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Symmetry Reduced and New Exact Nontraveling Wave Solutions of (2+1)-Dimensional Potential Boiti-Leon-Manna-Pempinelli Equation
With the aid of Maple symbolic computation and Lie group method, ()-dimensional PBLMP equation is reduced to some ()-dimensional PDE with constant coefficients. Using the homoclinic test technique and auxiliary equation methods, we obtain new exact nontraveling solution with arbitrary functions for the PBLMP equation.
In this paper, we will consider the potential Boiti-Leon-Manna-Pempinelli (PBLMP) equation where . By some transformations, the PBLMP equation (1) can be equivalent to the asymmetric Nizhnik-Novikov-Veselov (ANNV) system. In fact, the ANNV equation can be obtained from the inner parameter-dependent symmetry constraint of the KP equation  and may be considered as a model for an incompressible fluid . The Painlevé analysis, Lax pair, and some exact solutions have been studied for the PBLMP equation . Tang and Lou obtained the bilinear form of (1) and variable separation solutions including two arbitrary functions by the multilinear variable separation approach [4, 5].
In this paper, by means of Maple symbolic computation, we will use the Lie group method [6, 7], homoclinic test technique [8, 9] and so forth to reduce and solve the PBLMP equation. First, we will derive symmetry of (1). Then we use the symmetry to reduce (1) to some -dimensional PDE with constant coefficients. Finally, solving the reduced PDE by Homoclinic test technique and auxiliary equation methods [10, 11] implies abundant exact nontraveling wave periodic solutions for the PBLMP equation.
2. Symmetry of (1)
This section is devoted to Lie point group symmetries of (1). Let be the symmetry of (1). Based on Lie group theory , satisfies the following symmetry equation: To get some symmetries of (1), we take the function in the form where are functions of to be determined, and satisfies (1). Substituting (4) and (1) into (3), one can get where are arbitrary constants. are arbitrary functions of . is a arbitrary function of . Substituting (5) into (4), we obtain the symmetries of (1) as follows:
3. Symmetry Reduction of (1)
Based on the integrability of reduced equation of the symmetry (6), we consider the following three cases.
Case 1. Let in (6), then where is an arbitrary nonzero constant, . Solving the differential equation for , one gets Substituting (8) into (1), we get the following -dimensional nonlinear PDE with constant coefficients: Integrating (9) once with respect to and taking integration constant to zero yield
Case 3. Let in (6), then
Solving the equation for , we obtain
Substituting (15) into (1) yields a reduced PDE of (1) with constant coefficients:
Integrating (16) once with respect to and taking integration constant to zero yield
Combining the above results, we obtain some reduced equations of (1) expressed by (10), (13), and (17), respectively. Meanwhile many new explicit solutions of (1) from these reduced Equations. can be achieved. We omit other cases based on symmetries (6) here.
4. Solve Reduced PDE and Get Exact Nontraveling Wave Solutions of (1)
4.1. Solve Reduced PDE (10)
Now, we seek solutions of (10) by auxiliary equation method. Make transformation as follows: where are nonzero constants. Substituting (18) into (10) obtains an ordinary differential equation for as follows: where . Let , then (19) can be written as This is the fourth type of ellipse equation (12), its solutions are as follows: where is the integration constant. From the result of (21), some new exact solutions through of (1) can be obtained: Particularly, we assume , then the solution can be depicted by Figure 1(a). If , then can be depicted by Figures 1(b) and 2(a).
4.2. Solve Reduced PDE (13)
4.3. Solve Reduced PDE (17)
In this section, we use homoclinic test technique [8, 9] to (17) and transform the unknown function as follows: Substituting (26) into (17) and using the bilinear form, we can get where the Hirota operator is defined in . In this case we choose extended homoclinic test function where , and are real constants to be determined. Substituting (28) into (27) yields a set of algebraic equations as follows: Solving the above equations (29) yields where . Substituting (30)–(33) into (28) yields the solutions through of (1) as follows: when in (30); when in (30); when in (31) (see Figure 2(b)); when in (32); when in (33).
In this paper, a combination of Lie group method and homoclinic test technique and so forth is applied and thus the symmetries (6) are obtained. The (2+1)-dimensional potential Boiti-Leon-Manna-Pempinelli equation (1) is reduced to -dimensional nonlinear PDE of constant coefficients (10), (13), and (17). Further auxiliary equation method and homoclinic test technique are used and some new exact non-traveling wave solutions are obtained. And they include some special and strange structures to be further studied and other relevant solutions about symmetry (6) will be discussed later in another paper. Our results show that combining the Lie group method with homoclinic test technique and so forth is effective in finding nontraveling wave exact solutions of nonlinear evolution equations.
The authors would like to thank Professor Dai Zhengde for his helpful discussions. This work was supported by the Chinese Natural Science Foundation Grant no. 10971169 and the key research projects of Sichuan Provincial Educational Administration, no. 10ZA021.
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Copyright © 2013 Chen Han-Lin and Xian Da-Quan. 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.