Periodic Wave Solutions and Their Asymptotic Property for a Modified Fornberg–Whitham Equation

Abstract

Recently, periodic traveling waves, which include periodically symmetric traveling waves of nonlinear equations, have received great attention. This article uses some bifurcations of the traveling wave system to investigate the explicit periodic wave solutions with parameter α and their asymptotic property for the modified Fornberg–Whitham equation. Furthermore, when α tends to given parametric values, the elliptic periodic wave solutions become the other three types of nonlinear wave solutions, which include the trigonometric periodic blow-up solution, the hyperbolic smooth solitary wave solution, and the hyperbolic blow-up solution.