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

Abstract

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

Keywords

• Integrability,Lax representation,Fokas-Lenells equation,Hirota bilinear method,Soliton solutions

References

1. Абловиц М., Сигур Х. (1987). Солитоны и метод обратной задачи. М.: «Мир». Захаров В.Е., Манаков С.В., Новиков С.П., Питаевский Л.П. (1980). Введение в теорию интегрируемых систем. М.: «Наука». Jingsong H., Shuwei X., Kuppuswamy P. (2012). Rogue Waves of the Fokas-Lenells Equation. Journal of the Physical Society of Japan, 81(12). Lenells J., Fokas A.S. (2009). On a novel integrable generalization of the nonlinear Schrödinger equation. Nonlinearity, 22(1), 11-27. Myrzakulov R., Mamyrbekova G., Nugmanova G., Lakshmanan M. (2015). Integrable (2+1)-dimensional spin models with self-consistent potentials. Symmetry, 7(3), 1352-1375. Myrzakul Akbota, Myrzakulov R. (2017). Integrable motion of two interacting curves, spin systems and the Manakov system. International Journal of Geometric Methods in Modern Physics, 14(7). Борзых А.В. (2002). Метод Хироты и солитонные решения многомерного нелинейного уравнения Шредингера. Сибирский математический журнал, 43(2), 267-270. Matsuno Y. (2012). A direct method of solution for the Fokas-Lenells derivative nonlinear Schrödinger equation: II. Dark soliton solutions. Journal of Physics A: Mathematical and Theoretical, 45(47).