Quasi-periodic wave solutions and two-wave solutions of the KdV-Sawada-Kotera-Ramani equation

Abstract

The two-wave solutions of the KdV-Sawada-Kotera-Ramani equation are studied in this paper. By reducing this high-order wave equation into two associated solvable ordinary differential equations, we derive the two-wave solutions in the form u(x,t)=U(x−c1t)+V(x−c2t) which includes solitary wave solutions, periodic solutions and quasi-periodic wave solutions by letting c1=c2. We obtain a family of new exact two-wave solutions combined by a solitary wave and a periodic wave with two different wave speeds. These new exact two-wave solutions are neither periodic nor quasi-periodic wave solutions but approximating periodic wave solutions as time tends to infinity. The process of translation of the two-wave solution combined by two solitary wave solutions is illustrated by simulation. The approach presented in this work might be applied to study the bifurcation of multi-wave solutions of some important high-order nonlinear wave model equations.