Exact solitary wave and quasi-periodic wave solutions of the KdV-Sawada-Kotera-Ramani equation
- Lijun Zhang^{1, 2}Email author and
- Chaudry Masood Khalique^{2}
https://doi.org/10.1186/s13662-015-0510-y
© Zhang and Khalique 2015
Received: 30 December 2014
Accepted: 17 May 2015
Published: 26 June 2015
Abstract
In this paper we derive new exact solitary wave solutions and quasi-periodic traveling wave solutions of the KdV-Sawada-Kotera-Ramani equation by using a method which we introduce here for the first time. Firstly, we reduce the associated fourth-order nonlinear ordinary differential equation (ODE) into a solvable first-order nonlinear ODE to obtain new exact traveling wave solutions, including the solitary wave and periodic solutions. Furthermore, using the new method we derive the quasi-periodic wave solutions of this equation by assuming that the solutions of the corresponding higher-order ODE are the sum of the solutions of two solvable first-order nonlinear ODEs. This new method can be used to investigate the exact traveling wave solutions and quasi-periodic wave solutions of a general class of higher-order wave equations.
Keywords
1 Introduction
There are two classes of solitary waves, namely, embedded solitons and gas solitons that have been studied by many researches in the fields of nonlinear optics and water wave theory [2, 17–20]. In fact, soliton solutions are typically presented by homoclinic solutions to saddle-center equilibrium and saddle-saddle equilibrium, respectively, of the associated ODEs which describe traveling waves of the model PDEs. By using the method of dynamical systems and Congrove’s results [21], Li and Zhang [1] investigated the exact explicit gap soliton, embedded soliton, periodic, and quasi-periodic wave solutions of the KdV-Sawada-Kotera-Ramani equation.
Recently by using the sub-equation method and dynamical system analysis, the bifurcation and exact solutions of (1.4) were studied in [22, 23]. Following the idea and the results in [22], the bifurcations and exact traveling wave solutions to the KdV-Sawada-Kotera-Ramani equation are obtained in Section 1. The sub-equation method has been proposed and well applied in studying the exact solutions of nonlinear differential equations [6–11]. The main idea of the sub-equation method is to assume that the solutions to higher-order ODEs are polynomials of some functions satisfying a simpler equation. However, we observe that a family of solutions to (1.3) can be the sum of two solutions to a second-order ODE which can be reduced to a first-order nonlinear ODE. By using the exact solutions and bifurcations of this sub-equation which was derived in [23], some new traveling wave solutions and quasi-periodic traveling wave solutions of the KdV-Sawada-Kotera-Ramani equation are derived in Section 2.
2 A family of exact traveling wave solutions of the KdV- Sawada-Kotera-Ramani equation
2.1 Preliminaries
Equation (1.3) is a special case of (1.4), which is (1.6) in [22] with \(C=0\). Thus, from Theorem 2.1 and Theorem 2.2 in [22], we have the corresponding theorems regarding (1.3).
Theorem 2.1
Note that all the denominators in (2.2) are assumed to be nonzero. If the denominator of \(a_{i}\) in (2.2) is zero, then \(a_{i}\) can be arbitrary constant provided the numerator is also zero.
Theorem 2.2
Let \(h_{\pm}=\frac{2\Delta(-a_{2}\pm\sqrt{\Delta })+3a_{1}a_{2}a_{3}}{54a_{3}^{2}}\) and \(y_{e}^{\pm}=\frac{-a_{2}\pm\sqrt {\Delta}}{3a_{3}}\), where \(\Delta=a_{2}^{2}-3a_{1}a_{3}>0\), then the following conclusions hold:
2.2 A family of exact traveling wave solutions of (1.1) obtained from Theorems 2.1 and 2.2
By letting \(A=15\), \(B={a}/{b}\), \(D=15\), \(E=3{a}/{b}\), \(F=-{c}/{b}\), and \(G=g\), (1.3) is reduced to (1.2). Clearly \(D=15<3{A^{2}}/{40}\). Thus from Theorem 2.1, we know that \(y=y(\xi)\) solves (1.2) if it solves the first-order nonlinear ODE (2.1) with \(a_{3}=-1\), \(a_{2}=- {a}/(5b)\), \(a_{1}=(4a^{2}+25bc)/(75b^{2})\), and \(a_{0}=(375b^{3}g+8a^{3}+50abc)/(1{,}125b^{3})\). Note that g in (1.2) is an arbitrary constant, so \(a_{0}\) is also an arbitrary constant. Consequently, we obtain the solutions of (1.2) from Theorem 2.2 provided \(a_{2}^{2}-3a_{1}a_{3}=(a^{2}+5bc)/(5b^{2})>0\), i.e., \(c>-a^{2}/(5b)\) for \(b>0\) or \(c<- {a^{2}}/(5b)\) for \(b<0\).
Theorem 2.3
The KdV-Sawada-Kotera-Ramani equation (1.1) has the following traveling wave solutions with the wave speed c satisfying \(a^{2}+5bc>0\):
3 Exact quasi-periodic traveling wave solutions of the KdV- Sawada-Kotera-Ramani equation
In this section, we obtain a new family of exact traveling wave solutions of the KdV-Sawada-Kotera-Ramani equation (1.1), which includes the quasi-periodic solutions.
3.1 A new family of exact solutions and quasi-periodic solutions of (1.3)
Solving the first equation of system (3.5) for D gives \(D=A^{2}/15\) and thus \(a_{3}=-2A/15\). By substituting the value of \(a_{3}\) into the second equation and solving for E, we have \(E= AB/5\) and \(a_{2}=-{B}/{5}\). Then substituting the values of \(a_{3}\) and \(a_{2}\) into the third equation of (3.5) gives \(F-F_{2}+5F_{1}=4B^{2}/5\). In the same way, from system (3.3), we can obtain \(F-F_{1}+5F_{2}=4B^{2}/5\). Consequently, we can determine the two undetermined constants \(F_{1}\) and \(F_{2}\) as \(F_{1}=F_{2}= B^{2}/5- F/4\). We thus have the following theorem.
Theorem 3.1
3.2 Exact solutions and quasi-periodic solutions of the KdV-Sawada-Kotera-Ramani equation (1.1)
According to the above analysis and Theorem 3.1, we obtain the solutions of (1.2) and consequently the exact traveling wave solutions of the KdV-Sawada-Kotera-Ramani equation (1.1). We have the following theorem.
Theorem 3.2
The KdV-Sawada-Kotera-Ramani equation (1.1) admits the following traveling wave solutions: \(y_{ij}(x, t)=\phi_{i}(\xi, c)+\phi_{j}(\xi, c)\), \(i, j\in\{1,2,3,4,5,6\}\), with \(\xi=x-ct\). Here \(\phi_{i}\) s are determined by (3.10)-(3.15), respectively, and the wave speed c satisfies \(a^{2}+5bc>0\).
In fact, from Theorem 3.2, we can obtain the following three classes of solitary wave solutions; two families of periodic wave solutions, a family of quasi-periodic wave solutions, and some unbounded solutions.
4 Conclusion and discussion
In this paper, we studied the exact traveling wave solutions to the KdV-Sawada-Kotera-Ramani equation (1.1) via the sub-equation in the form \(({dy}/{d\xi})^{2}=a_{3}y^{3}+a_{2}y^{2}+a_{1}y+a_{0}\). The sub-equation of similar form, namely \(({dy}/{d\xi})^{2}=P_{m}(y)\), where \(P_{m}(y)\) is a polynomial of y, has been applied to investigate some nonlinear wave equations [4–11]. In all these papers, the solutions to the original equations are usually the polynomial functions of the solutions to the sub-equations. However, by using the new method introduced in this paper (the sum of two solutions to sub-equations), we obtained many new exact traveling wave solutions to the KdV-Sawada-Kotera-Ramani equation (1.1). Especially, some quasi-periodic wave solutions were derived by using this new method. Furthermore, we obtained a very general class of exact solutions of the KdV-Sawada-Kotera-Ramani equation (1.1), which included the solitary wave solutions, periodic and quasi-periodic traveling wave solutions and some unbounded traveling solutions as well. Our results are more general than those obtained previously in the literature. For example, the solutions (12) and (14) in [16] actually can be rewritten as our solutions (3.16) and (3.27), respectively. Unfortunately, (18) and (20) in [16] do not satisfy (1.1) and hence are not the solutions of (1.1).
It is well known that not only the exact solutions but also the bifurcations of the dynamical systems can be investigated by using the dynamical system theorem [24, 25]. The planar dynamical system method has been well applied in studying the traveling wave solutions of various nonlinear wave solutions [1, 22, 23, 26–31]. However, it is usually very difficult to study the systems in a higher-dimensional space unless they can be reduced to a two-dimensional space. Normally, the higher-order differential equations can be reduced to a lower-dimensional space provided that their first integrals can be derived [1, 21, 28]. Unfortunately, it is usually intractable to derive their first integrals. In this paper, we reduced the higher-order ODE into planar dynamical system by finding its lower-order sub-equation. Whether there are any other kinds of sub-equations possessed by this class of equations is still an open problem.
The method proposed in this paper can be applied to other nonlinear wave equations, especially to higher-order nonlinear wave equations. This might pave the way to the study of the exact traveling wave solutions of higher-order nonlinear wave equations. However, whether and how this method can be used to investigate the multiple-wave solutions of higher-order nonlinear wave equations will be the topic of our future study.
Declarations
Acknowledgements
L Zhang thanks the North-West University for the post-doctoral fellowship. This work is supported by the Nature Science Foundation of China (No. 11101371, No. 11422214).
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
References
- Li, JB, Zhang, Y: Homoclinic manifolds, center manifolds and exact solutions of four-dimensional traveling wave systems for two classes of nonlinear wave equations. Int. J. Bifurc. Chaos 21(2), 527-543 (2011) MATHView ArticleGoogle Scholar
- Yang, J: Dynamics of embedded solitons in the extended Korteweg-de Vries equations. Stud. Appl. Math. 106, 337-365 (2001) MATHMathSciNetView ArticleGoogle Scholar
- Inc, M, Kilic, B: Classification of traveling wave solutions for time-fractional fifth-order KdV-like equation. Waves Random Complex Media 24(2), 393-403 (2014) MathSciNetView ArticleGoogle Scholar
- Inc, M: Some special structures for the generalized nonlinear Schrödinger equation with nonlinear dispersion. Waves Random Complex Media 23(2), 77-88 (2013) MathSciNetView ArticleGoogle Scholar
- Kilic, B, Inc, M: The first integral method for the time fractional Kaup-Boussinesq system with time dependent coefficient. Appl. Math. Comput. 254, 70-74 (2015) MathSciNetView ArticleGoogle Scholar
- Ma, WX, Lee, JH: A transformed rational function method and exact solutions to the \((3+1)\)-dimensional Jimbo-Miwa equation. Chaos Solitons Fractals 42(3), 1356-1363 (2009) MATHMathSciNetView ArticleGoogle Scholar
- Zhang, S, Xia, T: A generalized new auxiliary equation method and its applications to nonlinear partial differential equations. Phys. Lett. A 363, 356-360 (2007) MATHMathSciNetView ArticleGoogle Scholar
- Li, H, Wang, KM, Li, JB: Exact traveling wave solutions for the Benjamin-Bona-Mahony equation by improved Fan sub-equation method. Appl. Math. Model. 37, 7644-7652 (2013) MathSciNetView ArticleGoogle Scholar
- Parkes, EJ, Duffy, BR, Abbott, PC: The Jacobi elliptic-function method for finding periodic-wave solutions to nonlinear evolution equations. Phys. Lett. A 295, 280-286 (2002) MATHMathSciNetView ArticleGoogle Scholar
- Feng, DH, Li, KZ: Exact traveling wave solutions for a generalized Hirota-Satsuma coupled KdV equation by Fan sub-equation method. Phys. Lett. A 375, 2201-2210 (2011) MATHMathSciNetView ArticleGoogle Scholar
- Fan, E: Uniformly constructing a series of explicit exact solutions to nonlinear equations in mathematical physics. Chaos Solitons Fractals 16, 819-839 (2003) MATHMathSciNetView ArticleGoogle Scholar
- Fan, E: Supersymmetric KdV-Sawada-Kotera-Ramani equation and its quasi-periodic wave solutions. Phys. Lett. A 374, 744-749 (2010) MATHMathSciNetView ArticleGoogle Scholar
- Hirota, R, Ito, M: Resonance of solitons in one dimension. J. Phys. Soc. Jpn. 52(3), 744-748 (1983) MathSciNetView ArticleGoogle Scholar
- Konno, K: Conservation laws of modified Sawada-Kotera equation in complex plane. J. Phys. Soc. Jpn. 61, 51-54 (1992) MathSciNetView ArticleGoogle Scholar
- Zhang, J, Zhang, J, Bo, L: Abundant travelling wave solutions for KdV-Sawada-Kotera equation with symbolic computation. Appl. Math. Comput. 203, 233-237 (2008) MATHMathSciNetView ArticleGoogle Scholar
- Qin, Z, Mu, G, Ma, H: \(G'/G\)-Expansion method for the fifth-order forms of KdV-Sawada-Kotera equation. Appl. Math. Comput. 222, 29-33 (2013) MathSciNetView ArticleGoogle Scholar
- Champneys, AR, Groves, MD, Woods, PD: A global characterization of gap solitary wave solutions to a coupled KdV system. Phys. Lett. A 271, 178-190 (2000) MathSciNetView ArticleGoogle Scholar
- Champneys, AR, Malomed, BA, Yang, J, Kaup, DJ: Embedded solitons: solitary wave in resonance with the linear spectrum. Physica D 152, 340-354 (2001) MathSciNetView ArticleGoogle Scholar
- Yang, J, Malomed, BA, Kaup, DJ, Champneys, AR: Embedded solitons: a new type of solitary wave. Math. Comput. Simul. 56, 585-600 (2001) MATHMathSciNetView ArticleGoogle Scholar
- Yagasaki, K, Wagenknecht, T: Detection of symmetric homoclinic orbits to saddle-centres in reversible systems. Physica D 214, 169-181 (2006) MATHMathSciNetView ArticleGoogle Scholar
- Cosgrove, MC: Higher-order Painlevé equations in the polynomial class I. Bureau symbol P2. Stud. Appl. Math. 104, 1-65 (2000) MATHMathSciNetView ArticleGoogle Scholar
- Zhang, LJ, Khalique, CM: Exact Solitary wave and periodic wave solutions of a class of higher-order nonlinear wave equations. (submitted) Google Scholar
- Zhang, LJ, Khalique, CM: Exact solitary wave and periodic wave solutions of the Kaup-Kuperschmidt equation. J. Appl. Anal. Comput. 5(3), 485-495 (2015) Google Scholar
- Guckenheimer, J, Holmes, P: Dynamical Systems and Bifurcations of Vector Fields. Springer, New York (1983) MATHView ArticleGoogle Scholar
- Chow, SN, Hale, JK: Method of Bifurcation Theory. Springer, New York (1981) Google Scholar
- Li, JB: Singular Traveling Wave Equations: Bifurcations and Exact Solutions. Science Press, Beijing (2013) Google Scholar
- Li, JB, Chen, GR: Bifurcations of traveling wave solutions for four classes of nonlinear wave equations. Int. J. Bifurc. Chaos 15(12), 3973-3998 (2005) MATHView ArticleGoogle Scholar
- Li, JB, Qiao, ZJ: Explicit solutions of the Kaup-Kuperschmidt equation through the dynamical system approach. J. Appl. Anal. Comput. 1(2), 243-250 (2011) MATHMathSciNetGoogle Scholar
- Zhang, LJ, Chen, LQ, Huo, XW: The effects of horizontal singular straight line in a generalized nonlinear Klein-Gordon model equation. Nonlinear Dyn. 72(4), 789-801 (2013) MATHMathSciNetView ArticleGoogle Scholar
- Rui, WG: Different kinds of exact solutions with two-loop character of the two-component short pulse equations of the first kind. Commun. Nonlinear Sci. Numer. Simul. 18, 2667-2678 (2013) MATHMathSciNetView ArticleGoogle Scholar
- Chen, AY, Wen, SQ, Huang, WT: Existence and orbital stability of periodic wave solutions for the nonlinear Schrödinger equation. J. Appl. Anal. Comput. 2(2), 137-148 (2012) MATHMathSciNetGoogle Scholar