Integrability, conservation laws and exact solutions for a model equation under non-canonical perturbation expansions. (English) Zbl 1504.35386
Summary: In this paper, the non-linear for the small long amplitude waves in two dimensional (2D) shallow water waves propagation with free surface are considered. The shallow water wave problem leads to the non-linear Hamiltonian model equation. Based on the binary Bell-polynomials approach, the bilinear form, bilinear Bäcklund transformation and multiple wave solutions are obtained. The conservation laws are constructed using two different techniques, namely, the Ibragimov’s theorem and the multiplier method. The Noether’s approach was applied to the non-linear Hamiltonian model equation to obtain the conservation laws. Also, we show that the non-linear Hamiltonian model equation is nonlinearly self-adjoint. Conserved quantities of Hamiltonian model equation are illustrated. Finally, with the help of the extended homogeneous balance method, and an exponential method, a set of new exact solutions for the non-linear Hamiltonian model equation are obtained.
MSC:
| 35Q35 | PDEs in connection with fluid mechanics |
| 35Q31 | Euler equations |
| 76B15 | Water waves, gravity waves; dispersion and scattering, nonlinear interaction |
| 35B35 | Stability in context of PDEs |
| 37K35 | Lie-Bäcklund and other transformations for infinite-dimensional Hamiltonian and Lagrangian systems |
| 35A30 | Geometric theory, characteristics, transformations in context of PDEs |
| 35J05 | Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation |
| 58J72 | Correspondences and other transformation methods (e.g., Lie-Bäcklund) for PDEs on manifolds |
Keywords:
Hamiltonian mechanics; Bäcklund transformation; multiple wave solution; conservation laws; extended \((G^\prime / G)\)-expansion method; \(\exp (- \Phi(\xi))\)-expansion methodReferences:
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