Lie Group Analysis for a (2+1)-dimensional Generalized Modified Dispersive Water-Wave System for the Shallow Water Waves

Shallow water waves refer to the waves with the bottom boundary affecting the movement of water quality points when the ratio of water depth to wavelength is small. Under investigation in this paper is a (2+1)-dimensional generalized modified dispersive water-wave (GMDWW) system for the shallow water waves. We obtain the Lie point symmetry generators and Lie symmetry groups for the GMDWW system via the Lie group method. Optimal system of the one-dimensional subalgebras is derived. According to that optimal system, we obtain certain symmetry reductions. Hyperbolic-function, trigonometric-function and rational solutions for the GMDWW system are derived via the polynomial expansion, Riccati equation expansion and (G′G) expansion methods.