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Classification, reduction, group-invariant solutions and conservation laws of the Gardner-KP equation. (English) Zbl 1172.76006

Summary: Symmetries and group invariant solutions to the Gardner-KP equation are obtained by using the direct symmetry method. At the same time, we find the corresponding Lie algebra, optimal system, classification and all similarity reductions. Our exact solutions generalize the corresponding results obtained by A.-M. Wazwaz [ibid. 204, No. 1, 162–169 (2008; Zbl 1159.35432)]. In addition, the conservation laws of Gardner-KP equation are also given.

MSC:

76B20 Ship waves
76M60 Symmetry analysis, Lie group and Lie algebra methods applied to problems in fluid mechanics
76M55 Dimensional analysis and similarity applied to problems in fluid mechanics
35Q51 Soliton equations

Citations:

Zbl 1159.35432
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References:

[1] Gardner, C.; Greene, J.; Kruskal, M.; Miura, R., Method for solving the Korteweg-de Vries equation, Phys. Rev. Lett., 19, 1095-1097 (1967) · Zbl 1061.35520
[2] Wazwaz, A. M., New solitons and kink solutions for the Gardner equation, Commun. Nonlinear Sci. Numer. Simul., 12, 8, 1395-1404 (2007) · Zbl 1118.35352
[3] Wazwaz, A. M., Solitons and singular solitons for the Gardner-KP equation, Appl. Math. Comput., 204, 162-169 (2008) · Zbl 1159.35432
[4] Olver, Peter J., Applications of Lie Groups to Differential Equations (1999), Springer-Verlag, pp. 199-206
[5] Zhang, H. Q., Extended Jacobi elliptic function expansion method and its applications, Commun. Nonlinear Sci. Numer. Simul., 12, 627-635 (2007) · Zbl 1111.35317
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