Solitary Wave Ansatz Research Articles

Exact Solution and Conservation Laws for Fifth-Order Korteweg-de Vries Equation

With the aid of Mathematica, new exact travelling wave solutions for fifth-order KdV equation are obtained by using the solitary wave ansatz method and the Wu elimination method. The derivation of conservation laws for a fifth-order KdV equation is considered.

New exact solutions to the compound KdV-Burgers system with nonlinear terms of any order

In this paper, we obtain some new exact solutions of compound KdV-Burgers system with nonlinear terms of any order by using the \((\frac{G^{\prime }}{G})\)-expand method. In addition, the solitary wave ansatz method is used to carry out this integration. Subsequently, it has been proved that the soliton solution exist only for the KdV-Burgers equation.

Envelope solitons for generalized forms of the phi-four equation

We consider two variants of the generalized phi-four equation with arbitrary constant coefficients and general values of the exponents in the dissipation and nonlinear terms. By using solitary wave ansatze in terms of sechp(x) and tanhp(x) functions respectively, we find the non-topological (bright) as well as topological (dark) soliton solutions for the considered models. The physical parameters in the soliton solutions are obtained as a function of the dependent model coefficients. The conditions of existence of solitons are presented. Further, we show that the obtained soliton solutions depend on the exponent of the wave function u(x,t), positive or negative, and on all the dependent model coefficients as well.

On soliton solutions for the Fitzhugh–Nagumo equation with time-dependent coefficients

We consider a generalized Fitzhugh–Nagumo equation exhibiting time-varying coefficients and linear dispersion term. By means of specific solitary wave ansatz and the tanh method, a new variety of soliton solutions are derived. The physical parameters in the soliton solutions are obtained as function of the time-dependent model coefficients. The conditions of existence and uniqueness of solitons are presented. These solutions may be useful to explain the nonlinear dynamics of waves in an inhomogeneous media that is described by the variable coefficients Fitzhugh–Nagumo equation. Clearly, adaptive methods are straightforward and concise and their applications for the Fitzhugh–Nagumo equation with t-dependent coefficients enable one to construct soliton-like solutions.

Abundant soliton and periodic wave solutions for the coupled Higgs field equation, the Maccari system and the Hirota–Maccari system

In this work, we explore a variety of solitary wave ansatze and periodic wave ansatze to some nonlinear equations. Three complex systems of nonlinear equations that appear in mathematical physics are investigated. We derive abundant soliton and periodic wave solutions for the coupled Higgs field equation, the Maccari system and the Hirota–Maccari system. The results obtained show that these three coupled equations exhibit the richness of explicit solutions: solitons, periodic and rational wave solutions.

Bright solitons of the variants of the Novikov–Veselov equation with constant and variable coefficients

In this paper, the existence of the bright soliton solution of four variants of the Novikov–Veselov equation with constant and time varying coefficients will be studied. We analyze the solitary wave solutions of the Novikov–Veselov equation in the cases of constant coefficients, time-dependent coefficients and damping term, generalized form, and in 1+N dimensions with variable coefficients and forcing term. We use the solitary wave ansatz method to derive these solutions. The physical parameters in the soliton solutions are obtained as functions of the dependent coefficients. Parametric conditions for the existence of the exact solutions are given. The solitary wave ansatz method presents a wider applicability for handling nonlinear wave equations.

Bright and dark soliton solutions for variable-coefficient diffusion–reaction and modified Korteweg–de Vries equations

In this paper, by using a solitary wave ansatz in the form of sechp and tanhp functions, we obtain the exact bright and dark soliton solutions for the considered model, respectively. The topological (dark) and non-topological (bright) soliton solutions to the variable-coefficient diffusion–reaction equation and the variable-coefficient modified Korteweg–de Vries equation with power law nonlinearity are obtained by using the solitary wave ansatz method. The parametric conditions for the formation of soliton pulses are determined. Note that it is always useful and desirable to construct exact analytical solutions, especially soliton-type envelope ones, for understanding most nonlinear physical phenomena.

Analytical and numerical solutions of the Schrödinger–KdV equation

The Schrodinger–KdV equation with power-law nonlinearity is studied in this paper. The solitary wave ansatz method is used to carry out the integration of the equation and obtain one-soliton solution. The G′/G method is also used to integrate this equation. Subsequently, the variational iteration method and homotopy perturbation method are also applied to solve this equation. The numerical simulations are also given.

Bright and dark soliton solutions for a new fifth-order nonlinear integrable equation with perturbation terms

In this paper, we use the solitary wave ansatz method to carry out the integration of a new fifth-order nonlinear equation having perturbation terms. The perturbation terms that are considered are the first-order dispersion term, the power law nonlinearity term, and the fifth- order dispersion term. Both bright and dark soliton solutions are obtained. The physical parameters in the soliton solutions: amplitude, inverse width, and velocity are obtained as functions of the dependent model coefficients. The conditions of the existence of the derived solitons are presented.

A one-soliton solution of the [formula omitted] equation with generalized evolution and time-dependent coefficients

We consider a nonlinear dispersive Zakharov–Kuznetsov (for short, Z K ( m , n , k ) ) equation. This equation governs the behavior of weakly nonlinear ion-acoustic waves in plasma comprising cold ions and hot isothermal electrons in the presence of a uniform magnetic field. A special form of this model with time-dependent coefficients of the nonlinear terms as well as the nonlinear dispersion terms is studied. Further, there are time-dependent linear attenuation and generalized evolution terms. The solitary wave ansatz is used to carry out the integration and an exact one-soliton solution is obtained. The parameters of the soliton envelope (amplitude, widths, velocity) are explicitly calculated in the course of the derivation of the exact solution, as functions of the varying model coefficients. The constraint relation between these time-dependent coefficients for the one-soliton solution to exist is established.

Existence of solitary wave solutions for the nonlinear Klein–Gordon equation coupled with Born–Infeld theory with critical Sobolev exponent

In this paper, we prove the existence of solutions for the nonlinear Klein–Gordon equation coupled with Born–Infeld theory under the electrostatic solitary wave ansatz by using variational methods.

1-Soliton solution of the D( m, n) equation with generalized evolution

This paper obtains the 1-soliton solution of the nonlinear dispersive Drinfel’d–Sokolov equation with power law nonlinearity. In the first case the soliton solution is without the generalized evolution. The solitary wave ansatz method is used to carry out the integration. Subsequently, the He’s semi-inverse variational principle is used to integrate the equation with power law nonlinearity. Parametric conditions for the existence of envelope solitons are given.

SOLITON SOLUTIONS FOR SEVENTH-ORDER KAWAHARA EQUATION WITH TIME-DEPENDENT COEFFICIENTS

In this work, a generalized seventh-order Kawahara equation characterized by time-dependent coefficients will be examined. The solitary wave ansatz will be used to obtain soliton solutions for this equation. The necessary conditions for the existence of the soliton solutions will also be derived.

Dark Solitons for a Generalized Korteweg-de Vries Equation with Time-Dependent Coefficients

We consider the evolution of long shallow waves in a convecting fluid when the critical Rayleigh number slightly exceeds its critical value within the framework of a perturbed Korteweg-de Vries (KdV) equation. In order to study the wave dynamics of nonlinear pulse propagation in an inhomogeneous KdV media, a generalized form of the considered model with time-dependent coefficients is presented. By means of the solitary wave ansatz method, exact dark soliton solutions are derived under certain parametric conditions. The results show that the soliton parameters (amplitude, inverse width, and velocity) are influenced by the time variation of the dependent model coefficients. The existence of such a soliton solution is the result of the exact balance among nonlinearity, third-order and fourth-order nonlinear dispersions, diffusion, dissipation, and reaction.

Soliton solutions and conservation laws of the Gilson–Pickering equation

This paper obtains the soliton solutions of the Gilson–Pickering equation. The G′/G method will be used to carry out the solutions of this equation and then the solitary wave ansatz method will be used to obtain a 1-soliton solution of this equation. Finally, the invariance and multiplier approach will be applied to recover a few of the conserved quantities of this equation.

Dark solitons for a combined potential KdV and Schwarzian KdV equations with t-dependent coefficients and forcing term

In this work we formally derive the dark soliton solutions for the combined potential KdV and Schwarzian KdV equations. The combined KdV and Schwarzian KdV equations with time-dependent coefficients and forcing term are then investigated to obtain dark soliton solutions. The solitary wave ansatz is used to carry out the analysis for both models.

Bright and dark solitons of the Rosenau-Kawahara equation with power law nonlinearity

In this paper, the topological (dark) as well as non-topological (bright) soliton solutions to the Rosenau-Kawahara equation with power law nonlinearity are obtained by the solitary wave ansatz method. A couple of conserved quantities are also calculated for the case of bright soliton solution.

Bright and dark solitons for a generalized Korteweg-de Vries–modified Korteweg-de Vries equation with high-order nonlinear terms and time-dependent coefficients

We consider a generalized Korteweg-de Vries–modified Korteweg-de Vries (KdV–mKdV) equation with high-order nonlinear terms and time-dependent coefficients. Bright and dark soliton solutions are obtained by means of the solitary wave ansatz method. The physical parameters in the soliton solutions are obtained as functions of the varying model coefficients. Parametric conditions for the existence of envelope solitons are given. In view of the analysis, we see that the method used is an efficient way to construct exact soliton solutions for such a generalized version of the KdV–mKdV equation with time-dependent coefficients and high-order nonlinear terms.

A study of shallow water waves with Gardner’s equation

In this paper the dynamics of solitary waves governed by Gardner’s equation for shallow water waves is studied. The mapping method is employed to carry out the integration of the equation. Subsequently, the perturbed Gardner equation is studied, and the fixed point of the soliton width is obtained. This fixed point is then classified. The integration of the perturbed Gardner equation is also carried out with the aid of He’s semi-inverse variational principle. Finally, Gardner’s equation with full nonlinearity is solved with the aid of the solitary wave ansatz method.

Dark solitons for a generalized nonlinear Schrödinger equation with parabolic law and dual-power law nonlinearities

The dynamics of dark solitons is studied within the framework of a generalized nonlinear Schrödinger equation. The specific cases of parabolic law and dual-power law nonlinearity are considered. The solitary wave ansatz method is used to carry out the integration. All the physical parameters in the soliton solutions are obtained as functions of the dependent model coefficients. Parametric conditions for the existence of envelope solitons are given. Copyright © 2011 John Wiley & Sons, Ltd.