In this paper, the family of BBM equation with strong nonlinear dispersive B(m, n) is considered. We apply the classical Lie method of infinitesimals. The symmetry reductions are derived from the optimal system of subalgebras and lead to systems of ordinary differential equations. We obtain for special values of the parameters of this equation, many exact solutions expressed by various single and combined nondegenerative Jacobi elliptic function solutions and their degenerative solutions (soliton, kink and compactons).

SUMMARYWe apply the Lie-group formalism and the nonclassical method due to Bluman and Cole to deduce symmetries of the generalized Kuramoto-Sivashinsky equation with dispersive effects. We make a full analysis of the symmetry reductions and we prove that the nonclassical method applied to the equation leads to new reductions, which cannot be obtained by Lie classical symmetries. Some new solutions can be derived.

Nonlinear partial differential equations are used to describe complex phenomena in various fields of science. In this work, we consider a generalized fourth‐order nonlinear wave equation from the point of view of the theory of symmetry reductions in partial differential equations. We derive classical symmetries, and we obtain the reductions from the optimal system of subalgebras. We derive all the low‐order conservation laws, and we search for multipliers of the reduced ordinary differential equations that are invariant under a symmetry group to reduce directly the order of the equations.

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