一类高阶非线性波方程的子方程与精确行波解

Sub-Equations and Exact Traveling Wave Solutions to a Class of High-Order Nonlinear Wave Equations

结合子方程和动力系统分析的方法研究了一类五阶非线性波方程的精确行波解.得到了这类方程所蕴含的子方程,并利用子方程在不同参数条件下的精确解,给出了研究这类高阶非线性波方程行波解的方法,并以Sawada-Kotera方程为例,给出了该方程的两组精确谷状孤波解和两组光滑周期波解.该研究方法适用于形如对应行波系统可以约化为只含有偶数阶导数、一阶导数平方和未知函数的多项式形式的高阶非线性波方程行波解的研究.

The exact traveling wave solutions to a class of 5th-order nonlinear wave equations were studied with the sub-equation method and the dynamic system analysis approach. The low- er-order sub-equations of this class of high-order nonlinear equations were first derived, then the traveling wave solutions were investigated via the various exact solutions to the sub-equa- tions under different parameter conditions. As an example, 2 families of exact valley-form soli- tary wave solutions and 2 families of smooth periodic traveling wave solutions to the Sawada- Kotera equation were presented. This method can be applied to study the traveling wave solu-tions to high-order nonlinear wave equations of which the corresponding traveling wave system can be reduced to the nonlinear ODEs involving only even-order derivatives, sum of squares of lst-order derivatives and polynomial of dependent variables.