Wronskian, Grammian and Pfaffian Solutions to Nonlinear Partial Differential Equations

Abstract

It is significantly important to search for exact soliton solutions to nonlinear partial differential equations (PDEs) of mathematical physics. Transforming nonlinear PDEs into bilinear forms using the Hirota differential operators enables us to apply the Wronskian and Pfaffian techniques to search for exact solutions for a (3+1)-dimensional generalized Kadomtsev-Petviashvili (KP) equation with not only constant coefficients but also variable coefficients under a certain constraint (ut + α 1(t)uxxy + 3α 2(t)uxuy)x +α 3 (t)uty -α 4(t)uzz + α 5(t)(ux + α 3(t)uy) = 0. However, bilinear equations are the nearest neighbors to linear equations, and expected to have some properties similar to those of linear equations. We have explored a key feature of the linear superposition principle, which linear differential equations have, for Hirota bilinear equations, while intending to construct a particular sub-class of N-soliton solutions formed by linear combinations of exponential traveling waves. Applications are given for the (3+1) dimensional KP, Jimbo-Miwa (JM) and BKP equations, thereby presenting their particular N-wave solutions. An opposite question is also raised and discussed about generating Hirota bilinear equations possessing the indicated N-wave solutions, and two illustrative examples are presented. Using the Pfaffianization procedure, we have extended the generalized KP equation to a generalized KP system of nonlinear PDEs. Wronskian-type Pfaffian and Gramm-type Pfaffian solutions of the resulting Pfaffianized system have been presented. Our results and computations basically depend on Pfaffian identities given by Hirota and Ohta. The Pl̈ucker relation and the Jaccobi identity for determinants have also been employed. A (3+1)-dimensional JM equation has been considered as another important example in soliton theory, uyt - uxxxy - 3(uxuy)x + 3uxz = 0. Three kinds of exact soliton solutions have been given: Wronskian, Grammian and Pfaffian solutions. The Pfaffianization procedure has been used to extend this equation as well. Within Wronskian and Pfaffian formulations, soliton solutions and rational solutions are usually expressed as some kind of logarithmic derivatives of Wronskian and Pfaffian type determinants and the determinants involved are made of functions satisfying linear systems of differential equations. This connection between nonlinear problems and linear ones utilizes linear theories in solving soliton equations. B̈acklund transformations are another powerful approach to exact solutions of nonlinear equations. We have computed different classes of solutions for a (3+1)-dimensional generalized KP equation based on a bilinear B̈acklund transformation consisting of six bilinear equations and containing nine free parameters. A variable coefficient Boussinesq (vcB) model in the long gravity water waves is one of the examples that we are investigating, ut + α 1 (t)uxy + α 2(t)(uw)x + α 3(t)vx = 0; vt + β1(t)(wvx + 2vuy + uvy) + β2(t)(uxwy - (uy)2) + β3(t)vxy + β4(t)uxyy = 0, where wx = uy. Double Wronskian type solutions have been constructed for this (2+1)-dimensional vcB model