Soliton solutions of fractional modified unstable Schrödinger equation using Exp-function method

Math Methods in App Sciences

2021

In this article, we derive a new local fractional Gardner equation based on the local fractional derivative. A novel traveling wave transform of the non‐differentiable type is utilized to convert the local fractional Gardner equation into a nonlinear local fractional ordinary differential equation. Then a new method called the Mittag–Leffler function based method is proposed for the first time ever to construct the abundant traveling wave solutions. By this method, three families (six sets) of the traveling wave solutions on Cantor set are obtained. Finally, the performances of the various solutions on Cantor set are presented through the three‐dimensional contours. It shows that the proposed method is straightforward, powerful and can provide more forms of the traveling wave solutions for the local fractional equations, which is expected to be helpful for the study of the traveling wave theory for local fractional equations.

Section: Preliminariesmentioning

confidence: 99%

On new abundant exact traveling wave solutions to the local fractional Gardner equation defined on Cantor sets

Math Methods in App Sciences

2021

“…The study of the soliton solutions can provide a theoretical reference for the research of nonlinear physical models, soliton control, optical switching equipment, and so on. Up to now, many effective and powerful methods have been obtained for constructing the soliton solutions of the NLPDEs: the extended trial equation method [1,2], generalized exponential rational function method [3], sine-Gordon expansion method [4][5][6], generalized (G′/G) expansion method [7], extended rational sinecosine and sinh-cosh method [8,9], F-expansion method [10][11][12], Cole-Hopf transformation [13,14], exp-function method [15][16][17], extended tanh-function method [18][19][20], Bäcklund transformation [21], Sardar subequation method [22,23], and so on [24][25][26][27][28][29][30][31]. In this paper, we aim to study the (2 + 1)-dimensional Heisenberg ferromagnetic spin chain equation (HFSCE) that is given by [32]:…”

Section: Introductionmentioning

confidence: 99%

Abundant Soliton Structures to the ( $2 + 1$ )-Dimensional Heisenberg Ferromagnetic Spin Chain Dynamical Model

Shi

Advances in Mathematical Physics

2023

In this paper, we aim to investigate the ( 2 + 1 )-dimensional Heisenberg ferromagnetic spin chain equation that is used to describe the nonlinear dynamics of magnets. Two recent effective technologies, namely, the variational method and subequation method, are employed to construct the abundant soliton solutions. By these two methods, diverse solutions such as the bright soliton, dark soliton, bright-dark soliton, perfect periodic soliton, and singular periodic soliton are successfully extracted. The numerical results are illustrated in the form of 3-D plots and 2-D curves by choosing proper parametric values to interpret the dynamics of wave profiles. Finally, the physical interpretation of the acquired results is elaborated in detail. The results obtained in this study are helpful to explain some physical meanings of some nonlinear physical models in electromagnetic waves.

“…Therefore, the study of the exact solution of the nonlinear evolution equation has become an important task [8][9][10]. At present, there are many ideal methods to find the exact solution of the partial differential equations (PDEs), such as the trial equation method [11,12], sine-Gordon expansion method [13,14], generalized (G /G)-expansion method [15,16], simplified extended tanh-function method [17,18], extended rational sine-cosine and extended rational sinh-cosh techniques [19,20], exp-function method [21][22][23][24], Sardar subequation method [25][26][27][28] and so on [29][30][31][32]. In this work, we focus on the modified Benjamin-Bona-Mahony equation (MBBME) that reads as follows [33]:…”

Section: Introductionmentioning

confidence: 99%

Variational Principle and Diverse Wave Structures of the Modified Benjamin-Bona-Mahony Equation Arising in the Optical Illusions Field

2022

Axioms

This study focuses on investigating the modified Benjamin-Bona-Mahony equation that is used to model the long wave in nonlinear dispersive media of the optical illusion field. Two effective techniques, the variational direct method and He’s frequency formulation method, are employed to seek the travelling wave solutions. Using these two techniques, abundant exact solutions such as the bright wave, bright-dark wave, bright-like wave, kinky-bright wave and periodic wave solutions, are obtained. The 3-D contours and 2-D curves are drawn to present the dynamic physical behaviors of the solutions by assigning the proper parameters. It shows that the proposed methods are effective but simple and only need one or two steps to construct the exact solutions, which are expected to provide some new insights to study the travelling wave solutions of the PDEs arising in physics.