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Many important physical situations such as fluid flows, marine environment,solid-state physics and plasma physics have been represented by shallow waterwave equation. In this article, we construct new solitary wave solutions forthe (3+1)-dimensional generalized shallow water wave (GSWW) equation by usingLie symmetry method. A variety of analytic (closed-form) solutions such as newperiodic solitary wave, cross-kink soliton and doubly periodic breather-typesolutions have been obtained by using invariance of the concerned(3+1)-dimensional GSWW equation under one-parameter Lie group oftransformations. Lie symmetry transformations have applied to generate thedifferent forms of invariant solutions of the (3+1)-dimensional GSWW equation.For different Lie algebra, Lie symmetry method reduces (3+1)-dimensional GSWWequation into various ordinary differential equations (ODEs) while one of theLie algebra, it is transformed into the well known (2+1)-dimensional BLMPequation. It is affirmed that the proposed techniques are convenient, genuineand powerful tools to find the exact solutions of nonlinear partialdifferential equations (PDEs). Under the suitable choices of arbitraryfunctions and parameters, 2D, 3D and contour graphics to the obtained resultsof GSWW equation are also analyzed graphically.

Lie symmetry analysis and new periodic solitary wave solutions of (3+1)-dimensional generalized shallow water wave equation