83 research outputs found
A Note On 3Solitary Wave Solutions of the Compound Burgers-Korteweg-de Vries Equation"
The goal of this note is to construct a class of traveling solitary wave
solutions for the compound Burgers-Korteweg-de Vries equation by means of a
hyperbolic ansatz
Traveling waves to a Burgers–Korteweg–de Vries-type equation with higher-order nonlinearities
AbstractIn this paper, first we survey some recent advances in the study of traveling wave solutions to the Burgers–Korteweg–de Vries equation and some comments are given. Then, we study a Burgers–Korteweg–de Vries-type equation with higher-order nonlinearities. A qualitative analysis to a two-dimensional autonomous system which is equivalent to the Burgers–KdV-type equation is presented, and indicates that under certain conditions, the Burgers–Korteweg–de Vries-type equation has neither nontrivial bell-profile solitary waves, nor periodic waves. Finally, a solitary wave solution is obtained by means of the first-integral method which is based on the ring theory of commutative algebra
Application of (G/G') -expansion method to the compound Kdv-burgers type equations
In this Letter, the (G'/G)-expansion method is proposed to seek exact
solutions of nonlinear evolution equations. For illustrative examples, we choose the
compound KdV-Burgers equation, the compound KdV equation, the KdV-Burgers
equation, the mKdV equation. The power of the employed method is confirmed
Towards new schemes: A Lie-group approach of the CBKDV and its derived equations
The aim of this paper is to propose methods that enable us to build new
numerical schemes, which preserve the Lie symmetries of the original
differential equations. To this purpose, the compound Burgers-Korteweg-de Vries
(\textit{CBKDV}) equation is considered. The particular case of the Burgers
equation is taken as a numerical example, and the resulting semi-invariant
scheme is exposed
Dynamics for the Compound Burgers-KdV equation
In this thesis, we study the Two-Dimensional Burgers-Korteweg-de Vries (2D-BKdV) equation and Two-Dimensional Compound Burgers-Korteweg-de Vries (2D-Compound BKdV) by analyzing the first integral equation, which indicates that under some particular conditions, the 2D-BKdV equation and 2D-Compound BKdV have exact traveling wave solutions. By using the elliptic integral and some transformations, traveling wave solution to the 2D-BKdV equation and 2DCompound BKdV are expressed explicitly
Meromorphic solutions of nonlinear ordinary differential equations
Exact solutions of some popular nonlinear ordinary differential equations are
analyzed taking their Laurent series into account. Using the Laurent series for
solutions of nonlinear ordinary differential equations we discuss the nature of
many methods for finding exact solutions. We show that most of these methods
are conceptually identical to one another and they allow us to have only the
same solutions of nonlinear ordinary differential equations
Homotopy Asymptotic Method and Its Application
As we all know, perturbation theory is closely related to methods used in the numerical analysis fields. In this chapter, we focus on introducing two homotopy asymptotic methods and their applications. In order to search for analytical approximate solutions of two types of typical nonlinear partial differential equations by using the famous homotopy analysis method (HAM) and the homotopy perturbation method (HPM), we consider these two systems including the generalized perturbed Kortewerg-de Vries-Burgers equation and the generalized perturbed nonlinear Schrödinger equation (GPNLS). The approximate solution with arbitrary degree of accuracy for these two equations is researched, and the efficiency, accuracy and convergence of the approximate solution are also discussed
Dispersive and diffusive-dispersive shock waves for nonconvex conservation laws
We consider two physically and mathematically distinct regularization
mechanisms of scalar hyperbolic conservation laws. When the flux is convex, the
combination of diffusion and dispersion are known to give rise to monotonic and
oscillatory traveling waves that approximate shock waves. The zero-diffusion
limits of these traveling waves are dynamically expanding dispersive shock
waves (DSWs). A richer set of wave solutions can be found when the flux is
non-convex. This review compares the structure of solutions of Riemann problems
for a conservation law with non-convex, cubic flux regularized by two different
mechanisms: 1) dispersion in the modified Korteweg--de Vries (mKdV) equation;
and 2) a combination of diffusion and dispersion in the mKdV-Burgers equation.
In the first case, the possible dynamics involve two qualitatively different
types of DSWs, rarefaction waves (RWs) and kinks (monotonic fronts). In the
second case, in addition to RWs, there are traveling wave solutions
approximating both classical (Lax) and non-classical (undercompressive) shock
waves. Despite the singular nature of the zero-diffusion limit and rather
differing analytical approaches employed in the descriptions of dispersive and
diffusive-dispersive regularization, the resulting comparison of the two cases
reveals a number of striking parallels. In contrast to the case of convex flux,
the mKdVB to mKdV mapping is not one-to-one. The mKdV kink solution is
identified as an undercompressive DSW. Other prominent features, such as
shock-rarefactions, also find their purely dispersive counterparts involving
special contact DSWs, which exhibit features analogous to contact
discontinuities. This review describes an important link between two major
areas of applied mathematics, hyperbolic conservation laws and nonlinear
dispersive waves.Comment: Revision from v2; 57 pages, 19 figure
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