Special Polynomials and Exact Solutions of the Dispersive Water Wave and Modified Boussinesq Equations

Exact solutions of the dispersive and modified equations are expressed in terms of special polynomials associated with rational solutions of the fourth Painleve equation, which arises as generalized scaling reductions of these equations. Generalized solutions that involve an infinite sequence of arbitrary constants are also derived which are analogues of generalized rational solutions for the Korteweg-de Vries, Boussinesq and nonlinear Schrodinger equations