Bilinear form, bilinear Bäcklund transformations, breather and periodic-wave solutions for a \((2+1)\)-dimensional shallow water equation with the time-dependent coefficients. (English) Zbl 1529.35516
Summary: Shallow water waves are seen in geophysical fluid dynamics, oceanography, coastal engineering and atmospheric science. In this paper, to describe the shallow water waves, we investigate a \((2+1)\)-dimensional shallow water equation with the time-dependent coefficients via symbolic computation. Based on the Hirota method and Painlevé integrable conditions in the existing literature, the bilinear form for that equation is hereby constructed. Via the bilinear form and exchange formulae, we build three bilinear Bäcklund transformations with certain soliton-like solutions. Via the extended homoclinic test approach, we derive the breather solutions and their asymptotic behaviors. We graphically show the breather waves. Periodic-wave solutions are worked out via the Hirota-Riemann method, and graphically displayed. Relation between the periodic-wave solutions and one-soliton solutions is discussed.
MSC:
| 35Q86 | PDEs in connection with geophysics |
| 35Q35 | PDEs in connection with fluid mechanics |
| 86A05 | Hydrology, hydrography, oceanography |
| 76B15 | Water waves, gravity waves; dispersion and scattering, nonlinear interaction |
| 76U60 | Geophysical flows |
| 35C08 | Soliton solutions |
| 35B10 | Periodic solutions to PDEs |
| 35B40 | Asymptotic behavior of solutions to PDEs |
| 37K35 | Lie-Bäcklund and other transformations for infinite-dimensional Hamiltonian and Lagrangian systems |
| 68W30 | Symbolic computation and algebraic computation |
Keywords:
shallow water waves; \((2+1)\)-dimensional shallow water equation with time-dependent coefficients; Hirota method; bilinear form; bilinear Bäcklund transformations; breather and periodic-wave solutions; symbolic computationReferences:
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