About this Journal Submit a Manuscript Table of Contents
International Journal of Differential Equations
Volume 2012 (2012), Article ID 596762, 11 pages
http://dx.doi.org/10.1155/2012/596762
Research Article

The Improved Riccati Equation Method and Exact Solutions to mZK Equation

College of Physics and Material Science, Anhui University, Hefei, Anhui 230039, China

Received 26 May 2012; Accepted 24 July 2012

Academic Editor: Giovany M. Figueiredo

Copyright © 2012 Xiaofeng Li. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We utilize the improved Riccati equation method to construct more general exact solutions to nonlinear equations. And we obtain the travelling wave solutions involving parameters, which are expressed by the hyperbolic functions, the trigonometric functions, and the rational functions. When the parameters are taken as special values, the method provides not only solitary wave solutions but also periodic waves solutions. The method appears to be easier and more convenient by means of a symbolic computation system. Of course, it is also effective to solve other nonlinear evolution equations in mathematical physics.

1. Introduction

More and more problems in the branches of modern mathematical physics are described in terms of suitable nonlinear models, and nonlinear physical phenomena are related to nonlinear equations, which are involved in many fields from physics to biology, chemistry, mechanics, and so forth. Nonlinear wave phenomena are very important in nonlinear sciences, in recent years, much effort has been spent on the construction of exact solutions of nonlinear partial differential equations. Many powerful and efficient methods have been presented to obtain the exact traveling wave solutions of nonlinear evolution equations, such as the Backlund transformation method [1, 2], Exp-function method [3, 4], homogeneous balance method [5, 6], tanh-function method [7, 8], the Jacobi elliptic function expansion [9, 10], and the 𝐺/𝐺-expansion method [11, 12]. A search of directly seeking for exact solutions of nonlinear equations has been more interesting because of the availability of symbolic computation, Mathematica or Maple. These computation systems are adequately utilized to perform some complicated and tedious algebraic and differential calculations on a computer. By using these methods and tools, one can successfully obtain exact solutions.

The ZK equation governs the behavior of weakly nonlinear ion-acoustic waves in plasma comprising cold ions and hot isothermal electrons in the presence of a uniform magnetic field. When the ion or electron plasma does not meet the Boltzmann distribution, Munro and Parkes derive the modified ZK equation (mZK equation), they also studied planar periodic two-dimensional long-wave perturbation wave solutions and the stability of traveling wave solutions in isolation [13, 14]. The mZK equation represents an anisotropic two-dimensional generalization of the KdV equation and can be derived in a magnetized plasma for small amplitude Alfven wave at a critical angle to undisturbed magnetic field. mZK equation is effectively applied to describe the evolution of various solitary waves in isothermal multicomponent magnetized plasma, similarly the description of stability of solitary waves of mZK equation has also appeared in [15]. The mZK equation has attracted the attention of many researchers in the past few years. For instance, from the mathematical point of view, local and global existence for the Cauchy problem was studied in [1618].

The 𝐺/𝐺-expansion method was proposed originally by Wang et al., which is one of the most effective direct methods to obtain travelling wave solutions for a large number of nonlinear evolution equations. This useful method is widely employed by many authors [11, 12]. The key ideas of the 𝐺/𝐺-expansion method are that the travelling wave solutions of nonlinear evolution equations can be expressed by polynomials in 𝐺/𝐺, where 𝐺 satisfies a second order linear differential equation, the degree of the polynomials can be determined by considering the homogeneous balance between the highest order partial derivatives and nonlinear terms appearing in nonlinear evolution equations considered, the coefficients of the polynomials can be obtained by solving a set of simultaneous algebraic equations resulted from the process of using the proposed method.

The paper is motivated by the desire to present a new method, named the improved Riccati equation method, so that it can be successfully applied to seeking the exact travelling wave solutions to the mZK equation. We will obtion two group values of coefficients regarding Riccati equation and nonlinear evolution equation. By contrast to both Riccati equation method and 𝐺/𝐺-expansion method, at this point, it is surely a meaningful improvement and innovation we have made to obtain much more abundant solutions. Following the description of the improved Riccati equation method, one can have access to exact solutions to nonlinear evolution equations smoothly.

2. Description of the Improved Riccati Equation Method

Step 1. We consider the nonlinear evolution equations in three independent variables 𝑥,𝑦,𝑡 and dependent variable 𝑢: 𝑁𝑢,𝑢𝑥,𝑢𝑡,𝑢𝑥𝑥,𝑢𝑥𝑡,𝑢𝑡𝑡,𝑢𝑦,𝑢𝑥𝑦,=0.(2.1)
We seek their traveling wave solutions in the following form 𝑢(𝑥,𝑦,𝑡)=𝑢(𝜉),𝜉=𝑘𝑥+𝑐𝑦+𝑑𝑡,(2.2) where 𝑘, 𝑐, and 𝑑 are arbitrary constants.
Equation (2.1) can be converted to an ordinary differential equation 𝑁1𝑢,𝑢,𝑢,=0.(2.3)

Step 2. In order to construct travelling wave solutions of nonlinear equations, it is reasonable to introduce the following ansatz 𝑢(𝑥,𝑦,𝑡)=𝑢(𝜉)=𝑛𝑖=𝑛𝑎𝑖𝑓𝑖(𝜉),(2.4) where 𝑎𝑖 are constants to be determined later, the balancing number 𝑛 is a positive integer which can be determined by balancing the highest order derivative terms with the highest power nonlinear terms in (2.3) and 𝑓(𝜉) satisfies the following elliptic equation: 𝑓(𝜉)=𝑝+𝑟𝑓(𝜉)+𝑞𝑓2(𝜉),(2.5) where 𝑝,𝑟,𝑞 are real parameters. And 𝑓(𝜉) can also be expanded to the following ansatz: 𝑓(𝜉)=𝑚𝑖=𝑚𝑏𝑖𝐺𝐺𝑚,(2.6) and 𝐺(𝜉) satisfies the following elliptic equations: 𝐺(𝜉)+𝜆𝐺(𝜉)+𝜇𝐺(𝜉)=0,(2.7) where 𝑏𝑖 are constants to be determined later, 𝜆,𝜇 are real parameters. 𝑚 is a positive integer which can be determined by balancing the highest order derivative terms with the highest power nonlinear terms in (2.5), and so we can get 𝑚=1.

Step 3. We substitute (2.6) and (2.7) into (2.5), equating the coefficients of all powers of (𝐺/𝐺) to zero, and we can get solutions of 𝑓(𝜉) with computerized symbolic computation.

Step 4. Then we substitute (2.4) and (2.5) into (2.3), equating the coefficients of all powers of 𝑓(𝜉) to zero, solving this set of algebraic equation with computerized symbolic computation, inserting these results and solutions of 𝑓(𝜉) into (2.4). Finally, setting𝜉=𝑘𝑥+𝑐𝑦+𝑑𝑡, we obtain the exact travelling wave solutions of (2.1).

3. Applications

We consider the modified Zakharov-Kuznetsov (mZK) equations in the following form: 𝑢𝑡+𝑢2𝑢𝑥+𝑢𝑥𝑥𝑥+𝑢𝑥𝑦𝑦=0.(3.1)

We also make the transformation 𝜉=𝑘𝑥+𝑠𝑦𝜔𝑡, where 𝜔,𝑘,𝑠 are constants to be determined later. Then (3.1) is reduced to the following: 𝜔𝑢+𝑘𝑢33+𝑘3+𝑘𝑠2𝑢=0.(3.2)

By balancing the highest order derivative terms and nonlinear terms in (3.2), so we get 𝑛=1. Then we can suppose that (3.2) has the solutions in the form: 𝑢(𝜉)=𝑎1𝑓1(𝜉)+𝑎0+𝑎1𝑓(𝜉).(3.3)

On substituting (2.5) and (3.3) into (3.2), collecting all terms with the same powers of 𝑓𝑖(𝜉) and setting each coefficient of the polynomials to zero, solving the over-determined algebraic equations by Mathematica, we can obtain the following results.

Set 1. 1𝜔=2𝑘4𝑝𝑞𝑟2𝑘2+𝑠2,𝑎1=±𝑝6𝑘2𝑠2,𝑎0=±32𝑟2𝑘2𝑠2,𝑎1=0,𝑘2+𝑠20,𝑘0,𝑝0,𝑟0.(3.4)

Set 2. 1𝜔=2𝑘4𝑝𝑞𝑟2𝑘2+𝑠2,𝑎1=0,𝑎0=±32𝑟2𝑘2𝑠2,𝑎1=±𝑞6𝑘2𝑠2,𝑘2+𝑠20,𝑘0,𝑞0,𝑟0.(3.5)

Similarly, we can also get the following result.

Case 1. 𝑟𝑞0,𝑝=2𝜆2+4𝜇4𝑞,𝑏0=𝜆𝑟2𝑞,𝑏11=𝑞,𝑏1=0.(3.6)

Case 2. 𝑟𝑞0,𝑝=2𝜆2+4𝜇4𝑞,𝑏0=𝜆𝑟2𝑞,𝑏1=0,𝑏1𝜆=𝑝+2𝑟2.4𝑞(3.7)

Using Case 1, substituting Sets 1, 2, and the general solutions of (2.6) into formula (3.3), we have three types of travelling wave solutions as follows (𝑐1 and 𝑐2 are arbitrary constants, 𝜉=𝑘𝑥+𝑠𝑦+(1/2)𝑘(4𝑝𝑞𝑟2)(𝑘2+𝑠2)𝑡).

When 𝑟24𝑝𝑞>0, we obtain the hyperbolic function solutions of (3.1) 𝑢1(𝜉)=±32𝑟2𝑘2𝑠2±𝑝6𝑘2𝑠2×𝜆+𝑟12𝑞𝑞𝑟24𝑝𝑞2𝑐1sinh𝑟24𝑝𝑞/2𝜉+𝑐2cosh𝑟2𝜉4𝑝𝑞/2𝑐2sinh𝑟24𝑝𝑞/2𝜉+𝑐1cosh𝑟2𝜉𝜆4𝑝𝑞/221,𝑢2(𝜉)=±32𝑟2𝑘2𝑠2±𝑞6𝑘2𝑠2×𝜆+𝑟12𝑞𝑞𝑟24𝑝𝑞2𝑐1sinh𝑟24𝑝𝑞/2𝜉+𝑐2cosh𝑟2𝜉4𝑝𝑞/2𝑐2sinh𝑟24𝑝𝑞/2𝜉+𝑐1cosh𝑟2𝜉𝜆4𝑝𝑞/22.(3.8)

If 𝑐10,𝑐21>𝑐22, then 𝑢(𝜉) becomes the solitary wave solutions of (3.1) as follows: 𝑢1(𝜉)=±32𝑟2𝑘2𝑠2±𝑝6𝑘2𝑠2×𝜆+𝑟12𝑞𝑞𝑟24𝑝𝑞2tanh𝑟24𝑝𝑞2𝜉+𝜉0𝜆21𝑢2(𝜉)=±32𝑟2𝑘2𝑠2±𝑞6𝑘2𝑠2×𝜆+𝑟12𝑞𝑞𝑟24𝑝𝑞2tanh𝑟24𝑝𝑞2𝜉+𝜉0𝜆2,(3.9) where 𝜉0=tanh1(𝑐2/𝑐1).

If 𝑐20,𝑐22>𝑐21, then 𝑢(𝜉) becomes the solitary wave solutions of (3.1) as follows: 𝑢1(𝜉)=±32𝑟2𝑘2𝑠2±𝑝6𝑘2𝑠2×𝜆+𝑟12𝑞𝑞𝑟24𝑝𝑞2coth𝑟24𝑝𝑞2𝜉+𝜉0𝜆21,𝑢2(𝜉)=±32𝑟2𝑘2𝑠2±𝑞6𝑘2𝑠2×𝜆+𝑟12𝑞𝑞𝑟24𝑝𝑞2coth𝑟24𝑝𝑞2𝜉+𝜉0𝜆2,(3.10) where 𝜉0=tanh1(𝑐1/𝑐2).

When 𝑟24𝑝𝑞=0, we get the rational function solutions of (3.1) 𝑢1(𝜉)=±32𝑟2𝑘2𝑠2±𝑝6𝑘2𝑠2𝜆+𝑟12𝑞𝑞𝑐2𝑐1+𝑐2𝜉𝜆21,𝑢2(𝜉)=±32𝑟2𝑘2𝑠2±𝑞6𝑘2𝑠2𝜆+𝑟12𝑞𝑞𝑐2𝑐1+𝑐2𝜉𝜆2.(3.11) When 𝑟24𝑝𝑞<0, we obtain the trigonometric function solutions of (3.1) 𝑢1(𝜉)=±32𝑟2𝑘2𝑠2±𝑝6𝑘2𝑠2×𝜆+𝑟12𝑞𝑞4𝑝𝑞𝑟22𝑐2cos4𝑝𝑞𝑟2/2𝜉𝑐1sin4𝑝𝑞𝑟2𝜉/2𝑐1cos4𝑝𝑞𝑟2/2𝜉+𝑐2sin4𝑝𝑞𝑟2𝜉𝜆/221,𝑢2(𝜉)=±32𝑟2𝑘2𝑠2±𝑞6𝑘2𝑠2×𝜆+𝑟12𝑞𝑞4𝑝𝑞𝑟22𝑐2cos4𝑝𝑞𝑟2/2𝜉𝑐1sin4𝑝𝑞𝑟2𝜉/2𝑐1cos4𝑝𝑞𝑟2/2𝜉+𝑐2sin4𝑝𝑞𝑟2𝜉𝜆/22.(3.12)

If 𝑐10,𝑐21>𝑐22, then 𝑢(𝜉) becomes the periodic wave solutions of (3.1) as follows: 𝑢1(𝜉)=±32𝑟2𝑘2𝑠2±𝑝6𝑘2𝑠2×𝜆+𝑟12𝑞𝑞4𝑝𝑞𝑟22𝜉tan04𝑝𝑞𝑟22𝜉𝜆21,𝑢2(𝜉)=±32𝑟2𝑘2𝑠2±𝑞6𝑘2𝑠2×𝜆+𝑟12𝑞𝑞4𝑝𝑞𝑟22𝜉tan04𝑝𝑞𝑟22𝜉𝜆2,(3.13) where𝜉0=tan1(𝑐2/𝑐1).

If 𝑐20,𝑐22>𝑐21, then 𝑢(𝜉) becomes the periodic wave solutions of (3.1) as follows: 𝑢1(𝜉)=±32𝑟2𝑘2𝑠2±𝑝6𝑘2𝑠2×𝜆+𝑟12𝑞𝑞4𝑝𝑞𝑟22cot4𝑝𝑞𝑟22𝜉+𝜉0𝜆21,𝑢2(𝜉)=±32𝑟2𝑘2𝑠2±𝑞6𝑘2𝑠2×𝜆+𝑟12𝑞𝑞4𝑝𝑞𝑟22cot4𝑝𝑞𝑟22𝜉+𝜉0𝜆2,(3.14) where 𝜉0=tan1(𝑐1/𝑐2).

Using Case 2, substituting Sets 1, 2, and the general solutions of (2.6) into formulate (3.3), we have three types of travelling wave solutions as follows (𝑐1 and 𝑐2 are arbitrary constants, 𝜉=𝑘𝑥+𝑠𝑦+(1/2)𝑘(4𝑝𝑞𝑟2)(𝑘2+𝑠2)𝑡).

When 𝑟24𝑝𝑞>0, we obtain the hyperbolic function solutions of (3.1) 𝑢1(𝜉)=±32𝑟2𝑘2𝑠2±𝑝6𝑘2𝑠2×𝜆𝑟+𝜇2𝑞𝑞𝑟24𝑝𝑞2𝑐1sinh𝑟24𝑝𝑞/2𝜉+𝑐2cosh𝑟2𝜉4𝑝𝑞/2𝑐2sinh𝑟24𝑝𝑞/2𝜉+𝑐1cosh𝑟2𝜉𝜆4𝑝𝑞/2211,𝑢2(𝜉)=±32𝑟2𝑘2𝑠2±𝑝6𝑘2𝑠2×𝜆𝑟+𝜇2𝑞𝑞𝑟24𝑝𝑞2𝑐1sinh𝑟24𝑝𝑞/2𝜉+𝑐2cosh𝑟2𝜉4𝑝𝑞/2𝑐2sinh𝑟24𝑝𝑞/2𝜉+𝑐1cosh𝑟2𝜉𝜆4𝑝𝑞/221.(3.15)

If 𝑐10,𝑐12>𝑐22, then 𝑢(𝜉) becomes the solitary wave solutions of (3.1) as follows: 𝑢1(𝜉)=±32𝑟2𝑘2𝑠2±𝑝6𝑘2𝑠2×𝜆𝑟+𝜇2𝑞𝑞𝑟24𝑝𝑞2tanh𝑟24𝑝𝑞2𝜉+𝜉0𝜆211,𝑢2(𝜉)=±32𝑟2𝑘2𝑠2±𝑞6𝑘2𝑠2×𝜆𝑟+𝜇2𝑞𝑞𝑟24𝑝𝑞2tanh𝑟24𝑝𝑞2𝜉+𝜉0𝜆21,(3.16) where 𝜉0=tanh1(𝑐2/𝑐1).

If 𝑐20,𝑐22>𝑐21, then 𝑢(𝜉) becomes the solitary wave solutions of (3.1) as follows: 𝑢1(𝜉)=±32𝑟2𝑘2𝑠2±𝑝6𝑘2𝑠2×𝜆𝑟+𝜇2𝑞𝑞𝑟24𝑝𝑞2coth𝑟24𝑝𝑞2𝜉+𝜉0𝜆211,𝑢2(𝜉)=±32𝑟2𝑘2𝑠2±𝑞6𝑘2𝑠2×𝜆𝑟+𝜇2𝑞𝑞𝑟24𝑝𝑞2coth𝑟24𝑝𝑞2𝜉+𝜉0𝜆21,(3.17) where 𝜉0=tanh1(𝑐1/𝑐2).

When 𝑟24𝑝𝑞=0, we get the rational function solutions of (3.1)𝑢1(𝜉)=±32𝑟2𝑘2𝑠2±𝑝6𝑘2𝑠2𝜆𝑟+𝜇2𝑞𝑞𝑐2𝑐1+𝑐2𝜉𝜆211,𝑢(3.18)2(𝜉)=±32𝑟2𝑘2𝑠2±𝑞6𝑘2𝑠2𝜆𝑟+𝜇2𝑞𝑞𝑐2𝑐1+𝑐2𝜉𝜆21.(3.19)

When 𝑟24𝑝𝑞<0, we obtain the trigonometric function solutions of (3.1) 𝑢1(𝜉)=±32𝑟2𝑘2𝑠2±𝑝6𝑘2𝑠2×𝜆𝑟+𝜇2𝑞𝑞4𝑝𝑞𝑟22𝑐2cos4𝑝𝑞𝑟2/2𝜉𝑐1sin4𝑝𝑞𝑟2𝜉/2𝑐1cos4𝑝𝑞𝑟2/2𝜉+𝑐2sin4𝑝𝑞𝑟2𝜉𝜆/221,𝑢2(𝜉)=±32𝑟2𝑘2𝑠2±𝑞6𝑘2𝑠2×𝜆𝑟+𝜇2𝑞𝑞4𝑝𝑞𝑟22𝑐2cos4𝑝𝑞𝑟2/2𝜉𝑐1sin4𝑝𝑞𝑟2𝜉/2𝑐1cos4𝑝𝑞𝑟2/2𝜉+𝑐2sin4𝑝𝑞𝑟2𝜉𝜆/221,(3.20)

If 𝑐10,𝑐21>𝑐22, then 𝑢(𝜉) becomes the periodic wave solutions of (3.1) as follows: 𝑢1(𝜉)=±32𝑟2𝑘2𝑠2±𝑝6𝑘2𝑠2×𝜆𝑟+𝜇2𝑞𝑞4𝑝𝑞𝑟22𝜉tan04𝑝𝑞𝑟22𝜉𝜆211,𝑢2(𝜉)=±32𝑟2𝑘2𝑠2±𝑞6𝑘2𝑠2×𝜆𝑟+𝜇2𝑞𝑞4𝑝𝑞𝑟22𝜉tan04𝑝𝑞𝑟22𝜉𝜆21,(3.21) where 𝜉0=tan1(𝑐2/𝑐1).

If 𝑐20,𝑐22>𝑐21, then 𝑢(𝜉) becomes the periodic wave solutions of (3.1) as follows: 𝑢1(𝜉)=±32𝑟2𝑘2𝑠2±𝑝6𝑘2𝑠2×𝜆𝑟+𝜇2𝑞𝑞4𝑝𝑞𝑟22cot4𝑝𝑞𝑟22𝜉+𝜉0𝜆211,𝑢2(𝜉)=±32𝑟2𝑘2𝑠2±𝑞6𝑘2𝑠2×𝜆𝑟+𝜇2𝑞𝑞4𝑝𝑞𝑟22cot4𝑝𝑞𝑟22𝜉+𝜉0𝜆21,(3.22) where 𝜉0=tan1(𝑐1/𝑐2).

4. Conclusion

In summary, the improved Riccati equation method has been proposed and used to find out exact solutions of nonlinear equations with the aid of Mathmatica software. Our method allows us carry out the solution process of nonlinear wave equations more systematically and conveniently by computer algebra systems such as Maple and Mathematica. We have successfully obtained some travelling wave solutions of the mZK equations. When the parameters are taken as special values, the solitary wave solutions and periodic wave solutions are obtained. We surely believe that these solutions will be of great importance for analyzing the nonlinear phenomena arising in applied physical sciences. The work shows that the improved Riccati equation method is sufficient, effective and suitable for solving other nonlinear evolution equations, it deserves further applying and studying as well.

Acknowledgment

During the work on this project, the author received invaluable advise and help from a lot of colleagues and friends. In particular, the author would like to thank Jiahua Han and Jiancheng Ji who cooperated with him.

References

  1. B. Tian and Y. T. Gao, “Spherical nebulons and Backlund transformation for a space or laboratory un-magnetized dusty plasma with symbolic computation,” The European Physical Journal D, vol. 33, pp. 59–65, 2005.
  2. L. Tian and J. Yin, “Stability of multi-compacton solutions and Bäcklund transformation in K(m,n,1),” Chaos, Solitons and Fractals, vol. 23, no. 1, pp. 159–169, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  3. J.-H. He and X.-H. Wu, “Exp-function method for nonlinear wave equations,” Chaos, Solitons and Fractals, vol. 30, no. 3, pp. 700–708, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  4. J.-H. He and L.-N. Zhang, “Generalized solitary solution and compacton-like solution of the Jaulent-Miodek equations using the Exp-function method,” Physics Letters A, vol. 372, no. 7, pp. 1044–1047, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  5. L. WangM, Y. B. Zhou, and Z. B. Li, “Application of homogeneous balance method to exact solutions of nonlinear equations in mathematical physics,” Physics Letters A, vol. 216, pp. 67–75, 1996.
  6. M. H. M. Moussa and R. M. El Shikh, “Two applications of the homogeneous balance method for solving the generalized Hirota-Satsuma coupled KdV system with variable coefficients,” International Journal of Nonlinear Science, vol. 7, no. 1, pp. 29–38, 2009. View at Zentralblatt MATH
  7. A. A. Soliman, “The modified extended tanh-function method for solving Burgers-type equations,” Physica A, vol. 361, no. 2, pp. 394–404, 2006. View at Publisher · View at Google Scholar
  8. M. A. Abdou and A. A. Soliman, “Modified extended tanh-function method and its application on nonlinear physical equations,” Physics Letters A, vol. 353, pp. 487–492, 2006.
  9. S. K. Liu, Z. T. Fu, S. D. Liu, and Q. Zhao, “Jacobi elliptic function expansion method and periodic wave solutions of nonlinear wave equations,” Physics Letters A, vol. 289, no. 1-2, pp. 69–74, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  10. Z. T. Fu, S. K. Liu, S. D. Liu, and Q. Zhao, “New Jacobi elliptic function expansion and new periodic solutions of nonlinear wave equations,” Physics Letters A, vol. 290, no. 1-2, pp. 72–76, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  11. M. Wang, X. Li, and J. Zhang, “The (G/G)-expansion method and travelling wave solutions of nonlinear evolution equations in mathematical physics,” Physics Letters A, vol. 372, no. 4, pp. 417–423, 2008. View at Publisher · View at Google Scholar
  12. A. Bekir, “Application of the (G/G)-expansion method for nonlinear evolution equations,” Physics Letters A, vol. 372, no. 19, pp. 3400–3406, 2008. View at Publisher · View at Google Scholar
  13. S. Munro and E. J. Parkes, “The derivation of a modified Zakharov-Kuznetsov equation and the stability of its solutions,” Journal of Plasma Physics, vol. 62, no. 3, pp. 305–317, 1999.
  14. S. Munro and E. J. Parkes, “Stability of solitary-wave solutions to a modified Zakharov-Kuznetsov equation,” Journal of Plasma Physics, vol. 64, no. 4, pp. 411–426, 2000.
  15. A. de Bouard, “Stability and instability of some nonlinear dispersive solitary waves in higher dimension,” Proceedings of the Royal Society of Edinburgh Section A, vol. 126, no. 1, pp. 89–112, 1996. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  16. F. Linares and A. Pastor, “Local and global well-posedness for the 2D generalized Zakharov-Kuznetsov equation,” Journal of Functional Analysis, vol. 260, no. 4, pp. 1060–1085, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  17. F. Ribaud and S. Vento, “A note on the Cauchy problem for the 2D generalized Zakharov-Kuznetsov equations,” Comptes Rendus Mathématique, vol. 350, no. 9-10, pp. 499–503, 2012. View at Publisher · View at Google Scholar
  18. F. Linares and A. Pastor, “Well-posedness for the two-dimensional modified Zakharov-Kuznetsov equation,” SIAM Journal on Mathematical Analysis, vol. 41, no. 4, pp. 1323–1339, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH