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
- 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).
Abstract
References
- 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).
Details
| Primary Language | English |
|---|---|
| Subjects | Engineering |
| Journal Section | Articles |
| Authors | |
| Publication Date | December 4, 2018 |
| Published in Issue | Year 2018Issue: 4 |
