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.     

Auto-Bäcklund transformations for a matrix partial differential equation

We derive auto-Bäcklund transformations, analogous to those of the matrix second Painlevé equation, for a matrix partial differential equation. We also then use these auto-Bäcklund transformations to derive matrix equations involving shifts in a discrete variable, a process analogous to the use of the auto-Bäcklund transformations of the matrix second Painlevé equation to derive a discrete matrix first Painlevé equation. The equations thus derived then include amongst other examples a semidiscrete matrix equation which can be considered to be an extension of this discrete matrix first Painlevé equation. The application of this technique to the auto-Bäcklund transformations of the scalar case of our partial differential equation has not been considered before, and so the results obtained here in this scalar case are also new. Other equations obtained here using this technique include a scalar semidiscrete equation which arises in the case of the second Painlevé equation, and which does not seem to have been thus derived previously.     

Existence of strong Bäcklund transformations in four or more dimensions and generalization of a family of Bäcklund transformations

A Bäcklund transformation between two (sets of) differential equations is strong if the transformation equations already imply the two equations. For each dimension n=2 ^k , k1, the existence of such strong transformations is proved by constructing a wide variety of them. A simple generalization of a known family of Bäcklund transformations is also given. One such provides a useful analogy for Yang's instanton equations.     

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.     

Bäcklund transformations for certain rational solutions of Painlevé VI

We introduce certain Bäcklund transformations for rational solutions of the Painlevé VI equation. These transformations act on a family of Painlevé VI tau functions. They are obtained from reducing the Hirota bilinear equations that describe the relation between certain points in the 3 component polynomial KP Grassmannian. In this way we obtain transformations that act on the root lattice of A⁵. We also show that this A⁵ root lattice can be related to the F₄⁽¹⁾ root lattice. We thus obtain Bäcklund transformations that relate Painlevé VI tau functions, parametrized by the elements of this F₄⁽¹⁾ root lattice.     

Von mises- and crocco-type hydrodynamical transformations: Order reduction of nonlinear equations,construction of Bäcklund transformations and of new integrable equations

Broad classes of nonlinear equations of mathematical physics are described that admit order reduction by applying the von Mises transformation (with the unknown function used as a new independent variable and with a suitable partial derivative used as a new dependent variable) and by applying the Crocco transformation (with the first and second partial derivatives used as new independent and dependent variables, respectively). Associated Bäcklund transformations are constructed that connect evolution equations of general form (their special cases include Burgers, Korteweg-de Vries, and Harry Dym type equations and many other nonlinear equations of mathematical physics). Transformations are indicated that reduce the order of hydrodynamic-type equations of higher orders. The generalized Calogero equation and a number of other new integrable nonlinear equations, reducible to linear equations, are considered.     

Soliton interactions,Bäcklund transformations,Lax pair for a variable-coefficient generalized dispersive water-wave system

Under investigation in this paper is a variable-coefficient generalized dispersive water-wave system, which can simulate the propagation of the long weakly non-linear and weakly dispersive surface waves of variable depth in the shallow water. Under certain variable-coefficient constraints, by virtue of the Bell polynomials, Hirota method and symbolic computation, the bilinear forms, one- and two-soliton solutions are obtained. Bäcklund transformations and new Lax pair are also obtained. Our Lax pair is different from that previously reported. Based on the asymptotic and graphic analysis, with different forms of the variable coefficients, we find that there exist the elastic interactions for u, while either the elastic or inelastic interactions for v, with u and v as the horizontal velocity field and deviation height from the equilibrium position of the water, respectively. When the interactions are inelastic, we see the fission and fusion phenomena.     

Factorization fo random Jacobi operators and Bäcklund transformations

Summary We show that a positive definite random Jacobi operatorL over an abstract dynamical systemT: XX can be factorized asL=D ², whereD is again a random Jacobi operator but defined over a new dynamical systemS: YY which is an integral extension ofT. An isospectral random Toda deformation ofL corresponds to an isospectral random Volterra deformation ofD. The factorization leads to commuting Bäcklund transformations which can be written explicitly in terms of Titchmarsh-Weyl functions. In the periodic case, the Bäcklund transformations are time 1 maps of a Toda flow with a time dependent Hamiltonian.This article was processed by the author using the Springer-Verlag TEX EconThe macro package 1991.     

Bäcklund transformations between four Lax-integrable 3D equations

Recently a classification of contactly-nonequivalent three-dimensional linearly degenerate equations of the second order was presented by E.V. Ferapontov and J. Moss. The equations are Lax-integrable. In our paper we prove that all these equations are connected with each other by appropriate Bäcklund transformations.     

Bäcklund transformations for nonlinear sigma models with values in Riemannian symmetric spaces

This work deals with Bäcklund transformations for the principal SL(n, ) sigma model together with all reduced models with values in Riemannian symmetric spaces. First, the dressing method of Zakharov, Mikhailov, and Shabat is shown, for the case of a meromorphic dressing matrix, to be equivalent to a Bäcklund transformation for an associated, linearly extended system. Comparison of this multi-Bäcklund transformation with the composition of ordinary ones leads to a new proof of the permutability theorem. A new method of solution for such multi-Bäcklund transformations (MBT) is developed, by the introduction of a soliton correlation matrix which satisfies a Riccati system equivalent to the MBT. Using the geometric structure of this system, a linearization is achieved, leading to a nonlinear superposition formula expressing the solution explicitly in terms of solutions of a single Bäcklund transformation through purely linear algebraic relations. A systematic study of all reductions of the system by involutive automorphisms is made, thereby defining the multi-Bäcklund transformations and their solution for all Riemannian symmetric spaces.Supported in part by the Natural Sciences and Engineering Research Council of Canada, and by the Fonds FCAC pour l'aide et le soutien à la recherche     

Recursion operator and bäcklund transformations for the Ruijsenaars-Toda lattice

We exhibit the recursion operator and the whole class of Bäcklund transformations for a relativistic version of the Toda lattice recently introduced by Ruijsenaars. These results allow us to prove the complete integrability of the system.     

Bäcklund transformations and local conservation laws for self-dual Yang-Mills fields with arbitrary gauge group

A process is described for deriving infinite sets of local conservation laws for self-dual Yang-Mills fields with arbitrary gauge group. This process utilizes a set of recently constructed Bäcklund transformations which leave the self-duality equation invariant.     

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.     

Laplace,Bäcklund and gauge transformations in the two-dimensional Toda lattice

The linear Bäcklund transformation (LBT) associated with the two-dimensional Toda lattice is shown to be equivalent to a sequence of Laplace transformations of a hyperbolic linear differential equation. When the Toda lattice is cut at a point, the corresponding Laplace invariant vanishes and the LBT can be integrated.     

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.     

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.     

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.