The recursion operator method for generalized symmetries and Bäcklund transformations of the Burgers’ equations

In this paper, the generalized symmetries of the second-order Burgers’ equation are obtained through the symmetry transformation method. The Bäcklund transformations (BTs) of the two equations are constructed by the recursion operator method. Then, the infinite number of exact solutions to these equations are investigated in terms of the generalized symmetries and Bäcklund transformations. Furthermore, the Bäcklund transformations and conservation law of the general Burgers’ equations are discussed.     

An affine interpretation of Bäcklund transformations

The paper is devoted to an affine interpretation of Bäcklundmaps (Bäcklund transformations are a particular case of Bäcklund maps) for second order differential equations with unknown function of two arguments. Note that up to now there are no papers where Bäcklund transformations are interpreted as transformations of surfaces in a space other than Euclidean space. In this paper, we restrict our considerations to the case of so-called Bäcklund maps of class 1. The solutions of a differential equation are represented as surfaces of an affine space with induced connection determining a representation of zero curvature. We show that, in the case when a second order partial differential equation admits a Bäcklund map of class 1, for each solution of the equation there is a congruence of straight lines in an affine space formed by the tangents to the affine image of the solution. This congruence is an affine analog of a parabolic congruence in Euclidean space. The Bäcklund map can be interpreted as a transformation of surfaces of an affine space under which the affine image of a solution of the differential equation is mapped into a particular boundary surface of the congruence.     

Reduction of the Sharma–Tasso–Olver equation and series solutions

This paper considers series solutions of the Sharma–Tasso–Olver (STO) equation. By using the extended homogenous balance method, we reduce the STO equation to a linear PDE and obtain Bäcklund transformation of it. Furthermore, the self-transformation of solutions for the STO equation is obtained. By the Bäcklund transformation and various series solutions of the PDE, abundant exact solutions of the STO equation are obtained including the multi-solitary wave solution, trigonometric function series solution, rational series solution and solution consisting of the three types of solutions.     

New version of Bäcklund transformations in Euclidean 3‐space

In this paper, we study the Bäcklund transformations for the adjoint curve in the Euclidean 3‐space. Firstly, it is obtained some essential equations of the Bäcklund transformation. After this, we give a new theorem, the Bäcklund transformations for the adjoint curve in Euclidean 3‐space.     

Bäcklund transformations for new fourth Painlevé hierarchies

We consider a system of equations defined using the Hamiltonian operator of the Boussinesq hierarchy, as well as two successive modifications thereof. We are able to reduce the order of these three systems and give Bäcklund transformations between the integrated equations. We also give auto-Bäcklund transformations for the two modified systems.Particular cases of two of the three equations considered correspond to generalized fourth Painlevé hierarchies and are new; these are particular cases of the two modified systems. Thus we obtain auto-Bäcklund transformations for these new fourth Painlevé hierarchies, as well as Bäcklund transformations between our hierarchies. Our results on reduction of order are also applicable in this special case, and include as a particular example a reduction of order for the scaling similarity reduction of the Boussinesq equation, a result which, remarkably, seems not to have been given previously.     

The Gross–Pitaevskii equation: Bäcklund transformations and admitted solutions

Bäcklund transformations are applied to study the Gross–Pitaevskii equation. Supported by previous results, a class of Bäcklund transformations admitted by this equation are constructed. Schwarzian derivative as well as its invariance properties turn out to represent a key tool in the present investigation. Examples and explicit solutions of the Gross–Pitaevskii equation are obtained.

     

On the integrability of the (1+1)-dimensional and (2+1)-dimensional Ito equations

Under investigation in this paper are the (1+1)-dimensional and (2+1)-dimensional Ito equations. With the help of the Bell polynomials method, Hirota bilinear method and symbolic computation, the bilinear representations, N-soliton solutions, bilinear Bäcklund transformations and Lax pairs of these two equations are obtained, respectively. In particular, we obtain a new bilinear form and N-soliton solutions of the (2+1)-dimensional Ito equation. The bilinear Bäcklund transformation and Lax pair of the (2+1)-dimensional Ito equation are also obtained for the first time. Copyright © 2014 John Wiley & Sons, Ltd.     

On Non-Abelian Coverings Over the Liouville Equation

Treating the hyperbolic Liouville equation as the flat connections equation on the semisimple Lie algebra A ₁, we investigate relationships between zero-curvature representations of the Liouville equation and its Bäcklund transformations provided by a special one-dimensional coverings. Formal deformations of these Bäcklund transformations and integration in nonlocal variables are studied.     

Lax pairs and Bäcklund transformations for a coupled Ramani equation and its related system

A coupled Ramani equation and its related system are proposed. By dependent variable transformation, they are transformed into bilinear equations. Lax pairs and Bäcklund transformations are presented for these two systems. Soliton solutions and rational solutions to the systems could be obtained.     

On the Nonlinear Schrödinger Equation on the Half Line with Homogeneous Robin Boundary Conditions

Boundary value problems for the nonlinear Schrödinger equations on the half line with homogeneous Robin boundary conditions are revisited using Bäcklund transformations. In particular: relations are obtained among the norming constants associated with symmetric eigenvalues; a linearizing transformation is derived for the Bäcklund transformation; the reflection‐induced soliton position shift is evaluated and the solution behavior is discussed. The results are illustrated by discussing several exact soliton solutions, which describe the soliton reflection at the boundary with or without the presence of self‐symmetric eigenvalues.     

Discretizations of the Landau-Lifshits equation

The relation between the Sklyanin chain and the Bäcklund transformations for the Landau-Lifshits equation is established. The stationary solutions of the chain determine an integrable mapping, which is a kind of classical Heisenberg spin chain. Some multifield generalizations are found.     

Ermakov–Pinney and Emden–Fowler Equations: New Solutions from Novel Bäcklund Transformations

We study the class of nonlinear ordinary differential equations y″ y = F(z, y²), where F is a smooth function. Various ordinary differential equations with a well-known importance for applications belong to this class of nonlinear ordinary differential equations. Indeed, the Emden–Fowler equation, the Ermakov–Pinney equation, and the generalized Ermakov equations are among them. We construct Bäcklund transformations and auto-Bäcklund transformations: starting from a trivial solution, these last transformations induce the construction of a ladder of new solutions admitted by the given differential equations. Notably, the highly nonlinear structure of this class of nonlinear ordinary differential equations implies that numerical methods are very difficult to apply.     

Integrable vector isotropic equations admitting differential substitutions of first order

A symmetry classification of integrable vector evolution equations of third order admitting Miura-type transformations is presented. We obtain the Bäcklund autotransformation for the new equation as well as differential substitutions relating the solutions of some integrable isotropic equations.     

Complex high order Toda and Volterra lattices1

Given a solution of a high order Toda lattice we construct a one parameter family of new solutions. In our method, we use a set of Bäcklund transformations such that each new generalized Toda solution is related to a generalized Volterra solution.     

Bäcklund maps and Lie–Bäcklund transformations as differential-geometric structures

This paper is an exposition of the author’s report prepared for the International Conference devoted to the centennial anniversary of G. F. Laptev (Laptev seminar–2009). In the first section, we consider Bäcklund transformations of second-order partial differential equations. In the present work, the theory of Bäcklund transformations is treated as a special branch of the theory of connections. The second section is devoted to differential-geometric structures generated by the so-called Lie–Bäcklund transformations (or, equivalently, contact transformations of higher order) that are a special case of diffeomorphisms between the manifolds of holonomic jets. Recall that it was G. F. Laptev who pointed out the possibility of considering differentiable mappings as differential-geometric structures.     

The painlevé property and coordinate transformations

In this paper, the question of conserving the Painlevé property of partial differential equations via coordinate transformations between partial differential equations is studied. Also, the effects of some types of transformations, like ordinary Bäcklund as well as auto-Bäcklund transformations of partial differential equations, are shown as well. Some features and comments, concerning higher order prolongations of these transformations as well as of the partial differential equations themselves, are given.     

Rational Solutions of Painlevé Equations

With Bäcklund transformations, we construct explicit solutions of Painlevé equations 2 and 4. Independently, we find solutions of degenerate cases of equations 3 and 5. The six Painlevé transcendents are referred to as 1–6.     

Spectral deformation and Bäcklund transformation of constrained Willmore surfaces

The class of constrained Willmore surfaces in space-forms forms a Möbius invariant class of surfaces with strong links to the theory of integrable systems. This paper is dedicated to an overview on the topic. We define a spectral deformation, by the action of a loop of flat metric connections, and Bäcklund transformations, by applying a dressing action. We establish a permutability between spectral deformation and Bäcklund transformation and verify that all these transformations corresponding to the zero multiplier preserve the class of Willmore surfaces. We show that, for special choices of parameters, both spectral deformation and Bäcklund transformation preserve the class of constrained Willmore surfaces admitting a conserved quantity, and, in particular, the class of constant mean curvature surfaces in 3-dimensional space-forms.     

Conditional Lie–Bäcklund symmetries and functionally generalized separable solutions to the generalized porous medium equations with source

The functionally generalized separable solutions of the generalized porous medium equations with power law and exponential diffusivity are studied by using the conditional Lie–Bäcklund symmetry method. The variant forms of the considered equations, which admit the linear conditional Lie–Bäcklund symmetries, are identified. A number of examples are considered and some exact solutions, defined on the polynomial, trigonometric and exponential invariant subspaces determined by the linear conditional Lie–Bäcklund symmetries, are constructed.     

Bäcklund transformations for the difference Hirota equation and the supersymmetric Bethe ansatz

We consider GL(K|M)-invariant integrable supersymmetric spin chains with twisted boundary conditions and demonstrate the role of Bäcklund transformations in solving the difference Hirota equation for eigenvalues of their transfer matrices. We show that the nested Bethe ansatz technique is equivalent to a chain of successive Bäcklund transformations “undressing” the original problem to a trivial one.