Exact closed form solutions of compound Kdv Burgers’ equation by using generalized (Gʹ/G) expansion method

In this investigation, the compound Korteweg-de Vries (Kd-V) Burgers equation with constant coefficients is considered as the model, which is used to describe the properties of ion-acoustic waves in plasma physics, and also applied for long wave propagation in nonlinear media with dispersion and dissipation. The aim of this paper to achieve the closed and dynamic closed form solutions of the compound KdV Burgers equation. We derived the completely new solutions to the considered model using the generalized (Gʹ/G)-expansion method. The newly obtained solutions are in form of hyperbolic and trigonometric functions, and rational function solutions with inverse terms of the trigonometric, hyperbolic functions. The dynamical representations of the obtained solutions are shown as the annihilation of three-dimensional shock waves, periodic waves, and multisoliton through their three dimensional and contour plots. The obtained solutions are also compared with previously exiting solutions with both analytically and numerically, and found that our results are preferable acceptable compared to the previous results.Publisher's Versio

Two classes of nonlocal Evolution Equations related by a shared Traveling Wave Problem

We consider reaction-diffusion equations and Korteweg-de Vries-Burgers (KdVB) equations, i.e. scalar conservation laws with diffusive-dispersive regularization. We review the existence of traveling wave solutions for these two classes of evolution equations. For classical equations the traveling wave problem (TWP) for a local KdVB equation can be identified with the TWP for a reaction-diffusion equation. In this article we study this relationship for these two classes of evolution equations with nonlocal diffusion/dispersion. This connection is especially useful, if the TW equation is not studied directly, but the existence of a TWS is proven using one of the evolution equations instead. Finally, we present three models from fluid dynamics and discuss the TWP via its link to associated reaction-diffusion equations

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

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

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

Convergence rate toward shock wave under periodic perturbation for generalized Korteweg-de Vries-Burgers equation

In this paper, a viscous shock wave under space-periodic perturbation of generalized Korteweg-de Vries-Burgers equation is investigated. It is shown that if the initial periodic perturbation around the viscous shock wave is small, then the solution time asymptotically tends to a viscous shock wave with a shift partially determined by the periodic oscillations. Moreover the exponential time decay rate toward the viscous shock wave is also obtained for some certain perturbations.Comment: 22 page

Modification of Truncated Expansion Method for Solving Some Important Nonlinear Partial Differential Equations

In this paper, we implemented modification of truncated expansion method for the exact solutions of the Konopelchenko-Dubrovsky equation the (n+1)-dimensional combined sinhcosh- Gordon equation and the Maccari system. Modification of truncated expansion method is a powerful solution method for obtaining exact solutions of nonlinear evolution equations. This method presents a wider applicability for handling nonlinear wave equations

Some results on the 1D linear wave equation with van der Pol type nonlinear boundary conditionsand the Korteweg-de Vries-Burgers equation

Many physical phenomena can be described by nonlinear models. The last few decades have seen an enormous growth of the applicability of nonlinear models and of the development of related nonlinear concepts. This has been driven by modern computer power as well as by the discovery of new mathematical techniques, which include two contrasting themes: (i) the theory of dynamical systems, most popularly associated with the study of chaos, and (ii) the theory of integrable systems associated, among other things, with the study of solitons. In this dissertation, we study two nonlinear models. One is the 1-dimensional vibrating string satisfying wtt − wxx = 0 with van der Pol boundary conditions. We formulate the problem into an equivalent first order hyperbolic system, and use the method of characteristics to derive a nonlinear reflection relation caused by the nonlinear boundary conditions. Thus, the problem is reduced to the discrete iteration problem of the type un+1 = F (un). Periodic solutions are investigated, an invariant interval for the Abel equation is studied, and numerical simulations and visualizations with different coefficients are illustrated. The other model is the Korteweg-de Vries-Burgers (KdVB) equation. In this dissertation, we proposed two new approaches: One is what we currently call First Integral Method, which is based on the ring theory of commutative algebra. Applying the Hilbert-Nullstellensatz, we reduce the KdVB equation to a first-order integrable ordinary differential equation. The other approach is called the Coordinate Transformation Method, which involves a series of variable transformations. Some new results on the traveling wave solution are established by using these two methods, which not only are more general than the existing ones in the previous literature, but also indicate that some corresponding solutions presented in the literature contain errors. We clarify the errors and instead give a refined result

A transformed rational function method and exact solutions to the 3+1 dimensional Jimbo-Miwa equation

A direct approach to exact solutions of nonlinear partial differential equations is proposed, by using rational function transformations. The new method provides a more systematical and convenient handling of the solution process of nonlinear equations, unifying the tanh-function type methods, the homogeneous balance method, the exp-function method, the mapping method, and the F-expansion type methods. Its key point is to search for rational solutions to variable-coefficient ordinary differential equations transformed from given partial differential equations. As an application, the construction problem of exact solutions to the 3+1 dimensional Jimbo-Miwa equation is treated, together with a B\"acklund transformation.Comment: 13 page