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

Meruyert Zhassybayeva; Kuralay Yesmakhanova

Research Article

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

Year 2018, Issue: 4, 61 - 66, 04.12.2018

Meruyert Zhassybayeva Kuralay Yesmakhanova

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).

Year 2018, Issue: 4, 61 - 66, 04.12.2018

Meruyert Zhassybayeva Kuralay Yesmakhanova

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).

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Details

Primary Language	English
Subjects	Engineering
Journal Section	Articles
Authors	Meruyert Zhassybayeva Kuralay Yesmakhanova
Publication Date	December 4, 2018
Published in Issue	Year 2018 Issue: 4

Cite

APA	Zhassybayeva, M., & Yesmakhanova, K. (2018). The Construction of the (2+1)-Dimensional Integrable Fokas-Lenells Equation and its Bilinear form by Hirota Method. The Eurasia Proceedings of Science Technology Engineering and Mathematics(4), 61-66.