Topological Soliton Solutions Research Articles

Topological soliton solutions for three shallow water waves models

In this article, we investigate three distinct physical structures for shallow water waves models by the improved ansatz method. The method was improved and can be used to obtain more generalized form topological soliton solutions than the original method. As a result, some new exact solutions of the shallow water equations are successfully established and the obtained results are exhibited graphically. The results showed that the improved ansatz method can be applied to solve other nonlinear differential equations arising from mathematical physics.

Topological and non-topological soliton solution of the 1 + 3 dimensional Gross-Pitaevskii equation with quadratic potential term

This paper carries out the integration of the 1 + 3 dimensional Gross-Pitaevskii equation (GPE) in presence of quadratic potential term to ob tain the 1-soliton solutions. The solitary wave ansatz method is employed to integrate the considered equation. Parametric conditions for the existence of the soliton solutions are determined. Both non-topological (bright) and topological (dark) solitonsolutions are reported and we observed that the existence condition for bright and dark soliton solutions are opposite to each other. Finally, the two integrals of motion of the governing model equation have been extracted

Non topological 1-soliton solution to resonant nonlinear schrodinger equation with kerr law nonlinearity

In this paper, using the methods of ansatz, sine-cosine and He’s semi-inverse variation, non-topological 1-soliton solution to Resonant Nonlinear Schrodinger Equation with Kerr law nonlinearity is obtained. The results show that these methods are very effective ones for finding exact solutions to various types of nonlinear evolution equations appearing in the studies of science and engineering.

Travelling wave solutions of generalized Klein–Gordon equations using Jacobi elliptic functions

This paper obtains exact travelling wave solutions of five various forms of the generalized nonlinear Klein–Gordon equations using Jacobi elliptic functions. Topological and non-topological soliton solutions are obtained as well as Jacobi elliptic function solutions. It is acquired constraint conditions for the existence of solitons.

Analytical soliton solutions of the (2 $$+$$ + 1)-dimensional sine-Gordon equation

In this letter, we investigated a new (2\(+\)1)-dimensional sine-Gordon equation. By the subsidiary ordinary differential equation method, some new explicit solutions are given. These solutions include hyperbolic function solutions and trigonometric function solutions amongst others. In particular, a topological 1-soliton solution is derived.

Topological and singular soliton solution to Kundu–Eckhaus equation with extended Kudryashov's method

In this paper, we apply the extended Kudryashov method to a nonlinear Schrödinger type equation called the Kundu–Eckhaus equation or the Eckhaus equation which was independently introduced by Wiktor Eckhaus and by Anjan Kundu in 1984–1985 to model the propagation of waves in dispersive media. The proposed method is direct, effective and takes full advantages of the Bernoulli and Riccati equations to construct new exact solutions of that model and can be extended to many nonlinear evolution equations in mathematical physics.

Application of G′/G-expansion method to Kuramoto-Sivashinsky equation

This paper obtains the solutions of the Kuramoto-Sivashinsky equation. The G′/G method is used to carry out the integration of this equation. Subsequently, its special case, will be integrated and topological 1-soliton solution will be obtained by the soliton ansatz method. The restrictions on the parameters and exponents are also identified.

Topological soliton solutions to the coupled Schrodinger–Boussinesq equation by the SEM

In this present study we consider the coupled Schrodinger–Boussinesq equation [1](1)iEt+Exx+β1E=NE,3Ntt−Nxxxx+3Nxx2+β2Nxx=E2xx,where β1 and β2 are real constants, and E(x, t) is a complex function, N(x, t) is a real function. In this paper, we enhanced new traveling wave solutions of coupled Schrodinger–Boussinesq equation via simplest equation method. Implementation of the method for searching exact solutions of the equation provided many new solutions which can be used to employ some practically physical and mechanical phenomena.

Maxwell–Higgs self-dual solitons on an infinite cylinder

We have studied the Maxwell–Higgs model on the surface of an infinite cylinder. In particular, we show that this model supports self-dual topological soliton solutions on the infinite tube. Finally, the Bogomol’nyi-type equations are studied from theoretical and numerical point of view.

Traveling wave solutions for nonlinear dispersive water-wave systems with time-dependent coefficients

In this article, the solitary wave and topological soliton solutions in the models that describe the propagation of surface water waves in a uniform channel are successfully constructed. The solitary wave ansatz is used to carry out these distinct solutions. The corresponding integrability criteria, also known as constraint conditions, naturally emerge from the analysis of these models.

Cabling in the Skyrme–Faddeev model

The Skyrme–Faddeev model is a three-dimensional nonlinear field theory that has topological soliton solutions, called hopfions, which are novel string-like solutions taking the form of knots and links. Solutions found thus far take the form of torus knots and links of these, however torus knots form only a small family of known knots. It is an open question whether any non-torus knot hopfions exist. In this paper we present a construction of knotted fields with the form of cable knots to which an energy minimization scheme can be applied. We find the first known hopfions which do not have the form of torus knots, but instead take the form of cable and hyperbolic knots.

Trajectories of DNA bubbles

The DNA molecule is considered as a complex dynamic system where nonlinear conformational waves can be activated and move along the polynucleotide chains. Local nonlinear distortions of the DNA structure named bubbles are studied with the help of the sine-Gordon equation modified by adding two terms that more accurately take into account heterogeneous nature of the DNA sequence. The model equation is solved numerically. Topological soliton solutions ϕ(z,t) having the form of kinks, are found. To obtain the trajectories of the bubbles we project the derivative of the function ϕ(z,t) on the plane (z,t). The approach is applied to artificial sequence consisting of n homogeneous regions separated by (n-1) boundaries, and to the sequence of plasmid pTTQ18. The obtained dependence of the bubble trajectories on the arrangement of the main functional regions (promoters, terminators and coding regions) is interpreted as an evidence of the existence of the relation between DNA dynamics and functioning.

Topological and non-topological soliton solutions to some time-fractional differential equations

This paper investigates, for the first time, the applicability and effectiveness of He’s semi-inverse variational principle method and the ansatz method on systems of nonlinear fractional partial differential equations. He’s semi-inverse variational principle method and the ansatz method are used to construct exact solutions of nonlinear fractional Klein–Gordon equation and generalized Hirota–Satsuma coupled KdV system. These equations have been widely applied in many branches of nonlinear sciences such as nonlinear optics, plasma physics, superconductivity and quantum mechanics. So, finding exact solutions of such equations are very helpful in the theoretical and numerical studies.

Single soliton solutions of the coupled nonlinear Klein–Gordon equations with power law nonlinearity

In this paper, the direct method is adopted to integrate the coupled nonlinear Klein–Gordon equations with power law nonlinearity. The exact single topological and non-topological soliton solutions of the one-coupled, two-coupled and N-coupled nonlinear Klein–Gordon equations with power law nonlinearity are proposed explicitly. Among these soliton solutions there are some constraint conditions which guarantee the existence of the solutions.

Singular solitons in optical metamaterials by ansatz method and simplest equation approach

This paper derives singular 1-soliton solution for optical metamaterials. There are two integration approaches that obtains the solution. These are the ansatz approach and the simplest equation approach. The second method also leads to an additional set of solutions that emerge as a by-product. These are topological soliton, rational solution and singular periodic solution. The constraint conditions for the existence of these solutions are also exhibited. The numerical simulation of a topological 1-soliton solution is also exhibited.

Topological and non-topological soliton solutions of Hamiltonian amplitude equation by He’s semi-inverse method and ansatz approach

Abstract This paper obtains the exact 1-soliton solution to the Hamiltonian amplitude equation. There are two types of integration architectures that are implemented in this paper. They are the He’s semi-inverse method and the ansatz method. These soliton solutions are obtained. There are constraint conditions that also fall out which must remain valid in order for the solitons and other solutions to exist.

Compactons and topological solitons of the Drinfel'd–Sokolov system

This paper presents the Drinfel’d–Sokolov system (shortly D(m, n)) in a detailed fashion. The Jacobi’s elliptic function method is employed to extract the cnoidal and snoidal wave solutions. The compacton and solitary pattern solutions are also retrieved. The ansatz method is applied to extract the topological 1-soliton solutions of the D(m, n) with generalized evolution. There are a couple of constraint conditions that will fall out in order to exist the topological soliton solutions.

Topological and Non-Topological Soliton Solutions of the Coupled Klein-Gordon-Schrodinger and the Coupled Quadratic Nonlinear Equations

In this paper, two solitary wave solutions are obtained for the Coupled Klein-Gordon-Schrodinger and Coupled quadratic nonlinear equations by the ansatz method. Both topological and non-topological solitary wave solutions are obtained.

Solitons in Optical Metamaterials by Functional Variable Method and First Integral Approach

AbstractThis paper addresses soliton solutions in optical metamaterials by functional variable method and first integral approach. These integration schemes lead to bright and singular 1-soliton solution. Topological soliton solution is also revealed. Additionally, as a byproduct, singular periodic solutions and continuous waves are obtained. The corresponding constraint conditions are also listed for these variety of solutions to exist. Finally, a numerical simulation of the bright 1-soliton solution is given.

Topological 1-soliton solution of nonlinear Schrödinger equation with dual-power law nonlinearity in nonlinear optical fibers

The nonlinear Schrodinger equation with dual-power law nonlinearity for describing the propagation of pulses through an optical fiber in an optical communication system is studied. The first integral method, which is based on the division theorem, is used for obtaining the exact 1-soliton solution of this equation. The exact 1-soliton solution is constructed through the established first integral.