Lie symmetry analysis, conservation laws and solitary wave solutions to a fourth-order nonlinear generalized Boussinesq water wave equation

Abstract A fourth-order nonlinear generalized Boussinesq water wave equation is studied in this work, which describes the propagation of long waves in shallow water. We employ Lie symmetry method to study its vector fields and optimal systems. Moreover, we derive its symmetry reductions and twelve families of soliton wave solutions by using the optimal systems, including hyperbolic-type, trigonometric-type, rational-type, Jacobi elliptic-type and Weierstrass elliptic-type solutions. Two of reduced equations are Painleve-like equations. Finally, the complete set of local conservation laws is presented with a detailed derivation by using the conservation law multiplier.

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