Abstract:

By transformation of a dependent variable, a nonlinear evolution equation (NLEE) is converted into a nonlinear partial differential equation (NPDE) with a polynomial type of a new dependent variable and its partial derivatives. A Jacobi elliptic function expansion method is proposed to construct the exact periodic solutions of several nonlinear equations—sine-Gordon equation and Dodd-Bullough-Mikhailov equation. Periodic solutions obtained with this method include the solitary solutions and the shock wave solutions. The method can also be applied to other nonlinear evolution equations.

Key words: nonlinear evolution equation; Jacobi elliptic function; exact periodic solution; solitary solution