The Construction of the (2+1)-Dimensional Integrable Fokas-Lenells Equation and its Bilinear form by Hirota Method

Abstract

Integrable nonlineardifferential equations are an important class of nonlinear wave equations thatadmit exact soliton of the solutions. In order to construct such equations tend to apply themethod of mathematical physics, the inverse scattering problem method (ISPM),which was discovered in 1967 by Gardner, Green, Kruskal, and Miura. This methodallows to solve more complicated problems.  One of these equations is the (1+1)-dimensional integrable Fokas-Lenellsequation, which was obtained by the bi-Hamiltonian method and appears as anintegrable generalization of the nonlinear Schrödinger equation. In this paper,we have examined the (1+1)-dimensional Fokas-Lenells equation and inorder to find more interesting solutions we have constructed the(2+1)-dimensional integrable Fokas-Lenells equation, whose integrability areensured by the existence of the Lax representation for it. In addition, usingthe Hirota’s method a bilinear form of the equation is constructed which wasfound by us, through which can be find its exact multisoliton solutions andbuild their graphs.