Soliton solutions and traveling wave solutions of the two-dimensional generalized nonlinear Schrödinger equations

Abstract

In this paper, the two-dimensional generalized nonlinear Schrödinger equations are introduced with the Lax pair. The existence of the Lax pair defines integrability for the partial differential equation, so the two-dimensional generalized nonlinear Schrödinger equations are integrable. Related to this development was the understanding that certain coherent structures called solitons play a basic role in nonlinear phenomena as fluid mechanics, nonlinear optics relativity, and lattice dynamics. Via the Hirota bilinear method, bilinear forms of the twodimensional generalized nonlinear Schrödinger equations are obtained. Based on which oneand two-soliton solutions are derived. Furthermore, to find traveling wave solutions the extended tanh method is applied. Through 2D and 3D plots, the dynamical behavior of the obtained solutions is studied. The generalized form of the nonlinear Schrödinger equations has a mathematical and physical interest because a fundamental model in the field of nonlinear science. The used methods are quite useful in the solution of nonlinear partial differential equations.

Точные Решения Двумерного Кирального Нелинейного Уравнения Шредингера

“…Соңғы бірнеше онжылдықта сызықтық емес теңдеулердің нақты шешімдері қарқынды түрде зерттелді. Қолданылатын негізгі әдістер: кері шашырау әдісі, Дарбу түрлендіруі, Хирота бисызықты әдісі және Ли әдісі [3][4][5][6][7].…”

Exploring the Dynamics of Dark and Singular Solitons in Optical Fibers Using Extended Rational Sinh–Cosh and Sine–Cosine Methods

“…Through this investigation, we aim to analyze the notable characteristics associated with these soliton solutions by adjusting the system parameters. Exploring solitary wave solutions for nonlinear equations is pivotal in unraveling various nonlinear physical phenomena [28][29][30][31][32][33][34][35]. Nonlinear wave phenomena manifest across a spectrum of engineering and scientific domains, encompassing fluid dynamics, plasma physics, optics, biology, condensed matter physics, and beyond [1,9].…”