Solitary and Periodic Waves in Coupled KdV Equations with Different Linear Dispersion Relations

Abstract

Based on the invariant expansion method, some reasonable approximate solutions of coupled Korteweg-de Vries (KdV) equations with different linear dispersion relations have been obtained. These solutions contain not only bell type soliton solutions but also periodic wave solutions that expressed by Jacobi elliptic functions. The results also show that if the arbitrary constants are selected suitably, the approximate solutions may become the exact ones.

Nonlocal Symmetry and Interaction Solutions of a Generalized Kadomtsev—Petviashvili Equation

“…For the generalized Kadomtsev-Petviashvili equation 1, we aim to look for its truncated Painlevé expansion solution in the following possible form (x, y, t)) , (27) where R(w) is a solution of the Riccati equation (29) and the function w must satisfy…”

“…Obviously, the Riccati equation (28) has a special solution R(w) = tanh(w), while the truncated Painlevé expansion solution (27) becomes…”