Lie symmetry and exact solution of (2+1)-dimensional generalized Kadomtsev-Petviashvili equation with variable coefficients

The simple direct method is adopted to find Non-Auto-Backlund transformation for variable coefficient non-linear systems. The (2+1)-dimensional generalized Kadomtsev-Petviashvili equation with variable coefficients is used as an example to elucidate the solution procedure, and its symmetry transformation and exact solutions are obtained

Extended Gram-type determinant, wave and rational solutions to two (3+1)-dimensional nonlinear evolution equations

a b s t r a c t New exact Grammian determinant solutions to two (3+1)-dimensional nonlinear evolution equations are derived. Extended set of sufficient conditions consisting of linear partial differential equations with variable-coefficients is presented. Moreover, a systematic analysis of linear partial differential equations is used for solving the representative linear systems. The bilinear Bäcklund transformations are also constructed for the equations. Also, as an application of the bilinear Bäcklund transforms, a new class of wave and rational solutions to the equations are explicitly computed

Wronskian, Grammian and Pfaffian Solutions to Nonlinear Partial Differential Equations

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

Pfaffian and Wronskian solutions to generalized integrable nonlinear partial differential equations

The aim of this work is to use the Pfaffian technique, along with the Hirota bilinear method to construct different classes of exact solutions to various of generalized integrable nonlinear partial differential equations. Solitons are among the most beneficial solutions for science and technology, from ocean waves to transmission of information through optical fibers or energy transport along protein molecules. The existence of multi-solitons, especially three-soliton solutions, is essential for information technology: it makes possible undisturbed simultaneous propagation of many pulses in both directions. The derivation and solutions of integrable nonlinear partial differential equations in two spatial dimensions have been the holy grail in the field of nonlinear science since the late 1960s. The prestigious Korteweg-de Vries (KdV) and nonlinear Schrödinger (NLS) equations, as well as the ,Kadomtsev-Petviashvili (KP) and Davey-Stewartson (DS) equations, are prototypical examples of integrable nonlinear partial differential equations in (1+1) and (2+1) dimensions, respectively. Do there exist Pfaffian and soliton solutions to generalized integrable nonlinear partial differential equations in (3+1) dimensions? In this dissertation, I obtained a set of explicit exact Wronskian, Grammian, Pfaffian and N-soliton solutions to the (3+1)-dimensional generalized integrable nonlinear partial differential equations, including a generalized KP equation, a generalized B-type KP equation, a generalized modified B-type KP equation, soliton equations of Jimbo-Miwa type, the nonlinear Ma-Fan equation, and the Jimbo-Miwa equation. A set of sufficient conditions consisting of systems of linear partial differential equations involving free parameters and continuous functions is generated to guarantee that the Wronskian determinant or the Pfaffian solves these generalized equations. On the other hand, as part of this dissertation, bilinear Bäcklund transformations are formally derived for the (3+1)-dimensional generalized integrable nonlinear partial differential equations: a generalized B-type KP equation, the nonlinear Ma-Fan equation, and the Jimbo-Miwa equation. As an application of the obtained Bäcklund transformations, a few classes of traveling wave solutions, rational solutions and Pfaffian solutions to the corresponding equations are explicitly computed. Also, as part of this dissertation, I would like to apply the Pfaffianization mechanism of Hirota and Ohta to extend the (3+1)-dimensional variable-coefficient soliton equation of Jimbo-Miwa type to coupled systems of nonlinear soliton equations, called Pfaffianized systems. Examples of the Wronskian, Grammian, Pfaffian and soliton solutions are explicitly computed. The numerical simulations of the obtained solutions are illustrated and plotted for different parameters involved in the solutions

Pfaffian and Wronskian solutions to generalized integrable nonlinear partial differential equations

The aim of this work is to use the Pfaffian technique, along with the Hirota bilinear method to construct different classes of exact solutions to various of generalized integrable nonlinear partial differential equations. Solitons are among the most beneficial solutions for science and technology, from ocean waves to transmission of information through optical fibers or energy transport along protein molecules. The existence of multi-solitons, especially three-soliton solutions, is essential for information technology: it makes possible undisturbed simultaneous propagation of many pulses in both directions. The derivation and solutions of integrable nonlinear partial differential equations in two spatial dimensions have been the holy grail in the field of nonlinear science since the late 1960s. The prestigious Korteweg-de Vries (KdV) and nonlinear Schrödinger (NLS) equations, as well as the ,Kadomtsev-Petviashvili (KP) and Davey-Stewartson (DS) equations, are prototypical examples of integrable nonlinear partial differential equations in (1+1) and (2+1) dimensions, respectively. Do there exist Pfaffian and soliton solutions to generalized integrable nonlinear partial differential equations in (3+1) dimensions? In this dissertation, I obtained a set of explicit exact Wronskian, Grammian, Pfaffian and N-soliton solutions to the (3+1)-dimensional generalized integrable nonlinear partial differential equations, including a generalized KP equation, a generalized B-type KP equation, a generalized modified B-type KP equation, soliton equations of Jimbo-Miwa type, the nonlinear Ma-Fan equation, and the Jimbo-Miwa equation. A set of sufficient conditions consisting of systems of linear partial differential equations involving free parameters and continuous functions is generated to guarantee that the Wronskian determinant or the Pfaffian solves these generalized equations. On the other hand, as part of this dissertation, bilinear Bäcklund transformations are formally derived for the (3+1)-dimensional generalized integrable nonlinear partial differential equations: a generalized B-type KP equation, the nonlinear Ma-Fan equation, and the Jimbo-Miwa equation. As an application of the obtained Bäcklund transformations, a few classes of traveling wave solutions, rational solutions and Pfaffian solutions to the corresponding equations are explicitly computed. Also, as part of this dissertation, I would like to apply the Pfaffianization mechanism of Hirota and Ohta to extend the (3+1)-dimensional variable-coefficient soliton equation of Jimbo-Miwa type to coupled systems of nonlinear soliton equations, called Pfaffianized systems. Examples of the Wronskian, Grammian, Pfaffian and soliton solutions are explicitly computed. The numerical simulations of the obtained solutions are illustrated and plotted for different parameters involved in the solutions