Analytic solutions of a microstructure PDE and the KdV and Kadomtsev–Petviashvili equations by invariant Painlevé analysis and generalized Hirota techniques

Abstract

Truncated Painlevé expansions, invariant Painlevé analysis, and generalized Hirota expansions are employed in combination to solve (‘partially reduce to quadrature’) the integrable KdV and KP equations, and a nonintegrable generalized microstructure (GMS) equation. Although the multisolitons of the KdV and KP equations are very well-known, the solutions obtained here for all the three NLPDEs are novel and non-trivial. The solutions obtained via invariant Painlevé analysis are all complicated rational functions, with arguments which themselves are confluent hypergeometric (KdV) or trigonometric (GMS) functions of various distinct non-traveling (KdV) and traveling wave variables. In some cases, this is slightly reminiscent of doubly-periodic elliptic function solutions when nonlinear ODE systems are reduced to quadratures. The solutions obtained by the use of recently-generalized Hirota-type expansions in the truncated Painlevé expansions are closer in functional form to conventional hyperbolic secant solutions, although with non-trivial traveling-wave arguments which are distinct for the three NLPDEs considered here.