Explicit exact solitary-wave solutions for compound KdV-type and compound KdV–Burgers-type equations with nonlinear terms of any order
Introduction
In this paper we consider the compound KdV-type equation with nonlinear terms of any orderand the compound KdV–Burgers-type equation with nonlinear terms of any orderThese equations arise in a variety of physical contexts and have been studied by many authors [1], [2], [3], [4], [5], [6], [7], [8]. The KdV-type equations have application in quantum field theory, plasma physics and solid-state physics. For example, the kink soliton can be used to calculate energy and momentum flow and topological charge in the quantum field. In [5] and [6], Dey and Coffey considered kink-profile solitary-wave solutions for Eq. (1.1) with p=1 and p=2. Eq. (1.2) with p=1 describes the propagation of undulant bores in shallow water and weakly nonlinear plasma waves with certain dissipative effects. The compound KdV–Burgers-type equation (1.2) with p>1 is a model for long-wave propagation in nonlinear media with dispersion and dissipation. The kink solution of Eq. (1.2) with p=1 was obtained in [8]. The stability of traveling-wave solutions of Eq. (1.2) with and p⩾1 are studied by Pego et al. [4]. In [7], we obtained kink- and bell-profile solitary-wave solutions of Eq. (1.1) with p=1, and kink solutions of Eq. (1.2) with p=1.
In this paper, the motivation is to use an approach to study both the compound KdV-type equation (1.1) and the compound KdV–Burgers-type equation (1.2), and find many possible solitary-wave solutions of these equations. Two kinds of solitary-wave solutions of (1.1) and kink solution of (1.2) are obtained by means of proper transformation which degrades the order of nonlinear terms and the undetermined coefficient method. A solitary-wave solution with negative velocity for the generalized KdV–Burgers equation ut+upux−αuxx+uxxx=0 is found.
This paper is organized as follows. Some basic results are presented in Section 2. In Section 3, we find the bell-profile solitary-wave solution of Eq. (1.1). The explicit kink-profile solitary-wave solutions of , are obtained in Section 4. Finally, a solitary-wave solution with negative velocity for the generalized KdV–Burgers equation is found in Section 5.
Section snippets
Basic results
Let u(x,t)=u(x−vt)≡u(ξ) be a traveling-wave solution for Eq. (1.2). Substituting u(ξ) in Eq. (1.2) yieldsIntegrating the above equation once, we havewhere k is an integrating constant.
Let C±=limξ→±∞u(ξ). Assume that asymptotic values of the solitary-wave solution of (1.2) haveand C± satisfies the following algebraic equationThen, the integrating
The bell-profile solitary-wave solutions to the compound KdV-type equation
By using the same deduction as of formula (2.4), we know that the following equationholds for the traveling-wave solutions of Eq. (1.1) with asymptotic conditions , . LetThen it follows from (3.1) that
Now, we assume that the solution of Eq. (3.3) has the following form:where and α are constants to
The kink-profile solitary-wave solution
LetThen we have from (2.4) thatNow, we assume that the solution of (4.2) has the following form:Thus, there are the following relations between parameters A,α and v:By solving system (4.4) of algebraic equations, two sets of solutions are obtained:
The kink-profile solitary-wave solution to the generalized KdV–Burgers equation
In this section, we consider the generalized KdV–Burgers equationsandand seek their kink-profile solitary-wave solution of the form , .
Notice that Eq. (5.1) is a special form of Eq. (1.2) in the case . Let p=m/2. Then the formulas , can be rewritten asandrespectively. Taking and p=m/2 in the system of algebraic equations (4.4), we have following
Acknowledgements
This work was supported by the National Natural Science Foundation of China and the Natural Science Foundation of Hunan Province.
References (8)
- et al.
Oscillatory instability of traveling waves for a KdV–Burgers equation
Phys D
(1993) Exact solutions for a compound KdV–Burgers equation
Phys Lett A
(1996)Wave propagation in nonlinear lattice, I
J Phys Soc Jpn
(1975)- et al.
Travelling wave solutions to the Korteweg–de Vries–Burgers equation
Proc R Soc Edinburgh Sect A
(1985)
Cited by (100)
Operational matrix for Atangana–Baleanu derivative based on Genocchi polynomials for solving FDEs
2020, Chaos, Solitons and FractalsCitation Excerpt :The Korteweg-de Vries equations have application in quantum field theory, solid-state physics and plasma physics. These equations have been studied by many authors and arise in a variety of physical contexts [18–21]. In [22], the authors have analyzed and extended the Korteweg-de Vries-Burgers equation with two levels of perturbation and combined the model with AB-derivative.
Periodic wave solutions and solitary wave solutions of generalized modified Boussinesq equation and evolution relationship between both solutions
2017, Journal of Mathematical Analysis and ApplicationsMathematical analysis of the generalized Benjamin and Burger-Kdv equations via the extended trial equation method
2014, Journal of the Association of Arab Universities for Basic and Applied SciencesQualitative analysis to traveling wave solutions of Zakharov-Kuznetsov- Burgers equation and its damped oscillatory solutions
2014, Applied Mathematics and ComputationBounded traveling wave solutions of variant boussinesq equation with a dissipation term and dissipation effect
2014, Acta Mathematica ScientiaCnoidal and snoidal wave solutions to coupled nonlinear wave equations by the extended Jacobi's elliptic function method
2013, Communications in Nonlinear Science and Numerical Simulation