H4-Solutions for the Olver–Benney equation

The Olver–Benney equation is a nonlinear fifth-order equation, which describes the interaction effects between short and long waves. In this paper, we prove the global existence of solutions of the Cauchy problem associated with this equation.

The Residual Symmetry and Consistent Tanh Expansion for the Benney System

The residual symmetry of the (2+1)-dimensional Benney system is derived from the truncated Painlevé expansion. Such residual symmetry is localised and the original Benney equation is extended into an enlarged system by introducing four new variables. By using Lies first theorem, we obtain the finite transformation for the localised residual symmetry. More importantly, we further localise the linear superposition of multiple residual symmetries and construct the nth Bäcklund transformation for the Benney system in the form of the determinant. Moreover, it is proved that the (2+1)-dimensional Benney system is consistent tanh expansion (CTE) solvable. The exact interaction solutions between solitons and any other types of potential Burgers waves are also obtained, which include soliton-error function waves, soliton-periodic waves, and so on.

A COMBINATION METHOD AND ITS APPLICATIONS TO NONLINEAR EVOLUTION EQUATIONS

In this paper, a combination method is developed to construct the explicit and exact solutions of nonlinear evolution equations (NEEs). As illustrative examples, the explicit and exact solutions of some physically significant NEEs, comprising the generalized BBM–Burgers equation, generalized KdV–Burgers equation and generalized Benney equation are successfully presented by this means.