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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:
76B20Ship waves
76M60Symmetry analysis, Lie group and algebra methods (fluid mechanics)
76M55Dimensional analysis and similarity (fluid mechanics)
35Q51Soliton-like equations
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 1103.35360 · doi:10.1103/PhysRevLett.19.1095
[2]Wazwaz, A. M.: New solitons and kink solutions for the gardner equation, Commun. nonlinear sci. Numer. simul. 12, No. 8, 1395-1404 (2007) · Zbl 1118.35352 · doi:10.1016/j.cnsns.2005.11.007
[3]Wazwaz, A. M.: Solitons and singular solitons for the gardner-KP equation, Appl. math. Comput. 204, 162-169 (2008) · Zbl 1159.35432 · doi:10.1016/j.amc.2008.06.011
[4]Olver, Peter J.: Applications of Lie groups to differential equations, (1999)
[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 · doi:10.1016/j.cnsns.2005.08.003
[6]Sirendaorejj; Jiang, S.: Auxiliary equations method for solving nonlinear partial differential equations, Phys. lett. A 309, 387-396 (2003) · Zbl 1011.35035 · doi:10.1016/S0375-9601(03)00196-8
[7]Ibragimov, N. H.: A new conservation theorem, J. math. Anal. appl. 333, 311-328 (2007) · Zbl 1160.35008 · doi:10.1016/j.jmaa.2006.10.078