Localized Symmetric and Asymmetric Solitary Wave Solutions of Fractional Coupled Nonlinear Schrödinger Equations

Abstract

The existence of solutions with localized solitary wave structures is one of the significant characteristics of nonlinear integrable systems. Darboux transformation (DT) is a well-known method for constructing multi-soliton solutions, using a type of localized solitary wave, of integrable systems, but there are still no reports on extending DT techniques to construct such solitary wave solutions of fractional integrable models. This article takes the coupled nonlinear Schrödinger (CNLS) equations with conformable fractional derivatives as an example to illustrate the feasibility of extending the DT and generalized DT (GDT) methods to construct symmetric and asymmetric solitary wave solutions for fractional integrable systems. Specifically, the traditional n-fold DT and the first-, second-, and third-step GDTs are extended for the fractional CNLS equations. Based on the extended GDTs, explicit solutions with symmetric/asymmetric soliton and soliton–rogon (solitrogon) spatial structures of the fractional CNLS equations are obtained. This study found that the symmetric solitary wave solutions of the integer-order CNLS equations exhibit asymmetry in the fractional order case.

Fractional Soliton and Semirational Solutions of a Fractional Two-Component Generalized Hirota Equation

“…It is shown that compared to integer-order solitons and solitrogons, the fractional ones obtained in this paper have significant characteristics. One is that fractional solitons and solitrogons tilt to a certain extent, gradually decelerate and widen until their velocities and wave widths stabilize and the tilt gradually disappears, and the other is the similar asymmetry [34] of fractional solitons and solitrogons. The deceleration propagation of fractional solitons and solitrogons is consistent with the anomalous diffusion phenomenon that occurs in the background of fractional dimensions.…”

“…These aspects are different from those of the traditional DT [29] and GDT [30]. Although we have some preliminary studies [33,34] on fractional DT and GDT, the starting point of this work is to reveal the novel dynamic characteristics dominated by fractional order of fractional soliton and semirational solutions from a new fractional integrable system.…”

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

“…where p 0 , p 1 , and p 2 are parameters. Plugging Equation (31) into Equations ( 9) and ( 10), we collect the coefficients of cos(µη), as follows:…”

“…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].…”

Modelling Symmetric Ion-Acoustic Wave Structures for the BBMPB Equation in Fluid Ions Using Hirota’s Bilinear Technique

“…Li and Guo explored optical solitons in the form of breathers, rogue waves, and semirational solutions on periodic backgrounds for the coupled Lakshmanan-Porsezian-Daniel equations [13], and Song et al studied Laguerre-Gaussian and Hermite-Gaussian solitons in the nonlocal nonlinear Schrödinger equation [14]. Zhang and Xu worked on the localized symmetric and asymmetric solitary wave solutions using the Darboux transformation [15]. There are many mathematical techniques to explore the soliton solution such as the direct algebraic method [16], the sine-Gordon expansion method [17], the new MEDA method [18,19], the Riccati equation mapping (REM) method [20], the Sardar subequation method [21], the Jacobi elliptic functions method [22].…”