Periodic and localized solutions of the long wave–short wave resonance interaction equation

Abstract

Abstract. In this paper, we investigate the (2+1) dimensional long wave-short wave resonance interaction (LSRI) equation and show that it possess the Painlevé property. We then solve the LSRI equation using Painlevé truncation approach through which we are able to construct solution in terms of three arbitrary functions. Utilizing the arbitrary functions present in the solution, we have generated a wide class of elliptic function periodic wave solutions and exponentially localized solutions such as dromions, multidromions, instantons, multi-instantons and bounded solitary wave solutions.

On exact solutions of the Bogoyavlenskii equation

“…The singular manifold method [9][10][11][12] is a powerful tool for obtaining exact solutions of nonlinear PDEs. Recently, some authors used it successfully to construct coherent structures for nonlinear PDEs [13,14]. According to the singular structure analysis for nonlinear PDEs [9][10][11][12], we truncate the Painlevé expansion of eq.…”

A New Integrable Variable-Coefficient 2+1-Dimensional Long Wave-Short Wave Equation and the Generalized Dressing Method

“…The soliton solution of long wave and short wave has been obtained in [2]. Radha et al in [3] derived periodic solutions and localized solutions of (1). Lai and Chow in [4] studied positon and dromion solutions of 2 + 1-dimensional long wave and short wave resonance interaction equations.…”

Combined wave solutions of the -dimensional generalized Nizhnik–Novikov–Veselov system

“…However, there are few works in the literatures to study the interactions among (elliptic) periodic waves and/or between the periodic waves and solitary waves because of the difficulties to find exact multiple (elliptic) periodic waves and/or periodicsolitary wave solutions though one knows a single solitary wave solution can be considered as a limit case of a single periodic wave solution. Recently, Radha [18] and Peng [19] discussed the (elliptic) periodic wave and interactions between the periodic waves by Painlevé truncation approach and MLVSA, respectively. Moreover, Lou [20] studied interactions among (elliptic) periodic wave and interactions between the periodic waves and solitary waves for nonintegrable (n+1)-dimensional sine-Gordon equation.…”