Compacton-like Solutions Research Articles

Transforming kinks into compactons in the [formula omitted]-sigma model

In this work, we investigate the solutions of vortices in the O(3)-sigma model with the gauge field governed by the Chern–Simons term and subject to a hyperbolic self-dual potential. We show that this model admits both topological and nontopological soliton solutions. Employing numerical analysis, we realize that the topological solutions of the model can be transformed into compacton-like solutions. On the other hand, after modifying the model by the introduction of a dielectric constant, an exciting feature appears; namely, the nontopological solutions can be transformed into kink-like solutions through the numerical variation of the dielectric constant. Furthermore, we discuss the degeneracy for the topological solitons in a given sector. Finally, we present the numerical solutions of the first model.

Compacton-Like Solutions in a Camassa–Holm Type Equation**Supported by the National Natural Science Foundation of China under Grant Nos. 11401274, 11271171 and Science and technology landing project of colleges and universities in Jiangxi Province under Grant Nos. KJLD14092, KJLD13093

This paper shows that a Camassa–Holm type equation can be reduced to Hamiltonian system by transformation of variables. The hamiltonian system is studied by making use of the dynamical systems theory and the qualitative behavior of degenerate singular points is presented. In particular, new type of compacton-like solutions is obtained by setting the partial differential equation under boundary condition .

Excitations and management of the nonlinear localized gap modes

We discuss about the theory of nonlinear localized excitations, such as soliton and compactons in the gap of the linear spectrum of the nonlinear systems. We show how the gap originates in the linear spectrum using examples of a few systems, such as nonlinear lattices, Bose–Einstein condensates in optical lattice and systems represented by coupled nonlinear evolution equations. We then analytically show the excitation of solitons and compacton-like solutions in the gap of the linear spectrum of a system of coupled Korteweg–de Vries (KdV) equations with linear and nonlinear dispersions. Finally, we discuss about the theory of Feshbach resonance management and dispersion management of the soliton solutions.

Explicit kink-like and compacton-like wave solutions for a generalized KdV equation

A generalized KdV equation $$\begin{aligned} u_t=u^2u_x-u^2u_{xxx}-4uu_xu_{xx}-(u_x)^3\\ \end{aligned}$$ is studied by using the bifurcation method of dynamical systems and a independent transformation. All possible explicit kink-like and compacton-like wave solutions are presented. The results of before references are enriched.

Kink-like wave and compacton-like wave solutions for generalized KdV equation

By using the bifurcation method of dynamical systems and numerical simulation approach of differential equations, we investigate generalized KdV equation \(u_t=u^{2}u_{x}-u^{2}u_{xxx}-4uu_xu_{xx}-(u_x)^3\). Two types of bounded traveling wave solutions are found, that is, the kink-like wave and compacton-like wave solutions. The planar graphs of these solutions are simulated by using software Mathematica; meanwhile, some interesting phenomena are revealed, that is, under some conditions, the periodic wave can become the kink-like wave and compacton-like wave, respectively, and the solitary wave can become the kink-like wave. The exact kink-like wave and compacton-like wave solutions with implicit or parameter expressions are given.

Compacton-like solutions for modified KdV and nonlinear Schrödinger equation with external sources

We present new types of compacton-like solutions for modified KdV and nonlinear Schrodinger equation with external sources, using a recently developed fractional transformation. In particular, we explicate these novel compactons for the trigonometric case, and compare their properties with those of the compactons and solitons in the case of modified KdV equation. Keeping in mind the significance of nonlinear Schrodinger equation with external source, for pulse propagation through asymmetric twin-core fibres, we hope that the newly found compacton may be launched in a long-haul telecommunication network utilizing asymmetric twin-core fibres.

Kink-Like Wave and Compacton-Like Wave Solutions for a Two-Component Fornberg-Whitham Equation

Using bifurcation method and numerical simulation approach of dynamical systems, we study a two-component Fornberg-Whitham equation. Two types of bounded traveling wave solutions are found, that is, the kink-like wave and compacton-like wave solutions. The planar graphs of these solutions are simulated by using software Mathematica; meanwhile, two new phenomena are revealed; that is, the periodic wave solution can become the kink-like wave or compacton-like wave solution under some conditions, respectively. Exact implicit or parameter expressions of these solutions are given.

Gap compactonlike solutions of coupled Kortweg–de Vries equations with linear and nonlinear dispersions

We show the existence of a type of excitation, which we term as "gap compactonlike," within the gap of the linear spectrum of a system of coupled Kortweg-de Vries equations with linear and nonlinear dispersions. Since the solutions lie in the gap region of the spectra, they avoid resonance with the linear oscillatory wave and, therefore, do not decay into radiations. These types of solutions are important in energy localization and transport in polymers and biopolymers, optical systems, etc.

Exact solutions of nonlinear dispersive K( m, n) model with variable coefficients

The variable-coefficient Korteweg-de Vries (KdV) equation with additional terms contributed from the inhomogeneity in the axial direction and the strong transverse confinement of the condense was presented to describe the dynamics of nonlinear excitations in trapped quasi-one-dimensional Bose–Einstein condensates with repulsive atom–atom interactions. To understand the role of nonlinear dispersion in this variable-coefficient model, we introduce and study a new variable-coefficient KdV with nonlinear dispersion (called vc-K( m, n) equation). With the aid of symbolic computation, we obtain its compacton-like solutions and solitary pattern-like solutions. Moreover, we also present some conservation laws for both vc-K +( n, n) equation and vc-K −( n, n) equation.

New abundant solutions for the Burgers equation

This paper applies a new modified Exp-function method to search for exact solitary solutions of the Burgers equation. Using the imaginary transformation suggested by He and Wu, solitary solutions can be easily converted into periodic or compacton-like solutions. Many new solutions are appearing for the first time in the open literature.

Compacton-like solutions of the hydrodynamic system describing relaxing media

We show the existence of a compacton-like solutions within the relaxing hydrodynamic-type model and perform numerical study of attracting features of these solutions.

Generalized solitary solution and compacton-like solution of the Jaulent–Miodek equations using the Exp-function method

A new generalized solitary solution of the Jaulent–Miodek equations is obtained using the Exp-function method. By a transformation, the solitary solution can be easily converted into a generalized compacton-like solution. The free parameters in the obtained generalized solutions might imply some meaningful results in physical process.

Solitary solutions, periodic solutions and compacton-like solutions using the Exp-function method

In this paper, the Exp-function method is used for seeking solitary solutions, periodic solutions, and compacton-like solutions of nonlinear differential equations. The combined KdV–MKdV equation and the Liouville equation are chosen to illustrate the effectiveness and convenience of the proposed method.

Compacton-like and kink-like waves for a higher-order wave equation of Korteweg–de Vries type

In this paper, the bifurcation method of the dynamical and numerical approach to differential equations to study higher order wave equations of Korteweg–de Vries (KdV) type is used. With this methodology we obtain the compacton-like and kink-like wave solutions of the high order KdV type equation. Their implicit expressions are given and their planar graphs are simulated. The results show that the numerical integrations are identical with the theoretical derivations.

Exp-function method for nonlinear wave equations

Abstract In this paper, a new method, called Exp-function method, is proposed to seek solitary solutions, periodic solutions and compacton-like solutions of nonlinear differential equations. The modified KdV equation and Dodd–Bullough–Mikhailov equation are chosen to illustrate the effectiveness and convenience of the suggested method.

Construction of solitary solution and compacton-like solution by variational iteration method

Variational iteration method is used to construct solitary solutions and compacton-like solutions for nonlinear dispersive equations. The chosen initial solution (trial function) can be in compacton form or in soliton form with some unknown parameters which can be determined in the solution procedure. The compacton-like solution can be converted to solitary solution by suitable choice of a parameter, and vice versa.

New compacton-like and solitary patterns-like solutions to nonlinear wave equations with linear dispersion terms

It has been shown that many fully nonlinear wave equations with nonlinear dispersion terms possess compacton solutions and solitary patterns solutions. In this paper, with the aid of Maple, the mKdV equation, the equation with a source term, the five order KdV-like equation and the KdV–mKdV equation are investigated using some new, generalized transformations. As a consequence, it is shown that these equations with linear dispersion terms admit new compacton-like solutions and solitary patterns-like solutions. These transformations can be also extended to other nonlinear wave equations with nonlinear dispersion terms to seek new compacton-like solutions and solitary patterns-like solutions.

A study on compacton-like solutions for the modified KdV and fifth order KdV-like equations

In this work we study compacton-like solutions for the modified KdV and fifth order KdV-like equations. We present compacton-like solutions of the integrable nonlinear equations. Although the behavior of the obtained solution is reminiscent of the solitons, yet its second derivative is discontinuous at the boundaries, a feature characterizing the compactons.

New Symmetry Reductions, Dromions-Like and Compacton Solutions for a 2D BS(m,n) Equations Hierarchy with Fully Nonlinear Dispersion**The project supported by National Key Basic Research Development Project Program of China under Grant No. G1998030600, Doctoral Foundation of China under Grant No. 98014119 and National Natural Science Foundation of China under Grant No. 10072013

We have found two types of important exact solutions, compacton solutions, which are solitary waves with the property that after colliding with their own kind, they re-emerge with the same coherent shape very much as the solitons do during a completely elastic interaction, in the and even models, and dromion solutions (exponentially decaying solutions in all direction) in many and models. In this paper, symmetry reductions in are considered for the break soliton-type equation with fully nonlinear dispersion (called equation) , which is a generalized model of break soliton equation , by using the extended direct reduction method. As a result, six types of symmetry reductions are obtained. Starting from the reduction equations and some simple transformations, we obtain the solitary wave solutions of equations, compacton solutions of equations and the compacton-like solution of the potential form (called ) . In addition, we show that the variable admits dromion solutions rather than the field itself in equation.

Abundant Symmetries and Exact Compacton-Like Structures in the Two-Parameter Family of the Estevez–Mansfield–Clarkson Equations**The project supported by National Key Basic Research Development Project Program of China under Grant No. G1998030600, Doctoral Foundation of China under Grant No. 98014119 and National Natural Science Foundation of China under Grant No. 10072013

The two-parameter family of Estevez–Mansfield–Clarkson equations with fully nonlinear dispersion (called E(m,n) equations), which is a generalized model of the integrable Estevez–Mansfield–Clarkson equation , is presented. Five types of symmetries of the E(m,n) equation are obtained by making use of the direct reduction method. Using these obtained reductions and some simple transformations, we obtain the solitary-like wave solutions of equation. In addition, we also find the compacton solutions (which are solitary waves with the property that after colliding with other compacton solutions, they reemerge with the same coherent shape) of E(3, 2) equation and for its potentials, say, , and compacton-like solutions of equations, respectively. Whether there exist compacton-like solutions of the other E(m,n) equation with is still an open problem.