Geometric Singular Perturbation Analysis to a Perturbed $(1 + 1)$-Dimensional Dispersive Long Wave Equation

Abstract

The existence of the solitary wave and the nonexistence of kink (anti-kink) wave solutions are studied for a perturbed $(1 + 1)$-dimensional dispersive long wave equation. The methods are based on the geometric singular perturbation {(GSP, for short)} approach, Melnikov method and bifurcation analysis. The results show that the solitary wave solution with a suitable wave speed $c$ and parameter ${\kappa}$ exists under the small singular perturbation. Interestingly, unlike solitary wave solutions, the kink (anti-kink) wave solution doesn't persist because the corresponding Melnikov function has no zeros. Further, numerical simulations are utilized to verify the correctness of our analytical results.