Bäcklund transformations and the infinite-dimensional symmetry group of the Kadomtsev-Petviashvili equation

The infinite-dimensional symmetry group of the potential Kadomtsev-Petviashvili (PKP) equation is found and used to obtain a Bäcklund transformation, involving two arbitrary functions of time. This transformation is then used to generate several different types of solutions from the zero solution of the PKP equation.     

B?cklund transformations,consistent Riccati expansion solvability,and soliton–cnoidal interaction wave solutions of Kadomtsev–Petviashvili equation

The famous Kadomtsev-Petviashvili(KP)equation 1 s a classical equation In soliton tneory.A Backlund transformation between the KP equation and the Schwarzian KP equation is demonstrated by means of the truncated Painlev6 expansion in this paper.One-parameter group transformations and one-parameter subgroup-invariant solutions for the extended KP equation are obtained.The consistent Riccati expansion(CRE) solvability of the KP equation is proved.Some interaction structures between soliton-cnoidal waves are obtained by CRE and several evolution graphs and density graphs are plotted.     

Unified approach to Miura,Bäcklund and Darboux Transformations for Nonlinear Partial Differential Equations

Abstract

This paper is an attempt to present and discuss at some length the Singular Manifold Method. This Method is based upon the Painlevé Property systematically used as a tool for obtaining clear cut answers to almost all the questions related with Nonlinear Partial Differential Equations: Lax pairs, Miura, Bäcklund or Darboux Transformations as well as τ -functions, in a unified way. Besides to present the basics of the Method we exemplify this approach by applying it to four equations in (1+1)-dimensions. Two of them are related with the other two through Miura transformations that are also derived by using the Singular Manifold Method.     

The residual symmetry,B?cklund transformations,CRE integrability and interaction solutions:(2+1)-dimensional Chaffee-Infante equation

In this paper, we consider the(2+1)-dimensional Chaffee–Infante equation, which occurs in the fields of fluid dynamics, high-energy physics, electronic science etc. We build B?cklund transformations and residual symmetries in nonlocal structure using the Painlevé truncated expansion approach. We use a prolonged system to localize these symmetries and establish the associated one-parameter Lie transformation group. In this transformation group, we deliver new exact solution profiles via the combi...     

Exact solutions of the Drinfel’d–Sokolov–Wilson equation using Bäcklund transformation of Riccati equation and trial function approach

In this paper, two integration schemes are employed to obtain solitons, singular periodic waves and other types of solutions of the Drinfel’d–Sokolov–Wilson equation. The two schemes studied in this paper are the Bäcklund transformation of Riccati equation and the trial function approach. The corresponding constraint conditions of the solutions are also given.     

Bäcklund transformations,kink soliton,breather- and travelling-wave solutions for a (3+1)-dimensional B-type Kadomtsev–Petviashvili equation in fluid dynamics

In this paper, we investigate a (3+1)-dimensional B-type Kadomtsev–Petviashvili (BKP) equation in fluid dynamics. Based on the Hirota method, we give a bilinear auto-Bäcklund transformation. Via the truncated Painlevé expansion, we get a Painlevé-type auto-Bäcklund transformation. With the aid of the symbolic computation, we derive some one- and two-kink soliton solutions. We present the oblique and parallel elastic interactions between the two-kink solitons. Via the extended homoclinic test technique, we construct some breather-wave solutions. Besides, we derive some lump solutions with the periods of the breather-wave solutions to the infinity. We observe that the shapes of a breather wave and a lump remain unchanged during the propagation. Based on the polynomial-expansion method, travelling-wave solutions are constructed.     

Exact PsTd invariant and PsTd symmetric breaking solutions,symmetry reductions and Bäcklund transformations for an AB–KdV system

In natural and social science, many events happened at different space–times may be closely correlated. Two events, A (Alice) and B (Bob) are defined as correlated if one event is determined by another, say, $B=\stackrel{?}{f}A$ for suitable $\stackrel{?}{f}$ operators. A nonlocal AB–KdV system with shifted-parity ($P_s$, parity with a shift), delayed time reversal ($T_d$, time reversal with a delay) symmetry where $B=\stackrel{?}{P_s}\stackrel{?}{T_d}A$ is constructed directly from the normal KdV equation to describe two-area physical event. The exact solutions of the AB–KdV system, including $P_s{T}_d$ invariant and $P_s{T}_d$ symmetric breaking solutions are shown by different methods. The $P_s{T}_d$ invariant solution show that the event happened at A will happen also at B. These solutions, such as single soliton solutions, infinitely many singular soliton solutions, soliton–cnoidal wave interaction solutions, and symmetry reduction solutions etc., show the AB–KdV system possesses rich structures. Also, a special Bäcklund transformation related to residual symmetry is presented via the localization of the residual symmetry to find interaction solutions between the solitons and other types of the AB–KdV system.     

Bäcklund–Darboux transformations and discretizations of N = 2 a = −2 supersymmetric KdV equation

The $N=2a=?2$ supersymmetric KdV equation is studied. A Darboux transformation and the corresponding Bäcklund transformation are constructed for this equation. Also, a nonlinear superposition formula is worked out for the associated Bäcklund transformation. The Bäcklund transformation and the related nonlinear superposition formula are used to construct integrable super semi-discrete and full discrete systems. The continuum limits of these discrete systems are also considered.     

Painlevé Analysis,Soliton Collision and B?cklund Transformation for the (3+1)-Dimensional Variable-Coefficient Kadomtsev–Petviashvili Equation in Fluids or Plasmas

In this paper, we investigate a(3+1)-dimensional generalized variable-coefficient Kadomtsev–Petviashvili equation, which can describe the nonlinear phenomena in fluids or plasmas. Painlev′e analysis is performed for us to study the integrability, and we find that the equation is not completely integrable. By virtue of the binary Bell polynomials,bilinear form and soliton solutions are obtained, and B¨acklund transformation in the binary-Bell-polynomial form and bilinear form are derived. Soliton collisions are graphically discussed: the solitons keep their original shapes unchanged after the collision except for the phase shifts. Variable coefficients are seen to affect the motion of solitons: when the variable coefficients are chosen as the constants, solitons keep their directions unchanged during the collision; with the variable coefficients as the functions of the temporal coordinate, the one soliton changes its direction.