
Zhenya Yan
Keyword(s):  
Schrödinger Equation ◽  
Exact Solutions ◽  
Nonlinear Wave ◽  
Schrodinger Equation ◽  
Wave Equations ◽  
Burgers Equation ◽  
Kdv Equation ◽  
Nonlinear Schrodinger Equation ◽  
Nonlinear Wave Equations ◽  
Nonlinear Schrödinger

The complex -symmetric nonlinear wave models have drawn much attention in recent years since the complex -symmetric extensions of the Korteweg–de Vries (KdV) equation were presented in 2007. In this review, we focus on the study of the complex -symmetric nonlinear Schrödinger equation and Burgers equation. First of all, we briefly introduce the basic property of complex symmetry. We then report on exact solutions of one- and two-dimensional nonlinear Schrödinger equations (known as the Gross–Pitaevskii equation in Bose–Einstein condensates) with several complex -symmetric potentials. Finally, some complex -symmetric extension principles are used to generate some complex -symmetric nonlinear wave equations starting from both -symmetric (e.g. the KdV equation) and non- -symmetric (e.g. the Burgers equation) nonlinear wave equations. In particular, we discuss exact solutions of some representative ones of the complex -symmetric Burgers equation in detail.

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