Positons of the modified Korteweg de Vries equation

The concept of positons, i.e. certain multiparametric solutions of the Korteweg de Vries equation with new properties, is extended to the modified Korteweg de Vries equation. It is shown that the essential features of positons carry over to this case; the collision of positons, the solitary-wave-positon interaction and simple generalizations are discussed in detail. Suggestions for future research and possible applications of the present work are sketched.     

Wronskian determinant solutions of the (3 + 1)-dimensional Jimbo-Miwa equation

A set of sufficient conditions consisting of systems of linear partial differential equations is obtained which guarantees that the Wronskian determinant solves the (3 + 1)-dimensional Jimbo-Miwa equation in the bilinear form. Upon solving the linear conditions, the resulting Wronskian formulations bring solution formulas, which can yield rational solutions, solitons, negatons, positons and interaction solutions.     

A second Wronskian formulation of the Boussinesq equation

A Wronskian formulation leading to rational solutions is presented for the Boussinesq equation. It involves third-order linear partial differential equations, whose representative systems are systematically solved. The resulting solutions formulas provide a direct but powerful approach for constructing rational solutions, positon solutions and complexiton solutions to the Boussinesq equation. Various examples of exact solutions of those three kinds are computed. The newly presented Wronskian formulation is different from the one previously presented by Li et al., which does not yield rational solutions.