- Research
- Open Access
Symmetry reduction and exact solutions of two higher-dimensional nonlinear evolution equations
- Yongyi Gu^{1}Email authorView ORCID ID profile and
- Jianming Qi^{2}
https://doi.org/10.1186/s13660-017-1587-5
© The Author(s) 2017
- Received: 1 August 2017
- Accepted: 12 December 2017
- Published: 21 December 2017
Abstract
In this paper, symmetries and symmetry reduction of two higher-dimensional nonlinear evolution equations (NLEEs) are obtained by Lie group method. These NLEEs play an important role in nonlinear sciences. We derive exact solutions to these NLEEs via the \(\exp(-\phi (z))\)-expansion method and complex method. Five types of explicit function solutions are constructed, which are rational, exponential, trigonometric, hyperbolic and elliptic function solutions of the variables in the considered equations.
Keywords
- nonlinear evolution equations
- symmetry
- \(\exp(-\phi(z))\)-expansion method
- complex method
- exact solutions
- meromorphic function
MSC
- 30D35
- 34A05
1 Introduction
We note that equation (4) includes many famous NLEEs as its special cases. For instance, if \(a_{1}=a_{3}=a_{4}=a_{6}=0\), then equation (4) is the Korteweg-de Vries equation [2, 3]. If \(a_{2}=a_{4}=a_{5}=0\), then equation (4) is the \((2+1)\) dimensional ZK-MEW equation [4]. If \(a_{3}=a_{4}=a_{6}=0\), then equation (4) is the Gardner equation [5]. If \(a_{4}=a_{5}=a_{6}=0\), then equation (4) is the modified Zakharov-Kuznetsov equation [6].
In recent years, it has aroused widespread interest in the study of NLEEs [7–13]. Equations (3) and (4) are very meaningful higher-dimensional NLEEs which can describe many dynamic processes and important phenomena in engineering and physics. The YTSF equation is a mostly used model for investigating the dynamics of solitons and nonlinear waves in fluid dynamics, plasma physics and weakly dispersive media [13]. Zakharov and Kuznetsov [14] proposed the ZK equation to describe nonlinear ion-acoustic waves in a plasma comprised of cold ions and hot isothermal electrons in the presence of a uniform magnetic field. Many physical phenomena, in the purely dispersive limit, are governed by this type of equation, such as the long waves on a thin liquid film [15], the Rossby waves in a rotating atmosphere [16], and the isolated vortex of drift waves in a three-dimensional plasma [17]. The gZK equation is of a generalized setting of ZK equation. Seeking exact solutions of NLEEs is an interesting and significant subject. Over the past few years, many powerful methods for constructing the solutions of NLEEs have been used, for instance, the Bäcklund transform method [18], direct algebraic method [19], modified simple equation method [20], Lie group method [21, 22], \(\exp(-\phi(z))\)-expansion method [8, 9, 23, 24], and so on. Recently, Yuan et al. [25–27] introduced the complex method to find the exact solutions of NLEEs in mathematical physics. In this paper, we study symmetries, symmetry reduction of the two higher-dimensional NLEEs, and then we obtain their exact solutions via the \(\exp(-\phi(z))\)-expansion method and complex method.
2 Description of the methods
2.1 Description of the \(\exp(-\phi(z))\)-expansion method
Step 3. Substituting equation (7) into equation (6) and accounting the function \(\exp(-\phi(z))\), we obtain a polynomial of \(\exp(-\phi(z))\). Equating all the coefficients of the same power of \(\exp(-\phi(z))\) to zero yields a set of algebraic equations. By solving the algebraic equations, we get the values of \(C_{n}\neq0\), δ, μ, and then we substitute them into equation (7) along with equations (9)-(15) to complete the determination of the solutions of equation (5).
2.2 Description of the complex method
Set \(p, q\in{\mathbb {N}}^{*}\), and the meromorphic solutions w of equation (16) have at least one pole. If equation (16) has exactly p distinct meromorphic solutions, and their multiplicity of the pole at \(z=0\) is q, then equation (16) is said to satisfy the \(\langle p,q \rangle\) condition. It might not be easy to show that the \(\langle p,q \rangle\) condition of equation (16) holds, so we need the weak \(\langle p,q \rangle \) condition as follows.
If a meromorphic function g is a rational function of z, or a rational function of \(e^{\alpha z}\), \(\alpha\in{\mathbb {C}}\), or an elliptic function, then we say that g belongs to the class W.
Lemma 2.1
By the above definitions and lemma, we now present the complex method.
Step 1. Insert the transformation \(T: u(x,y,t)\rightarrow w(z)\) defined by \((x,y,t)\rightarrow z \) into a given PDE to yield a nonlinear ODE.
Step 2. Insert (17) into the ODE to determine whether the weak \(\langle p,q \rangle\) condition holds.
Step 3. Insert the indeterminate solutions introduced in Lemma 2.1 into the ODE, and then get meromorphic solutions of the ODE with a pole at \(z=0\).
Step 4. Obtain meromorphic solutions \(w(z-z_{0})\) by Lemma 2.1 and the addition formula.
Step 5. Inserting the inverse transformation \(T^{-1}\) into the meromorphic solutions, we get the exact solutions for the original PDE.
3 Symmetries and symmetry reduction
3.1 Symmetries
3.2 Symmetry reduction
4 Exact solutions
4.1 Exact solutions of gZK equation via the \(\exp(-\phi (z))\)-expansion method
4.2 Exact solutions of gZK equation via the complex method
Inserting (17) into equation (39) we have \(p=2\), \(q=1\), \(\beta_{{-1}}=\pm\sqrt{\frac{-6 ( a_{2} {k}^{2}+a_{3} {k}^{2}+2 {l}^{2} )}{a_{1}}}\), \(\beta_{{0}}=-\frac{a_{5}}{2a_{1}}\), \(\beta_{{1}}=-\frac{a_{5}^{2}}{24a_{1}^{2}}\sqrt {\frac{-6a_{1}}{ a_{2} {k}^{2}+a_{3} {k}^{2}+2 {l}^{2}}}\), \(\beta _{{2}}=-\frac{12a_{1}^{2}\gamma-a_{5}^{3}+6a_{1}a_{5}}{48a_{1}^{2}( a_{2} {k}^{2}+a_{3} {k}^{2}+2 {l}^{2})}\) and \(\beta_{3}\) is an arbitrary constant.
Therefore, equation (39) is a second order BBEq and satisfies the weak \(\langle2,1 \rangle\) condition. Hence, by Lemma 2.1, we see that meromorphic solutions of equation (39) belong to W. We will show meromorphic solutions of equation (39) in the following.
4.3 Exact solutions of YTSF equation via the \(\exp(-\phi (z))\)-expansion method
4.4 Exact solutions of YTSF equation via the complex method
Inserting (17) into equation (47) we have \(p=1\), \(q=2\), \(\beta_{{-2}}=-2k\), \(\beta_{{-1}}=0\), \(\beta_{{0}}=-\frac {4lk+4k^{2}+3r^{2}}{6k^{2}l}\), \(\beta_{{1}}=0\), \(\beta_{{2}}=-\frac {16k^{4}+32lk^{3}+(16l^{2}-12l\gamma+24r^{2})k^{2}+24lkr^{2}+9r^{4}}{120k^{5}l^{2}}\), and \(\beta_{3}\) is an arbitrary constant.
Therefore, equation (47) is a second order BBEq and satisfies the weak \(\langle1,2 \rangle\) condition. Hence, by Lemma 2.1, we see that meromorphic solutions of equation (47) belong to W. We will show meromorphic solutions of equation (47) in the following.
4.5 Comparison
Implementing the \(\exp(-\phi(z))\)-expansion method, we found seven solutions for the gZK and YSFT equation, respectively. Using the complex method, we found five solutions for the gZK equation and three solutions for the YSFT equation. Rational solutions \(w_{17}(z)\) and \(w_{27}(z)\) are obtained via the \(\exp(-\phi(z))\)-expansion method, and \(W_{r,1}(z)\) and \(W_{r}(z)\) are obtained via the complex method. If we let \(c=-z_{0}\), then \(w_{17}(z)\) is equivalent to \(W_{r,1}(z)\), and \(w_{27}(z)\) is equivalent to \(W_{r}(z)\). For getting rational solutions, these two methods are in good agreement. Rational solutions \(W_{r,2}(z)\) and simply periodic solutions \(W_{s,2}(z)\) and \(W_{s}(z)\) are new and cannot be degenerated successively through elliptic function solutions. From the results, we can find more solutions by the \(\exp(-\phi(z))\)-expansion method, whereas we can obtain elliptic function solutions just by the complex method. These two methods are very useful tools in finding the exact solutions of NLEEs.
5 Computer simulations
- (1)By employing the complex method, we are capable to obtain simply periodic solutions \(W_{s,1}(z)\) and \(W_{s,2}(z)\) of the gZK equation. The solutions \(W_{s,1}(z)\) and \(W_{s,2}(z)\) come from hyperbolic function. Figure 1 shows the shape of solutions \(W_{s,2}(z)\) for \(k=1\), \(l=1\), \(\alpha=1\), \(a_{1}=-6\), \(a_{2}=1\), \(a_{3}=1\), \(a_{5}=-24\), and \(z_{1}=1\) within the interval \(-2\pi\leq\xi, \eta\leq2\pi\). Note that they have two distinct generation poles which are showed by Figure 1.
- (2)By using the complex method, we achieve to obtain simply periodic solutions \(W_{s}(z)\) of the YSTF equation. The solutions \(W_{s}(z)\) are in terms of the hyperbolic function solution. The solutions \(W_{s}(z)\) in Figure 2 of the YSTF equation are represented the singular soliton solution for the parameters \(k=1\), \(l=1\), \(r=1\), \(\alpha=1\) and \(y=0\) within the interval \(-2\pi\leq\xi, \eta\leq2\pi\).
6 Conclusions
In this article, we utilize Lie group analysis to obtain symmetries and symmetry reduction for two higher-dimensional NLEEs. In this way, we can reduce the dimension of the NLEEs, which is relevant in the fields of mathematical physics and engineering. Five types of explicit function solutions are constructed by the \(\exp(-\phi(z))\)-expansion method and complex method. It demonstrates these methods are very efficient and powerful to seek the exact solutions of NLEEs. We can apply the idea of this study to other NLEEs.
Declarations
Acknowledgements
This work was supported by the NSF of China (11271090, 11701111); the NSF of Guangdong Province (2016A030310257); the Foundation for Young Talents in Educational Commission of Guangdong Province (2015KQNCX116). Thanks to the Joint PHD Program of Guangzhou University and Curtin University. Thanks to the editors and referees with their very useful suggestions and helpful comments.
Authors’ contributions
All authors typed, read and approved the final manuscript.
Competing interests
The authors declare that they have no competing interests.
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
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