On a geometrical property of the Bäcklund transformation of the sine-Gordon equation

We show how a solution of the Bäcklund transformation equations for the sine-Gordon equation determines an explicit isometry from the underlying surface to the upper half plane. We then use the action on the upper half to find explicitly the general solution of the equations.     

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.     

Bilinear Bcklund transformation and explicit solutions for a nonlinear evolution equation

The bilinear form of two nonlinear evolution equations are derived by using Hirota derivative. The B\"{a}cklund transformation based on the Hirota bilinear method for these two equations are presented, respectively. As an application, the explicit solutions including soliton and stationary rational solutions for these two equations are obtained.     

Connection formulae for the Kametake and Bäcklund transformation for the toda lattice

It is shown that the Kametake transformation for the Toda lattice is equivalent to the Bäcklund transformation in its standard form.     

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.     

Bäcklund transformations for the generalized sine-gordon equation in 2+1 and 3+1 dimensions

We construct Bäcklund transformations for the generalized sine-Gordon equations in 2+1 and 3+1 dimensions. The connection of these equations with the nonlinear model is considered.     

On the permutability of Bäcklund transformations: 1 infinitesimal Bäcklund transformations of the sine-Gordon equation

Letu′=B _a u be the Bäcklund transformation of the sine-Gordon equation, we prove that $$B_{a + \varepsilon } B_a^{ - 1} u = u + \varepsilon \sum\limits_{n = 0}^\infty {2D^{ - 1} } \frac{{\delta G_{n + 1} }}{{\delta u}}a^{2n} ,$$ where {G _n} is an infinite set of conserved densities of the sine-Gordon equation and η ⁿ ≡D ^?1δG _n /δu are just the symmetries obtained by Olver [17]. Basing upon this expansion, we prove the equivalence between the permutability of the infinitesimal Bäcklund transformations and the involution of the conserved densities of the sine-Gordon equation.     

Gauge and Bäcklund transformations for the generalized sine-Gordon equation and its η-dependent modified equation

We study the generalized sine-Gordon hierarchy and its associated-dependent modified sine-Gordon hierarchy. Two Bäcklund transformations for these two families are constructed. One of them is a generalization of the Bäcklund transformations of Wadatiet al. and the other one is new. Gauge transformations of a relevant AKNS system are employed to reduce the integration of these equations via the Bäcklund transformations to quadratures. Three generations of explicit solutions of the sine-Gordon equation are presented.     

On Bianchi and Bäcklund Transformations of Two-Dimensional Surfaces in E4

In this paper we discuss the existence and properties of the Bianchi transformations for pseudospherical surfaces in E ⁴. The results of the paper show that the theory of Bianchi transformations in the discussed case is essentially different from the well-known case of pseudospherical surfaces in E ³ (in general n-manifolds of constant and negative curvature in E ^{2 n − 1}). This revised version was published online in July 2006 with corrections to the Cover Date.     

On Bianchi permutability of Bäcklund transformations for asymmetric quad-equations

We prove the Bianchi permutability (existence of superposition principle) of Bäcklund transformations for asymmetric quad-equations. Such equations and their Bäcklund transformations form 3D consistent systems of a priori different equations. We perform this proof by using 4D consistent systems of quad-equations, the structural insights through biquadratics patterns and the consideration of super-consistent eight-tuples of quad-equations on decorated cubes.     

Solitons,Bäcklund transformation and Lax pair for a variable-coefficient generalized Boussinesq system in the shallow water

Under investigation in this paper is a variable-coefficient generalized Boussinesq system, which describes the propagation of long weakly non-linear and weakly dispersive surface waves in the shallow water. Bilinear form, Bäcklund transformation and Lax pair are derived by virtue of the Bell polynomials. One- and two-soliton solutions are constructed via the Hirota method. Propagation characteristics and collision behaviors of the solitons are discussed through the graphical analysis. Elastic collisions of the two bell- and periodic-shaped solitons are obtained. Head-on collision between the two solitons occurs when the sign of the two wave numbers are different; Otherwise, the overtaking collision happens.     

Bäcklund transformation for the first flows of the relativistic Toda hierarchy and associated properties

In this communication we study a class of one parameter dependent auto-Bäcklund transformations for the first flow of the relativistic Toda lattice and also a variant of the usual Toda lattice equation. It is shown that starting from the Hamiltonian formalism such transformations are canonical in nature with a well defined generating function. The notion of spectrality is also analyzed and the separation variables are explicitly constructed.     

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.     

Auto-Bäcklund transformations and solitary wave solutions for the nonlinear evolution equation

In the present work, according to the concept of extended homogeneous balance method and with help of Maple, we get auto-Bäcklund transformations for a (2 + 1)-dimensional nonlinear evolution equation. Subsequently, by using these auto-Bäcklund transformation, exact explicit solutions of this equation are obtained.     

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.     

Universal invariance properties of Painlevé analysis and Bäcklund transformation in nonlinear partial differential equations

We define two transformations, independent of the polynomial PDE, making all the compatibility conditions at resonances and all the equations defining the Bäcklund transformation invariant by homographic transformation of the expansion function. The resulting dramatic shortening of invariant equations makes their solution feasible by hand, leading to new analytic solutions.