Construction Of Soliton Solutions Research Articles

Dynamic behavior of optical solitons to the Coupled-Higgs equation through an efficient method

In this study, through the [Formula: see text]-expansion method, we extract soliton solutions to the coupled-Higgs equation. The studied nonlinear model is known to describe Higgs mechanism. The Higgs mechanism is essential to explain the generation mechanism of the property “mass” for gauge bosons. The proposed method is one of the most powerful methods for constructing soliton solutions for nonlinear partial differential equations. The obtained wave solutions include exponential, hyperbolic, and distinct structures of complex function solutions. The presented results may be helpful in explaining the physical features of various nonlinear physical phenomena. In order to analyze the dynamic behavior of all obtained solutions, we plot three-dimensional and two-dimensional graphs for the obtained solutions.

New exact solutions for the reaction-diffusion equation in mathematical physics

We utilize three mathematical techniques, specifically exp(−φ(ξ))-expansion (EXP), sine-cosine (SCT) and Riccati–Bernoulli sub-ODE techniques (RRT) to obtain soliton solutions of the reaction-diffusion equation. These methods are applied to construct new exact-periodic and soliton solutions for the reaction-diffusion equation. Furthermore, these mentioned methods give a robust, efficient mathematical tool for solving various models of nonlinear partial differential equations in nonlinear science. Indeed the reported results are very vital in explaining the physical phenomena of the proposed model.

Soliton solutions of the nonlinear Schrödinger equation with defect conditions

Motivated by a recent development in the derivation of soliton solutions for initial-boundary value problems via the dressing method, we reconsider solutions to the nonlinear Schrödinger (NLS) equation on two half-lines connected via integrable defect conditions. The dressing method to construct soliton solutions is applied, while preserving the spectral boundary constraint with a time-dependent defect matrix. Moreover, N-soliton solutions are explicitly constructed and it is proven that the solitons are transmitted through the defect independently of one another.

Short comment on the construction of soliton solutions

This short comment continues the discussions raised by Zayed and Shohib (2020) and Zayed and El-Ganaini (2020). The authors showed, by very simple calculation, that the exp(−ϕ(ξ))- and the tan(ϕ∕2)-expansion techniques recently proposed in some articles are equivalent to the (G′∕G)-expansion and the generalizing Riccati equation mapping methods, respectively. In this regard, we review and justify that many techniques used for constructing exact soliton solutions to nonlinear partial differential equations are particular cases of an older Riccati-expansion approach.

Solitons and other solutions to (n+1)-dimensional modified Zakharov–Kuznetsov equation by exp-function method

In this paper, the exp-function method is applied to construct exact solutions and soliton solutions of (n+1)-dimensional modified Zakharov–Kuznetsov (mZK) equation. Solitary and topological soliton solutions are obtained. These solutions are significant importance for studying waves in plasma physics.

General soliton and (semi‐)rational solutions to the nonlocal Mel'nikov equation on the periodic background

AbstractIn this paper, the Hirota's bilinear method and Kadomtsev‐Petviashvili hierarchy reduction method are applied to construct soliton, line breather and (semi‐)rational solutions to the nonlocal Mel'nikov equation with nonzero boundary conditions. These solutions are expressed as Gram‐type determinants. When N is even, soliton, line breather and (semi‐)rational solutions on the constant background are derived while these solutions are located on the periodic background for odd N. Regularity of these solutions and their connections with the local Mel'nikov equation are analyzed for proper choices of parameters that appear in the solutions. The dynamics of the solutions are discussed in detail. All possible configurations of soliton and lump solutions are found for . Several interesting dynamical behaviors of semi‐rational solutions are observed. It is shown that certain lumps may exhibit fusion and fission phenomena during their interactions with solitons while some lump may change its direction of movement after it collides with solitons.

Analytic solutions for a new model of the(3+1)-Boussinesq equation

In this paper, we have applied the mapping method to solve the (3+1)-dimensional Boussinenesq equation where we have obtained exact solutions for evolution equation to construct exact periodic and soliton solutions of nonlinear partial differential evolution equation. Many have obtained new families of exact traveling wave solutions, but the Boussinesq equation is successfully. These solutions may be significantly important for the explanation of some practical physical problems. New exact travelling wave solutions are obtained and expressed in terms of hyperbolic functions, trigonometric functions, rational functions and elliptic functions. It is shown that the mapping method provides a powerful mathematical tool for solving a great many nonlinear partial differential equations in mathematical physics.

Exact Solutions of Rosenzweig-Macarthur (RM) Model Equations by Using Exp Function Method

He’s exp function method promising for finding exact solutions of nonlinear evolution equations in mathematical physics. The exact solutions of the Rosenzweig-MacArthur (RM) model equations is obtain by using the exp-function method. The method is by transformation used to construct solitary and soliton solutions of nonlinear evolution equations. Thefree parameters in the obtained generalized solutions might imply some meaningful results in physical process.

Soliton solutions in geometrically nonlinear Cosserat micropolar elasticity with large deformations

We study the fully nonlinear dynamical Cosserat micropolar elasticity problem in three dimensions with various energy functionals dependent on the microrotation R¯ and the deformation gradient tensor F. We derive a set of coupled nonlinear equations of motion from first principles by varying the complete energy functional. We obtain a double sine–Gordon equation and construct soliton solutions. We show how the solutions can determine the overall deformational behaviours and discuss the relations between wave numbers and wave velocities thereby identifying parameter values where the soliton solution does not exist.

Soliton solutions of the sine-Gordon equation on the half line

We provide a direct method for constructing soliton solutions of the sine-Gordon equation in the laboratory coordinates on the half line in the presence of integrable boundary conditions derived in Sklyanin’s classical work (Sklyanin, 1987). Explicit examples, including boundary bound states, are presented.

Multisoliton solutions of the two-component Camassa–Holm system and their reductions

We develop a systematic procedure for constructing soliton solutions of an integrable two-component Camassa–Holm (CH2) system. The parametric representation of the multisoliton solutions is obtained by using a direct method combined with a reciprocal transformation. The properties of the solutions are then investigated in detail focusing mainly on the smooth one- and two-soliton solutions. The general N-soliton case is described shortly. Subsequently, we show that the CH2 system reduces to the CH equation and the two-component Hunter–Saxton (HS2) system by means of appropriate limiting procedures. The corresponding expressions of the multisoliton solutions are presented in parametric forms, reproducing the existing results for the reduced equations. Last, we discuss the reduction from the HS2 system to the HS equation.

Application of the Exp-Function Method for Solving a Partial Differential Equation Arising in Problems of Hydrodynamics

The aim of present paper is to obtain the analytical solution of Harry Dym equation by using Exp-function method (EFM). He’s Exp-function method is used to construct solitary and soliton solutions of nonlinear evolution equation. The method is straightforward and concise, and its applications are promising. This method is developed for searching exact travelling wave solutions of nonlinear partial differential equations. The EFM presents a wider applicability for handling nonlinear wave equations. Also, it is shown that the EFM, with the help of symbolic computation, provides a straightforward and powerful mathematical tool for solving nonlinear evolution equations.

Soliton solutions and dynamical evolutions of a generalized AKNS system in the framework of inverse scattering transform

It is important and interesting to derive integrable systems or construct soliton solutions of nonlinear partial differential equations (PDEs). In this paper, a new and more general Ablowitz–Kaup–Newell–Segur (AKNS) system with Lax integrability is derived and solved in the framework of inverse scattering transform (IST). To be specific, the known AKNS’s linear spectral problem and its time evolution equation are first generalized by embedding a nonisospectral parameter whose varying with time obeys the rational function of spectral parameter. Based on the generalized AKNS spectral problem and time evolution equation, we then derive a generalized AKNS system with infinite number of terms. Finally, in the case of reflectionless potentials, explicit n-soliton solutions of the generalized AKNS system are formulated through the IST method. It is shown that the dynamical evolutions of the obtained one-soliton solutions and two-soliton solutions possess time-varying speeds and amplitudes in the process of propagations.

The Classification of the Single Traveling Wave Solutions to the Modified Fornberg–Whitham Equation

The aim of present paper is to obtain the analytical solution of modified Fornberg–Whitham equation by using \(\tan (\phi /2)\)-expansion and \(\tanh (\phi /2)\)-expansion methods. These methods are used to construct solitary and soliton solutions of nonlinear evolution equation. These methods are straightforward and concise, and their applications are promising. These methods are developed for searching exact traveling wave solutions of nonlinear partial differential equations. The exact particular solutions containing five types hyperbolic function solution, trigonometric function solution, exponential solution, logarithmic solution and rational solution. The expansion methods presents a wider applicability for handling nonlinear wave equations. Also, their are shown that the expansion methods, with the help of symbolic computation, provide a straightforward and powerful mathematical tool for solving nonlinear evolution equations.

Improvement of the Exp-function method for solving the BBM equation with time-dependent coefficients

In this article, we establish the exact solutions for the BBM equation with time-dependent coefficients. The Exp-function method (EFM) and improvement of the Exp-function method (IEFM) are used to construct solitary and soliton solutions of nonlinear evolution equations. These methods are developed for searching exact travelling wave solutions of nonlinear partial differential equations. The exact particular solutions are of four types: the hyperbolic function solution, trigonometric function solution, exponential solution and rational solution. It is shown that the EFM and IEFM, with the help of symbolic computation, provide a straightforward and powerful mathematical tool for solving nonlinear evolution equations in mathematical physics.

Constructing soliton solutions and super-bilinear form of lattice supersymmetric KdV equation

The Hirota bilinear form and multisoliton solution for semidiscrete and fully discrete (difference–difference) versions of the supersymmetric Korteweg–de Vries (KdV) equation found by Xue et al (2013 J. Phys. A: Math. Theor 46 502001) are presented. The solitonic interaction term displays a fermionic dressing factor as in the continuous supersymmetric case. Using bilinear equations it is also shown that a new integrable semidiscrete (and fully discrete) version of supersymmetric KdV can be constructed with a simpler bilinear form but a more complicated interaction dressing. Its continuum limit is also computed.

Noncommutative Multisoliton Solutions of a Supersymmetric Chiral Model

Multisoliton configurations of a superextended and Moyal-type noncommutative deformed modified2+1chiral model have been constructed with the dressing method by Lechtenfeld and Popov several years ago. These configurations have no-scattering property. A two-soliton configuration with nontrivial scattering was constructed soon after that. More multisoliton solutions with general pole data of the superextended and noncommutative Ward model will be constructed in this paper. The method is the supersymmetric and noncommutative extension of Dai and Terng’s in constructing soliton solutions of the Ward model.

Exp-Function Method for a Generalized MKdV Equation

Under investigation in this paper is a generalized MKdV equation, which describes the propagation of shallow water in fluid mechanics. In this paper, we have derived the exact solutions for the generalized MKdV equation including the bright soliton, dark soliton, two-peak bright soliton, two-peak dark soliton, shock soliton and periodic wave solution via Exp-function method. By figures and symbolic computations, we have discussed the propagation characteristics of those solitons under different values of those coefficients in the generalized MKdV equation. The method constructing soliton solutions in this paper may be useful for the investigations on the other nonlinear mathematical physics model and the conclusions of this paper can give theory support for the study of dynamic features of models in the shallow water.

Hirota's method and the search for integrable partial difference equations. 1. Equations on a 3 × 3 stencil

Hirota's bilinear method (‘direct method’) has been very effective for constructing soliton solutions to many integrable equations. The construction of one-soliton solution (1SS) and two-soliton solution (2SS) is possible even for non-integrable bilinear equations, but the existence of a generic three-soliton solution (3SS) imposes severe constraints and is in fact equivalent to integrability. This property has been used before in searching for integrable partial differential equations, and in this paper we apply it to two-dimensional (2D) partial difference equations defined on a 3 × 3 stencil. We also discuss how the obtained equations are related to projections and limits of the 3D master equations of Hirota and Miwa, and find that sometimes a singular limit is needed.

Application of the Exp-function method for solving the combined KdV-mKdV and Gardner-KP equations

Purpose: In this article, we establish the exact solutions for the combined KdV-mKdV and Gardner-KP equations. Methods: The Exp-function method (EFM) is used to construct solitary and soliton solutions of nonlinear evolution equations. Results: This method is developed for searching the exact travelling wave solutions of nonlinear partial differential equations. Conclusions: It is shown that the EFM, with the help of symbolic computation, provides a straightforward and powerful mathematical tool for solving nonlinear evolution equations in mathematical physics.