New exact special solutions with solitary patterns for Boussinesq-like B(m, n) equations with fully nonlinear dispersion

In this paper, a new approach, the extend sinh–cosh method, is proposed, to investigate the exact solutions with solitary patterns of the Boussinesq-like equations with fully nonlinear dispersion, B(m, n) equations: utt + (um)xx − (un)xxxx = 0. The new exact special solutions with solitary patterns of the equations are found by our new method. The two special cases, B(2, 2) and B(3, 3), are chosen to illustrate the concrete scheme of our approach in B(m, n) equations. The nonlinear equations B(m, n) are addressed for two different cases, namely when m = n being odd and even integers. An entirely new general formulas for the solutions of B(n, n) equations are established. The general formulas for the solutions of B(n, n) equations with all integer n > 1 can be extended to the case of B(m, n) equations with m = n being noninteger. Our results include not only some known results in literature as special cases but also some new exact special solutions with solitary patterns. The method presented by this paper is suitable for studying exact special solutions with solitary patterns of some other equations.