Traveling-Wave Solutions for Korteweg–de Vries–Burgers Equations through Factorizations

Abstract

Travelling-wave solutions of the standard and compound form of Kortewegde Vries-Burgers equations are found using factorizations of the corresponding reduced ordinary differential equations. The procedure leads to solutions of Bernoulli equations of nonlinearity 3/2 and 2 (Riccati), respectively. Introducing the initial conditions through an imaginary phase in the travelling coordinate, we obtain all the solutions previously reported, some of them being corrected here, and showing, at the same time, the presence of interesting details of these solitary waves that have been overlooked before this investigation.

Trial Equation Method Based on Symmetry and Applications to Nonlinear Equations Arising in Mathematical Physics

“…These solutions can be also written in the form given by (26) and (27) in the paper by Cornejo-Perez et al [23]. …”

Solutions of a Class of Duffing Oscillators with Variable Coefficients

“…A certain class of nonlinear second-order ODE's can be solved by a factorization technique under some conditions (see for instance [12][13][14]16]). These equations frequently appear when looking for travelling wave solutions of partial nonlinear equations, such as KortewegdeVries-Burgers [12,13], Kadomtsev-Petviashvili [12], or Benjamin-Bona-Mahony [14].…”

“…These equations frequently appear when looking for travelling wave solutions of partial nonlinear equations, such as KortewegdeVries-Burgers [12,13], Kadomtsev-Petviashvili [12], or Benjamin-Bona-Mahony [14]. In some cases the factorizations are directly related to first integrals of the equation.…”

“…However, when the coefficients are constant they have been investigated using many procedures: the elliptic averaging [9], the sinh-Gordon equation expansion [10], the homotopy-perturbation [11], and others mentioned in [9][10][11]. In this paper we will deal with this equation by means of the factorization method, which in general is applicable to nonlinear ordinary differential equations (ODE's) when the coefficients do not depend on the independent variable [12][13][14][15]. However, as we shall see, the case of variable coefficients can be also factorized by transforming the initial equation into a canonical form [16].…”

Classifying Traveling Wave Solutions to the Zhiber-Shabat Equation

“…Many methods, such as the transformed rational function method [2], the multiple exp-function algorithm [3] and the factorization method [4], have been proposed to find exact traveling wave solutions to nonlinear partial differential equations. At the same time, Ma has obtained complexiton solutions, a kind of multi-wave solutions, to some nonlinear partial differential equations [5,6].…”