In this article, (3+1)-dimensional generalized Shallow Water-like (gSWl) equation is discussed. The infinitesimal generators of the equation are derived by using the Lie symmetry analysis method. The optimal system is obtained based on the adjoint table of the generators of the equation. Exact solutions of the equation are constructed by applying symmetry reduction, Exp−ϕ(ξ) expansion method, Exp-function expansion method, Riccati equation method, and G′/G expansion method. For analyzing the dynamical behavior of the solutions, we derive the physical structures of dark soliton, kink wave, and periodic solutions via numerical simulations.
In this paper, the logarithmic Monge–Ampère flow evolution equation is studied by the classical Lie symmetry analysis method. First, the optimal system was obtained by the adjoint representation table. Second, by performing symmetric transformation on the optimal system, the corresponding ordinary differential equations are obtained, and the Jacobian elliptic function solution, the periodic solution and the power series solution are constructed. Finally, the dynamical behaviors of the solutions are described by choosing arbitrary parameters.
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