Kink Profile Solitary Wave Solutions Research Articles

Asymptotic stability of kink profile solitary wave solutions for generalized gardner equation with dissipative term

In this paper, we study the asymptotic stability of the monotone decreasing kink profile solitary wave solutions of the generalized Gardner equation with dissipative term. Firstly, we obtain the estimation of the first-order and second-order derivatives for monotone decreasing kink profile solitary wave solutions, then by using anti-derivative strategy, with the help of prior estimate and Young inequality, we overcome the difficulties caused by two higher-order nonlinear terms of the equation in the estimation, and prove the asymptotic stability of monotone decreasing kink profile solitary wave solution in the sense of [Formula: see text] norm.

Asymptotic Stability of Monotone Decreasing Kink Profile Solitary Wave Solutions for Generalized KdV-Burgers Equation

In this paper, we focus on studying the asymptotic stability of the monotone decreasing kink profile solitary wave solutions for the generalized KdV-Burgers equation. We obtain the estimate of the first-order and second-order derivatives for monotone decreasing kink profile solitary wave solutions, and overcome the difficulties caused by high-order nonlinear terms in the generalized KdV-Burgers equation in the estimate by using L2 energy estimating method and Young inequality. We prove that the monotone decreasing kink profile solitary wave solutions are asymptotically stable in H1. Moreover, we obtain the decay rate of the perturbation ψ in the sense of L2 and L∞ norm, respectively, which are (1 + t) −1/2 and (1 + t) −1/4 by using Gargliado-Nirenberg inequality.

Bounded traveling wave solutions for MKdV-Burgers equation with the negative dispersive coefficient

This paper studies the bounded traveling wave solutions of MKdV-Burgers equation with the negative dispersive coefficient by the theory of planar dynamical systems, undetermined coefficients method. The global phase portraits under the different parameter conditions, as well as the existent number and conditions of the bounded traveling wave solutions are obtained for the dynamical system corresponding to the traveling wave solutions of MKdV-Burgers equation. The relation is investigated between the profiles of the bounded traveling wave solutions and dissipative coefficient. And a critical value characterizing the scale of dissipative effect, is given, which is different from the one proposed by R.F. Bikbaev in his article. Focusing on the open issue proposed by R.F. Bikbaev, based on the bell and kink profile solitary wave solutions of MKdV-Burgers equation we presented, approximate damped oscillatory solutions of MKdV-Burgers equation are obtained according to the evolution relation of orbits corresponding to the approximate damped oscillatory solutions in the global phase portraits.

Traveling wave solutions of compressible fluid equations and orbital stability

In this paper, we discuss the existence of traveling wave solutions for compressible fluid equations by applying the theory and method of planar dynamical system, and obtain explicit expressions for all bounded traveling wave solutions by undetermined coefficient method, including kink and bell profile traveling wave solutions, as well as periodic wave solutions. We prove the kink profile solitary wave solution, both sides of which asymptotic values are not zero, is orbitally stable by the theory of Grillakis-Shatah-Strauss orbital stability.

Bounded traveling wave solutions of variant boussinesq equation with a dissipation term and dissipation effect

This article studies bounded traveling wave solutions of variant Boussinesq equation with a dissipation term and dissipation effect on them. Firstly, we make qualitative analysis to the bounded traveling wave solutions for the above equation by the theory and method of planar dynamical systems, and obtain their existent conditions, number, and general shape. Secondly, we investigate the dissipation effect on the shape evolution of bounded traveling wave solutions. We find out a critical value r* which can characterize the scale of dissipation effect, and prove that the bounded traveling wave solutions appear as kink profile waves if |r|≥r*; while they appear as damped oscillatory waves if |r|≥r*. We also obtain kink profile solitary wave solutions with and without dissipation effect. On the basis of the above discussion, we sensibly design the structure of the approximate damped oscillatory solutions according to the orbits evolution relation corresponding to the component u(ξ) in the global phase portraits, and then obtain the approximate solutions (u˜(ξ),H˜(ξ)). Furthermore, by using homogenization principle, we give their error estimates by establishing the integral equation which reflects the relation between exact and approximate solutions. Finally, we discuss the dissipation effect on the amplitude, frequency, and energy decay of the bounded traveling wave solutions.

Qualitative analysis to traveling wave solutions of Zakharov–Kuznetsov–Burgers equation and its damped oscillatory solutions

The theory of planar dynamical systems is applied in this paper to carry out a qualitative analysis to the planar dynamical system corresponding to the bounded traveling wave solution of the Zakharov–Kuznetsov–Burgers equation, and obtain the existence and uniqueness of the bounded traveling wave solutions. According to the discussions on the relationships between the shapes of bounded traveling wave solutions and the dissipation coefficient d, a critical value d0 is found for arbitrary traveling wave speed v and integral constant g. This equation has a unique monotone kink profile solitary wave solution as the dissipation coefficient d satisfies |d¯|>d0; while it has a unique damped oscillatory solution as |d¯|<d0. This paper also presents the exact bell profile solitary wave solution as d=0. Furthermore, we appropriately design the structure of the damped oscillatory solution in light of the evolution relationships of the solution orbit in the global phase portraits to which the damped oscillatory solution corresponds and obtain its approximate solution by means of the undetermined coefficients method. Finally, based on the integral equation that reflects the relationships between the approximate damped oscillatory solution and its exact solution, the error estimate is given for the approximate damped oscillatory solution. The error is infinitesimal decreasing in exponential form.

Shape analysis and damped oscillatory solutions for a class of nonlinear wave equation with quintic term

This paper aims at analyzing the shapes of the bounded traveling wave solutions for a class of nonlinear wave equation with a quintic term and obtaining its damped oscillatory solutions. The theory and method of planar dynamical systems are used to make a qualitative analysis to the planar dynamical system which the bounded traveling wave solutions of this equation correspond to. The shapes, existent number, and conditions are presented for all bounded traveling wave solutions. The bounded traveling wave solutions are obtained by the undetermined coefficients method according to their shapes, including exact expressions of bell and kink profile solitary wave solutions and approximate expressions of damped oscillatory solutions. For the approximate damped oscillatory solution, using the homogenization principle, its error estimate is given by establishing the integral equation, which reflects the relation between the exact and approximate solutions. It can be seen that the error is infinitesimal decreasing in the exponential form.

Qualitative analysis and explicit traveling wave solutions for the Davey–Stewartson equation

In this paper, we use the bifurcation method of dynamical systems to study the traveling wave solutions for the Davey–Stewartson equation. A number of traveling wave solutions are obtained. Those solutions contain explicit periodic wave solutions, periodic blow‐up wave solutions, unbounded wave solutions, kink profile solitary wave solutions, and solitary wave solutions. Relations of the traveling wave solutions are given. Some previous results are extended. Copyright © 2013 John Wiley &amp; Sons, Ltd.

Qualitative Analysis and Travelling Wave Solutions for the Chaffee–Infante Equation

This paper employs the theory of planar dynamical systems and undetermined coefficient method to study travelling wave solutions to the Chaffee–Infante equation. By qualitative analysis, global phase portraits of the dynamic system corresponding to the equation are obtained for different parameter conditions. Furthermore, the relations between the properties of travelling wave solutions and the diffusion coefficient λ of the equation are investigated. In addition, all possible kink profile solitary wave solutions and approximate damped oscillatory solutions to the equation are obtained by using undetermined coefficient method. Error estimates indicate that the approximate solutions are meaningful. Based on these studies, the main contribution in this paper is to reveal the diffusion effect on travelling wave solutions to the Chaffee–Infante equation.

Shape Analysis and Solution to a Class of Nonlinear Wave Equation with Cubic Term

AbstractIn this paper, we analyze the relation between the shape of the bounded traveling wave solutions and dissipation coefficient of nonlinear wave equation with cubic term by the theory and method of planar dynamical systems. Two critical values which can characterize the scale of dissipation effect are obtained. If dissipation effect is not less than a certain critical value, the traveling wave solutions appear as kink profile; while if it is less than this critical value, they appear as damped oscillatory. All expressions of bounded traveling wave solutions are presented, including exact expressions of bell and kink profile solitary wave solutions, as well as approximate expressions of damped oscillatory solutions. For approximate damped oscillatory solution, using homogenization principle, we give its error estimate by establishing the integral equation which reflects the relations between the exact and approximate solutions. It can be seen that the error is an infinitesimal decreasing in the exponential form.

Traveling Wave Solutions for the Generalized Zakharov Equations

We use the bifurcation method of dynamical systems to study the traveling wave solutions for the generalized Zakharov equations. A number of traveling wave solutions are obtained. Those solutions contain explicit periodic wave solutions, periodic blow‐up wave solutions, unbounded wave solutions, kink profile solitary wave solutions, and solitary wave solutions. Relations of the traveling wave solutions are given. Some previous results are extended.

The Characteristic Function Method and Its Application to (1 + 1)-Dimensional Dispersive Long Wave Equation

In this paper, the characteristic function method is applied to seek traveling wave solutions of nonlinear partial differential equations in a unified way. We consider the Wu-Zhang equation (which describes (1 + 1)-dimensional disper-sive long wave). The equations governing the wave propagation consist of a pair of non linear partial differential equations. The characteristic function method reduces the system of nonlinear partial differential equations to a system of nonlinear ordinary differential equations which is solved via the shooting method, coupled with Rungekutta scheme. The results include kink-profile solitary wave solutions, periodic wave solutions and rational solutions. As an illustrative example, the properties of some soliton solutions for Wu-Zhang equation are shown by some figures.

New explicit solutions for (2 + 1)-dimensional soliton equation

In this Letter, we study (2 + 1)-dimensional soliton equation by using the bifurcation theory of planar dynamical systems. Following a dynamical system approach, in different parameter regions, we depict phase portraits of a travelling wave system. Bell profile solitary wave solutions, kink profile solitary wave solutions and periodic travelling wave solutions are given. Further, we present the relations between the bounded travelling wave solutions and the energy level h. Through discussing the energy level h, we obtain all explicit formulas of solitary wave solutions and periodic wave solutions.

Approximate Damped Oscillatory Solutions for Generalized KdV‐Burgers Equation and Their Error Estimates

We focus on studying approximate solutions of damped oscillatory solutions of generalized KdV‐Burgers equation and their error estimates. The theory of planar dynamical systems is employed to make qualitative analysis to the dynamical systems which traveling wave solutions of this equation correspond to. We investigate the relations between the behaviors of bounded traveling wave solutions and dissipation coefficient, and give two critical values λ1 and λ2 which can characterize the scale of dissipation effect, for right and left‐traveling wave solution, respectively. We obtain that for the right‐traveling wave solution if dissipation coefficient α ≥ λ1, it appears as a monotone kink profile solitary wave solution; that if 0 &lt; α &lt; λ1, it appears as a damped oscillatory solution. This is similar for the left‐traveling wave solution. According to the evolution relations of orbits in the global phase portraits which the damped oscillatory solutions correspond to, we obtain their approximate damped oscillatory solutions by undetermined coefficients method. By the idea of homogenization principle, we give the error estimates for these approximate solutions by establishing the integral equations reflecting the relations between exact and approximate solutions. The errors are infinitesimal decreasing in the exponential form.

A series of new exact solutions of nonlinear evolution equations

AbstractWith the aid of computer symbolic computation system Maple, the generalized auxiliary equation method is first applied to two nonlinear evolution equations, namely, the nonlinear elastic rod equation and (2 + 1)‐dimensional Boiti‐Leon‐Pempinelli equation. As a results, some new types of exact traveling wave solutions are obtained which include bell and kink profile solitary wave solutions, and triangular periodic wave solutions and singular solutions. The method is straightforward and concise, and it can also be applied to other nonlinear evolution equations in mathematical physics. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2010

Qualitative analysis and solutions of bounded traveling wave for B-BBM equation

In this paper, we apply the theory of planar dynamical systems to carry out qualitative analysis for the dynamical system corresponding to B-BBM equation, and obtain global phase portraits under various parameter conditions. Then, the relations between the behaviors of bounded traveling wave solutions and the dissipation coefficient µ are investigated. We find that a bounded traveling wave solution appears as a kink profile solitary wave solution when µ is more than the critical value obtained in this paper, while a bounded traveling wave solution appears as a damped oscillatory solution when µ is less than it. Furthermore, we explain the solitary wave solutions obtained in previous literature, and point out their positions in global phase portraits. In the meantime, approximate damped oscillatory solutions are given by means of undetermined coefficients method. Finally, based on integral equations that reflect the relations between the approximate damped oscillatory solutions and the implicit exact damped oscillatory solutions, error estimates for the approximate solutions are presented.

Exact travelling wave solutions for the dissipative (2 + 1)-dimensional AKNS equation

This paper employs the theory of planar dynamical systems and undetermined coefficient method to study travelling wave solutions of the dissipative (2 + 1)-dimensional AKNS equation. By qualitative analysis, global phase portraits of the dynamic system corresponding to the equation are obtained under different parameter conditions. Furthermore, the relations between the properties of travelling wave solutions and the dissipation coefficient r of the equation are investigated. In addition, the possible bell profile solitary wave solution, kink profile solitary wave solutions and approximate damped oscillatory solutions of the equation are obtained by using undetermined coefficient method. Error estimates indicate that the approximate solutions are meaningful. Based on above studies, a main contribution in this paper is to reveal the dissipation effect on travelling wave solutions of the dissipative (2 + 1)-dimensional AKNS equation.

Qualitative analysis and solutions of bounded travelling waves for the fluidized-bed modelling equation

We apply the theory of planar dynamical systems to carry out a qualitative analysis for the planar dynamical system corresponding to the fluidized-bed modelling equation. We obtain the global phase portraits of this system under various parameter conditions and the existence conditions of bounded travelling-wave solutions of this equation. According to the discussion on relationships between the behaviours of bounded travelling-wave solutions and the dissipation coefficients ε and δ, we find a critical value λ0 for arbitrary travelling-wave velocity υ. This equation has a unique damped oscillatory solution as ∥ε + δυ∥ &lt; λ0 and ∥ε + δυ∥ ≠ 0, while it has a unique monotone kink profile solitary-wave solution as ∥ε + δυ∥ &gt; λ0. By means of the undetermined coefficients method, we obtain the exact bell profile solitary-wave solution and monotone kink profile solitary-wave solution. Meanwhile, we obtain the approximate damped oscillatory solution. We point out the positions of these solutions in the global phase portraits. Finally, based on integral equations that reflect the relationships between the approximate damped oscillatory solutions and the implicit exact damped oscillatory solutions, error estimates for the approximate damped oscillatory solutions are presented.

Exact travelling wave solutions of the dissipative coupled Korteweg-de Vries equation

This paper employs the theory of planar dynamical systems and undetermined coefficient method to study travelling wave solutions of the dissipative coupled Korteweg-de Vries equation. The possible kink profile solitary wave solutions and approximate damped oscillatory solutions of the equation are obtained by using undetermined coefficient method. Error estimates indicate that the approximate solutions are meaningful.

Some Exact Solutions of Variable Coefficient Cubic-Quintic Nonlinear Schrödinger Equation with an External Potential

By constructing appropriate transformations and an extended elliptic sub-equation approach, we find some exact solutions of variable coefficient cubic-quintic nonlinear Schrödinger equation with an external potential, which include bell and kink profile solitary wave solutions, singular solutions, triangular periodic wave solutions and so on.