Traveling wave solutions of the generalized nonlinear evolution equations

doi:10.1016/j.amc.2008.11.048

Applied Mathematics and Computation

Volume 210, Issue 2, 15 April 2009, Pages 551-557

https://doi.org/10.1016/j.amc.2008.11.048 Get rights and content

Abstract

Solitary wave solutions for a family of nonlinear evolution equations with an arbitrary parameter in the exponents are constructed. Some of the obtained solutions seem to be new.

Introduction

In this paper we present a method, which enables one to construct traveling wave solutions for the following family of nonlinear partial differential equations: $M_{α_{1}, \dots, α_{N}}^{(N + 1)} [u] : \frac{\partial u}{\partial t} + \sum_{i = 1}^{N} α_{i} \frac{\partial^{i + 1} u}{\partial x^{i + 1}} + u^{m} \frac{\partial u}{\partial x} = 0, N ⩾ 1 .$ The equation $M_{α_{1}, \dots, α_{N}}^{(N + 1)} [u]$ is of order N + 1 and depends on N + 1 parameters denoted by α₁, …, α_N, m. This family contains several famous partial differential equations, such as the Korteweg–de Vries equation [1], [2], [3], [4], [5], the Korteweg–de Vries–Burgers equation [6], the Kuramoto–Sivashinsky equation [7], [8], [9], [10], [11], [12] and their generalizations [13], [14], [15], [16], [17]. Various members of the family (1.1) at N = 3 have been intensively studied in recent years. For example, see [18], [19], [20], [21], [22], [23], [24], [25], [26], [27], [28].

Our approach is based on constructing Bäcklund transformations between the traveling wave reductions of the equation $M_{α_{1}, \dots, α_{N}}^{(N + 1)} [u]$ and the Nth-order ordinary differential equation with two remarkable properties. First of all, this equation no longer contains the terms with m in the exponents, i.e. the parameter m moves from the exponents to the coefficients. Second, its mero-morphic solutions have only simple poles. All these makes the equation in question extremely convenient for further analysis. In particular, this equation admits application of various methods for constructing exact solutions. In this paper, we will use the simplest equation method suggested and developed in works [29], [30], [31], [32], [33] accordingly. The simplest equation method is based on expressing exact solutions of the given equation through solutions of another equation, called the simplest equation.

This paper is organized as follows. In Section 2, we describe our method: we derive the Bäckhand transformations and introduce the simplest equation. In Sections 3 Solitary wave solutions of the generalized Burgers equation, 4 Solitary wave solutions of the generalized Korteweg–de Vries–Burgers equation, 5 Solitary wave solutions of the generalized Kuramoto–Sivashinsky equation, 6 Solitary wave solutions of the fifth-order equation, we give several specific examples for the most interesting equations of the family (1.1).

Section snippets

Method applied

First of all let us show that there exist Bäckhand transformations between the traveling wave reductions of the equation $M_{α_{1}, \dots, α_{N}}^{(N + 1)} [u]$ and another ordinary differential equation with the properties discussed in Section 1. Making the substitution: $u (x, t) = y (z), z = x - C_{0} t$ in the equation $M_{α_{1}, \dots, α_{N}}^{(N + 1)} [u]$ and integrating the result yields the following Nth-order equation: $P_{α_{1}, \dots, α_{N}}^{(N)} [y] : \sum_{i = 1}^{N} α_{i} \frac{d^{i} y}{{dz}^{i}} - C_{0} y + \frac{y^{m + 1}}{m + 1} = 0, N ⩾ 1,$ provided that m ≠ −1. The constant of integration is set to be zero. Suppose y(z) is a

Solitary wave solutions of the generalized Burgers equation

The first member of family (1.1) is the generalized Burgers equation: $u_{t} + α u_{xx} + u^{m} u_{x} = 0, α \neq 0 .$ This equation possesses the traveling wave reduction (2.1) with y(z) satisfying the equation ${ay}_{z} - C_{0} y + \frac{1}{m + 1} y^{m + 1} = 0 .$ Eq. (3.2) is the Bernulli equation. The equalities $\begin{matrix} w = \frac{y_{z}}{y}, \\ y^{m} = (m + 1) {C_{0} - α w}, \end{matrix}$ provide Bäckhand transformations between Bernulli equation (3.2) and the Riccati equation: $α w_{z} + m α w^{2} - {mC}_{0} w = 0 .$ Following the procedure suggested in the previous section we find the coefficients a and b: $a = - \frac{C_{0}}{2 α}, b = \frac{m^{2} C_{0}^{2}}{4 α^{2}} .$ The general

Solitary wave solutions of the generalized Korteweg–de Vries–Burgers equation

Let us look for exact solutions of the generalized Korteweg–de Vries–Burgers equation: $u_{t} - α u_{xx} + β u_{xxx} + u^{m} u_{x} = 0, β \neq 0 .$ Using traveling wave reduction (2.1), we obtain the following second-order ordinary differential equation: $β y_{zz} - α y_{z} - C_{0} y - \frac{1}{m + 1} y^{m + 1} = 0, β \neq 0$ The Bäcklund transformations: $\begin{matrix} w = \frac{yz}{y}, \\ y^{m} = (m + 1) {C_{0} + α w - β (w_{z} + w^{2})}, \end{matrix}$ relate solutions of Eq. (4.2) and of the equation: $β w_{zz} - (m - 2) β {ww}_{z} - α w_{z} - m β w^{3} + m α w^{2} + {mC}_{0} w = 0 .$ The latter equation possesses solutions of the type (2.10) with $a = \frac{α}{β (m + 4)}, b = \frac{α^{2} m^{2}}{4 β^{2} (m + 4)^{2}},$ provided that: $C_{0} = - 2 α^{2}$

Solitary wave solutions of the generalized Kuramoto–Sivashinsky equation

In this section, we will construct exact solutions of the generalized Kuramoto–Sivashinsky equation: $u_{t} + α u_{xx} + β u_{xxx} + γ u_{xxxx} + u^{m} u_{x} = 0, γ \neq 0 .$ This equation possesses the traveling wave reduction (2.1) with y(z) satisfying the equation: $γ y_{zzz} + β y_{zz} + α y_{z} - C_{0} y + \frac{1}{m + 1} y^{m + 1} = 0 .$ The Bäcklund transformations: $\begin{matrix} w = \frac{yz}{y}, \\ y^{m} = (m + 1) {C_{0} - α w - β (w_{z} + w^{2}) - γ (w_{zz} + 3 {ww}_{z} + w^{3})}, \end{matrix}$ relate solutions of this equation and of the equation: $γ w_{zzz} - γ {mww}_{zz} + 3 γ {ww}_{zz} + 3 {gw}^{2} w_{z} - 3 γ {mw}^{2} w_{z} + 3 γ w_{z}^{2} + 2 β {ww}_{z} + β w_{zz} - {mw}^{4} γ - {mw}^{3} β - β {mww}_{z} - m α w^{2} + α w_{z} + {mwC}_{0} = 0 .$ Following the procedure

Solitary wave solutions of the fifth-order equation

In this section, we consider the following nonlinear fifth-order evolution equation: $u_{t} + α u_{xx} + β u_{xxx} + γ u_{xxxx} + δ u_{xxxxx} + u^{m} u_{x} = 0, δ \neq 0 .$ Again making the substitution (2.1) in (6.1) and integrating the result one gets $δ y_{zzzz} + γ y_{zzz} + β y_{zz} + α y_{z} + \frac{1}{m + 1} y^{m + 1} - C_{0} y = 0 .$ Eq. (2.7) for the function w(z) can be written as $δ w_{zzzz} + 4 δ {ww}_{zzz} - δ {mww}_{zzz} + 4 δ w^{3} w_{z} + 12 δ {ww}_{z}^{2} + 10 δ w_{z} w_{zz} - 4 δ {mw}^{2} w_{zz} - δ {mw}^{5} - 6 δ w^{3} {mw}_{z} + 6 δ w^{2} w_{zz} + 3 γ {ww}_{zz} - 3 δ {ww}_{z}^{2} m + 2 β {ww}_{z} - γ {mww}_{zz} - 3 γ {mw}^{2} w_{z} + 3 γ w^{2} w_{z} - α {mw}^{2} + α w_{z} - {mw}^{3} β - δ γ {mw}^{4} - β {mww}_{z} + 3 γ {wz}^{2} + γ w_{zzz} + β w_{zz} + {mC}_{0} w = 0 .$ This equation in the case m ≠ −8/3

Conclusion

In this paper, we have introduced an algorithm of finding traveling wave solutions for a family of nonlinear partial differential equation (1.1). This family contains a number of equations with mathematical and physical applications. The algorithm is based on constructing Bäcklund transformations between the traveling wave reductions of (1.1) and ordinary differential equations more simple for further analysis.

We would like to note that the same procedure can be applied to the equation $P_{α_{1}, \dots, α_{N}}^{(}$

Acknowledgement

This work was supported by the International Science and Technology Center under Project B 1213.

References (33)

D. Michelson
Elementary particles as solutions of the Sivashinsky equation
Physica D
(1990)
D. Lu et al.
New solitary wave and periodic wave solutions for general types of KdV and KdV–Burgers equations
Commun. Nonlinear Sci. Numer. Simul.
(2009)
Y.Z. Peng
Exact travelling wave solutions for the Zakharov–Kuznetsov equation
Appl. Math. Comput.
(2008)
M. Qin et al.
An effective method for finding special solutions of nonlinear differential equations with variable coefficients
Phys. Lett. A
(2008)
N.A. Kudryashov
Exact solutions of the generalized Kuramoto–Sivashinsky equation
Phys. Lett. A
(1990)
N.A. Kudryashov
On types of nonlinear nonintegrable equations with exact solutions
Phys. Lett. A
(1991)
S. Zhang
New exact solutions of the KdV–Burgers–Kuramoto equation
Phys. Lett. A
(2006)
S.A. Khuri
Traveling wave solutions for nonlinear differential equations: a unified ansatse approach
Chaos Solitons Fract.
(2007)
J. Nickel
Traveling wave solutions to the Kuramoto–Sivashinsky equation
Chaos Solitons Fract.
(2007)
N.A. Kudryashov
Exact solitary waves of the Fisher equation
Phys. Lett. A
(2005)

N.A. Kudryashov

Simplest equation method to look for exact solutions of nonlinear differential equations

Chaos Solitons Fract.

(2005)

N.A. Kudryashov et al.

Polygons of differential equation for finding exact solutions

Chaos Solitons Fract.

(2007)

N.A. Kudryashov et al.

Extended simplest equation method for nonlinear differential equations

Appl. Math. Comput.

(2008)

D.J. Korteweg et al.

On the change of form of long waves advancing in a rectangular canal and on a new type of long stationary waves

Philos. Mag.

(1895)

N.J. Zabuski et al.

Integration of “solitons” in a collisionless plasma and the recurrence of initial states

Phys. Rev. Lett.

(1965)

C.S. Gardner et al.

Method for solving the Korteweg–de Vries equations

Phys. Rev. Lett.

(1967)

Cited by (45)

Breather-type and multi-wave solutions for (2+1)-dimensional nonlocal Gardner equation
2021, Applied Mathematics and Computation
In this work, different kinds of solutions including breather-type and multi-wave solutions are obtained for the $(2 + 1)$ -dimensional Gardner equation by using bilinear form, the extended homoclinic test approach and three-wave method. We obtained the coefficient conditions in solution ansatz for the existing of breather and multi-wave solutions. By selecting appropriate values of the parameter, three dimensional, contour and density plots of solutions are drawn in order to better understand the dynamic behaviors of considered physical phenomena.
Classifying algebraic invariants and algebraically invariant solutions
2020, Chaos, Solitons and Fractals
A concept of algebraic invariants and algebraically invariant solutions for autonomous ordinary differential equations and systems of autonomous ordinary differential equations is considered. A variety of known exact solutions of autonomous ordinary differential equations and a great number of traveling wave solutions of famous partial differential equations are in fact algebraically invariant solutions. A method, which can be used to find all irreducible algebraic invariants, is introduced. As an example, all irreducible algebraic invariants for the traveling wave reductions of the dispersive Kuramoto–Sivashinsky equation and the modified Kuramoto–Sivashinsky equation are classified. Novel solutions of the modified Kuramoto–Sivashinsky equation are obtained.
Investigating the tangent dispersive solitary wave solutions to the Equal Width and Regularized Long Wave equations
2020, Journal of King Saud University - Science
Citation Excerpt :
Nonlinear partial differential equations play important roles in modeling varieties of physical problems (Wazwaz, 2009). Such equations including the known evolution equations (Kudryashov and Demina, 2009; Malfliet, 2004; Novikov and Veselov, 1986) occur frequently in many nonlinear sciences. Further, the Equal Width (EW) and Regularized Long Wave (RLW) equations being important class of evolution equations happen to take parts in optics, propagation of various waves, transmission of nonlinear waves with dispersion processes and in many branches of nonlinear sciences, see Hamdi et al. (2003), Evans and Raslan (2005), Lu et al. (2018), Fan (2012), Korkmaz (2016), Morrison et al. (1984) and Ramos (2007).
In this paper, we analytically construct certain dispersive solitary wave solutions to the Equal Width (EW) and Regularized Long Wave (RLW) equations using the Modified Extended Tanh Expansion Method. The study also analyze the effect of $U_{x}$ being the major difference between the two equations after restricting METEM to only tangent function solutions for one-to-one comparison. The Mathematica software is used for the computations as well as the graphical illustrations, respectively.
The effect of perturbed advection on a class of solutions of a non-linear reaction-diffusion equation
2016, Applied Mathematics and Computation
In this work, the traveling wave solutions of a one-dimensional reaction-diffusion equation with advection are studied. The traveling wave solutions are obtained using the G′/G-expansion method. The shock thickness and spectral stability have been discussed for the obtained solution in the parameter interval. The essential spectra of the perturbed and linearized differential operator about the traveling antikink and kink solutions at the equilibrium states are obtained. The point spectrum is calculated using Evans function with Lie midpoint method and Magnus method. It is shown that, for a symmetric potential well, the traveling kink and antikink solutions which connect the stable equilibrium states of the system are stable. It is observed that the perturbation on the advection exhibits contrasting effect on the solution properties (shock thickness and the eigenvalue) of kink and antikink solutions. Variation of the reaction coefficient leads to instability of the solutions, unlike the diffusion coefficient which enhances the stability. On the other hand, the variation of reaction and diffusion coefficients show the monotonic effect on the shock thickness of the traveling kink and antikink solutions. This study is expected to be useful in analyzing the slow or fast invasion and stability of the population movement in different steady states.
Analytical and numerical solutions of the generalized dispersive Swift-Hohenberg equation
2016, Applied Mathematics and Computation
The generalization of the Swift–Hohenberg equation is studied. It is shown that the equation does not pass the Kovalevskaya test and does not possess the Painlevé property. Exact solutions of the generalized Swift–Hohenberg equation which are very useful to test numerical algorithms for various boundary value problems are obtained. The numerical algorithm which is based on the Crank–Nicolson–Adams–Bashforth scheme is developed. This algorithm is tested using the exact solutions. The selforganization processes described by the generalization of the Swift–Hohenberg equation are studied.
On integrability and quasi-periodic wave solutions to a (3+1)-dimensional generalized KdV-like model equation
2016, Applied Mathematics and Computation
Citation Excerpt :
It is well known that the integrable properties and exact solutions of nonlinear evolutions equations (NLEEs) play a vital role in the study of nonlinear phenomena and mathematics. There are a lot of methods to look for special solutions of NLEEs, such as Lie group [1], Hirota bilinear method [2], Darboux transformation [3], the inverse scattering transformation [4] and so on [5–9]. The bilinear method developed by Hirota offers a canonical way to investigate the integrability and exact solutions of nonlinear equations.
Under investigation in this paper is the integrability of a (3+1)-dimensional generalized KdV-like model equation, which can be reduced to several integrable equations. With help of Bell polynomials, an effective method is presented to succinctly derive the bilinear formalism of the equation, based on which, the soliton solutions and periodic wave solutions are also constructed by using Riemann theta function. Furthermore, the Bäcklund transformation, Lax pairs, and infinite conservation laws of the equation can easily be derived, respectively. Finally, the relationship between periodic wave solutions and soliton solutions are systematically established. It is straightforward to verify that these periodic waves tend to soliton solutions under a small amplitude limit.

View all citing articles on Scopus

View full text

Traveling wave solutions of the generalized nonlinear evolution equations

Abstract

Introduction

Section snippets

Method applied

Solitary wave solutions of the generalized Burgers equation

Solitary wave solutions of the generalized Korteweg–de Vries–Burgers equation

Solitary wave solutions of the generalized Kuramoto–Sivashinsky equation

Solitary wave solutions of the fifth-order equation

Conclusion

Acknowledgement

Physica D

Commun. Nonlinear Sci. Numer. Simul.

Appl. Math. Comput.

Phys. Lett. A

Phys. Lett. A

Phys. Lett. A

Phys. Lett. A

Chaos Solitons Fract.

Chaos Solitons Fract.

Phys. Lett. A

Chaos Solitons Fract.

Chaos Solitons Fract.

Appl. Math. Comput.

On the change of form of long waves advancing in a rectangular canal and on a new type of long stationary waves

Philos. Mag.

Integration of “solitons” in a collisionless plasma and the recurrence of initial states

Phys. Rev. Lett.

Method for solving the Korteweg–de Vries equations

Phys. Rev. Lett.