Bäcklund transformations of the axially symmetric stationary Einstein-Maxwell equations II

We obtained a Bäcklund transformation of the axially symmetric stationary Einstein-Maxwell equation which is a generalization of Neugebauer's I₁ transformation in the case of the Ernst equations. This transformation is shown to generate the generalized Ehlers transformation and the Harrison transformation.     

A variational approach to bäcklund and infinitesimal bäcklund transformation for Kadomtsev-Petviashvili equation

A new form of the Bäcklund transformation for the Kadomtsev-Petviashvili equation is obtained through the variational formalism. At no stage we use the equation of motion for the deduction of the Bäcklund transformation. So we follow the method of Steudel for the derivation of the infinite number of conservation laws using Noether's theorem in three dimensions and the corresponding infinitesimal Bäcklund transformation.     

Bäcklund transformations,nonlocal symmetry and exact solutions of a generalized (2+1)-dimensional Korteweg–de Vries equation

In this paper, nonlocal residual symmetry of a generalized (2+1)-dimensional Korteweg–de Vries equation is derived with the aid of truncated Painlevé expansion. Three kinds of non-auto and auto Bäcklund transformations are established. The nonlocal symmetry is localized to a Lie point symmetry of a prolonged system by introducing auxiliary dependent variables. The linear superposed multiple residual symmetries are presented, which give rise to the $n$th Bäcklund transformation. The consistent Riccati expansion method is employed to derive a Bäcklund transformation. Furthermore, the soliton solutions, fusion-type $N$-solitary wave solutions and soliton–cnoidal wave solutions are gained through Bäcklund transformations.     

Ableitung einer kontinuierlichen Mannigfaltigkeit von Erhaltungssätzen der modifizierten Korteweg-de-Vries-Gleichung nach dem Noetherschen Satz

Derivation of a Continuous Set of Conservation Laws for the Modified Korteweg-de Vries Equation by Noether's Theorem A method developed recently to derive a continuous set of conservation laws from extended Bäcklund transformations by means of Noether's theorem is applied to the modified Korteweg-de Vries (m. KdV) equation that describes Alfvén waves in a plasma. The corresponding conserved currents are equivalent to those found by WADATI , SANUKI and KONNO . It is shown that the extended Bäcklund transformation B?_α for the m. KdV equation, which coincides with that for the sine-Gordon equation, by MIURA'S transformation becomes the extended Bäcklund transformation β_x for the Korteweg-de Vries equation where x = 1/2α.     

Bäcklund transformations and soliton-type solutions for σ models with values in real Grassmannian spaces

A Bäcklund transformation for the two-dimensional σ model with values in oriented real Grassmannian spaces is constructed using the known Bäcklund transformation for the SO(n) principal models. The construction provides a natural way to linearize the differential system of the Bäcklund transformation. In the case of the S ⁿ≈SO(n+1/SOn) model, the Backlund transformation reduces to that of Pohlmeyer. The one-soliton solution for S ²～SO(3)/SO(2) is obtained analytically and plots of one-soliton and two-soliton solutions are displayed.     

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.     

A method of obtaining the N-soliton solution of the Boussinesq equation in terms of a wronskian

By means of a Bäcklund transformation for the Boussinesq equation in bilinear form, the wronskian form of the N-soliton solution of that equation is deduced. The solution is then verified by direct substitution in the equation.     

一类非线性耦合系统的复合型双孤子新解

首先给出一种函数变换,把一类非线性耦合系统化为两个第一种椭圆方程组.然后利用第一种椭圆方程的新解与B?cklund变换,构造了一类非线性耦合系统的无穷序列复合型双孤子新解.     

Bäcklund transformation of matrix equations and a discrete matrix first Painlevé equation

We show that the known auto-Bäcklund transformation for the matrix second Painlevé equation can be generalized to a much wider class of equations. This auto-Bäcklund transformation is an involution and so cannot be used on its own to generate an infinite sequence of different solutions, although for particular equations a second auto-Bäcklund transformation allows this to be done. We also give a Bäcklund transformation for this general class of matrix equations. For the matrix second Painlevé equation we also give a coalescence limit, and a construction of special integrals and of a discrete matrix first Painlevé equation.     

Darboux Transformation for the Vector Sine-Gordon Equation and Integrable Equations on a Sphere

We propose a method for construction of Darboux transformations, which is a new development of the dressing method for Lax operators invariant under a reduction group. We apply the method to the vector sine-Gordon equation and derive its Bäcklund transformations. We show that there is a new Lax operator canonically associated with our Darboux transformation resulting an evolutionary differential-difference system on a sphere. The latter is a generalised symmetry for the chain of Bäcklund transformations. Using the re-factorisation approach and the Bianchi permutability of the Darboux transformations, we derive new vector Yang–Baxter map and integrable discrete vector sine-Gordon equation on a sphere.     

On Bianchi’s Bäcklund transformation of quadrics

We investigate basic features of Bianchi’s Bäcklund transformation of quadrics and obtain it under weaker minimal assumptions: the only Bäcklund transformation with defining surface is Bianchi’s Bäcklund transformation of quadrics.     

COMMUTATIVITY OF PFAFFIANIZATION AND BÄCKLUND TRANSFORMATIONS: THE LEZNOV LATTICE

In this paper, we first obtain Wronskian solutions to the Bäcklund transformation of the Leznov lattice and then derive the coupled system for the Bäcklund transformation through Pfaffianization. It is shown the coupled system is nothing but the Bäcklund transformation for the coupled Leznov lattice introduced by J. Zhao etc. [1]. This implies that Pfaffianization and Bäcklund transformation is commutative for the Leznov lattice. Moreover, since the two-dimensional Toda lattice constitutes the Leznov lattice, it is obvious that the commutativity is also valid for it.     

The exact solution to Boussinesq equation through a limiting procedure

It is shown that if a new bilinear Bäcklund transformation to Boussinesq equation has been reduced by the gauge transformation, it always has another set of new N-soliton solution by performing an appropriate limiting procedure on the already known ones.     

On the modified discrete KP equation with self-consistent sources

The modified discrete KP equation is the Bäcklund transformation for the Hirota’s discrete KP equation or the Hirota-Miwa equation. We construct the modified discrete KP equation with self-consistent sources via source generation procedure and clarify the algebraic structure of the resulting coupled modified discrete KP system by presenting its discrete Gram-type determinant solutions. It is also shown that the commutativity between the source generation procedure and Bäcklund transformation is valid for the discrete KP equation. Finally, we demonstrate that the modified discrete KP equation with self-consistent sources yields the modified differential-difference KP equation with self-consistent sources through a continuum limit. The continuum limit of an explicit solution to the modified discrete KP equation with self-consistent sources also gives the explicit solution for the modified differential-difference KP equation with self-consistent sources.     

A Constructive Approach to the Soliton Solutions of Integrable Quadrilateral Lattice Equations

Scalar multidimensionally consistent quadrilateral lattice equations are studied. We explore a confluence between the superposition principle for solutions related by the Bäcklund transformation, and the method of solving a Riccati map by exploiting two known particular solutions. This leads to an expression for the N-soliton-type solutions of a generic equation within this class. As a particular instance we give an explicit N-soliton solution for the primary model, which is Adler’s lattice equation (or Q4).     

Discrete bäcklund transformation

A discretized Bäcklund transformation is given for the discrete sine-Gordon equation, and the general soliton solution is obtained.     

Bäcklund transformations applied to the cylindrical Korteweg-de Vries equation

A hierarchy of solutions generated by the Bäcklund transformation for the cylindrical KdV equation is derived in terms of Airy functions. The 2nth solution is given explicitly and shown to reduce to the n-soliton solution obtained by Calogero and Degasperis when the eigenvalues are taken equal in pairs. The solution corresponds to that obtained for the second Painlevé equation by Airault when the eigenvalues tend to zero.     

Supersymmetric two-boson equation: Bilinearization and solutions

A bilinear formulation for the supersymmetric two-boson equation is derived. As applications, some solutions are calculated for it. We also construct a bilinear Bäcklund transformation.     

Dynamics days : Twente,June 7–9, 1983 annual informal workshop on nonlinear dynamics

A hierarchy of isospectral deformation equations related to the Kaup-Newell spectral problem is obtained. The generalized gauge transformations of the eigenmatrix of the spectral problem are used to generate Bäcklund and self-Bäcklund transformations. In a special case, the Bäcklund transformation of the null solution for all the equations in the hierarchy has been explicitly solved.     

A DIRECT PROCEDURE ON THE INTEGRABILITY OF NONISOSPECTRAL AND VARIABLE-COEFFICIENT MKdV EQUATION

An elementary and systematic method based on binary Bell polynomials is applied to nonisospectral and variable-coefficient MKdV (vcMKdV) equation. The bilinear representation, bilinear Bäcklund transformation, Lax pair and infinite local conservation laws are obtained step by step, without too much clever guesswork.