Bäcklund Transformations and Inverse Scattering Solutions for Some Pseudospherical Surface Equations

In is known that the equations [u_t ? g(u)u_x]x = ±g′(u) describe pseudo-spherical surfaces, i.e. that these equations are the integrability conditions for the structural equations of such surfaces, provided g satisfies g′ + µg = θ. In this paper we obtain self-Bäcklund transformations for these equations by a geometric method, and show how the inverse scattering method generates global solutions.     

On the Solution of the Generalized Wave and Generalized Sine-Gordon Equations

The generalized wave equation and generalized sine-Gordon equations are known to be natural multidimensional differential geometric generalizations of the classical two-dimensional versions. In this paper we associate a system of linear differential equations with these equations and show how the direct and inverse problems can be solved for appropriately decaying data on suitable lines. An initial-boundary-value problem is solved for these equations.     

New Bilinear Bäcklund Transformation and Lax Pair for the Supersymmetric Two‐Boson Equation

In this paper, I introduce a class of super Bell polynomials, which are found to play an important role in the characterization of super supersymmetric equations. An effective approach based on the use of the super Bell polynomials is developed to systematically investigate the bilinearization, Bäcklund transformation, and Lax pair for supersymmetric equations. I take a supersymmetric two‐boson equation to illustrate this procedure. A new bilinear Bäcklund transformation and a Lax pair with both fermionic and bosonic parameters are given. In addition, a kind of exact solitons for the equation are further constructed with the help of the bilinear Bäcklund transformation.     

Bäcklund transformations for higher order Painlevé equations

We present a new generalized algorithm which allows the construction of Bäcklund transformations (BTs) for higher order ordinary differential equations (ODEs). This algorithm is based on the idea of seeking transformations that preserve the Painlevé property, and is applied here to ODEs of various orders in order to recover, amongst others, their auto-BTs. Of the ODEs considered here, one is seen to be of particular interest because it allows us to show that auto-BTs can be obtained in various ways, i.e. not only by using the severest of the possible restrictions of our algorithm.     

Bäcklund Transformations and Superposition Principles in Nonlinear Elastodynamics

A Bäcklund transformation is presented which allows the construction of model nonlinear elastic laws associated with a reduction of the uniaxial Lagrangian elastodynamic system to the canonical form adopted for the classical Hooke's law. A nonlinear superposition principle for the iterative generation of such elastic laws is recorded.     

Conditional Lie–Bäcklund Symmetries and Invariant Subspaces to Nonlinear Diffusion Equations with Convection and Source

The conditional Lie–Bäcklund symmetry method is used to study the invariant subspace of the nonlinear diffusion equations with convection and source terms. We obtain a complete list of canonical forms for such equations which admit higher order conditional Lie–Bäcklund symmetries and multidimensional invariant subspaces. The functionally generalized separable solutions to the resulting equations are constructed due to the corresponding symmetry reductions. For most of the cases, they are reduced to solving finite‐dimensional dynamical systems.     

On a Painlevé II Model in Steady Electrolysis: Application of a Bäcklund Transformation

A model equation of Painlevé II type was introduced by Bass in 1964 in connection with a boundary value problem which describes the electric field distribution in a region x > 0 occupied by an electrolyte. This is possibly the earliest explicit physical application of a Painlevé equation to be found in the literature. Here we return to this problem informed by the subsequent discovery of a Bäcklund transformation for Painlevé II. This enables us to construct exact representations for the electric field and ion distributions for boundary value problems wherein the ratio of fluxes of the positive and negative ions at the boundary adopts one of an infinite sequence of values.     

Energy Scattering for the Generalized Davey—Stewartson Equations

Abstract Considring the generalized Davey-Stewartson equation i-△u+λ│u│~pu+μE(│u│~q)│u│~(q-2)u=0,where λ>0,μ≥0,E=F~(-1)(ξ_1~2│ξ│~2)F,we obtain the existence of scattering operator in ∑(R~n):u{u∈H~1(R~n):│x│u∈L~2(R~n)}.     

Soliton solutions and Bäcklund transformation for the Kupershmidt five-field lattice: a bilinear approach

The Kupershmidt five-field lattice is considered in this paper. By a dependent variable transformation, the Kupershmidt lattice is transformed into a bilinear form by the introduction of three auxiliary variables. We present a Bäcklund transformation and a nonlinear superposition formula for the Kupershmidt lattice. As an application of the results, soliton solutions are derived.     

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.     

A Supersymmetric AKNS Problem and Its Darboux‐Bäcklund Transformations and Discrete Systems

In this paper, we consider a supersymmetric AKNS spectral problem. Two elementary and a binary Darboux transformations are constructed. By means of reductions, Darboux and Bäcklund transformations are given for the supersymmetric modified Korteweg‐de Vries, sinh‐Gordon, and nonlinear Schrödinger equations. These Darboux and Bäcklund transformations are adopted for the constructions of integrable discrete super systems, and both semidiscrete and fully discrete systems are presented. Also, the continuum limits of the relevant discrete systems are worked out.     

Conditional Lie‐Bäcklund Symmetry of Evolution System and Application for Reaction‐Diffusion System

Conditional Lie‐Bäcklund symmetry (CLBS) method is developed to study system of evolution equations. It is shown that reducibility of a system of evolution equations to a system of ordinary differential equations can be fully characterized by the CLBS of the considered system. As an application of the approach, a class of two‐component nonlinear diffusion equations is studied. The governing system and the admitted CLBS can be identified. As a consequence, exact solutions defined on the polynomial, exponential, trigonometric, and mixed invariant subspaces are constructed due to the corresponding symmetry reductions.     

The vector generalization of the Landau–Lifshitz equation: Bäcklund transformation and solutions

The Bäcklund transformations (BTs) for vector generalization of the Landau–Lifshitz equation are presented. Also periodic and soliton-like solutions from BTs are computed.     

Forced Nonlinear Evolution Equations and the Inverse Scattering Transform

The Korteweg—de Vries and nonlinear Schrödinger equations with an external forcing of distribution type are considered. The reflection coefficient is found to satisfy a nonlinear equation of a certain characteristic form which also appears in the semi-infinite problem.     

Bäcklund transformation of fractional Riccati equation and its applications to the space–time FDEs

In this paper, the Bäcklund transformation of fractional Riccati equation is presented to establish traveling wave solutions for two nonlinear space–time fractional differential equations in the sense of modified Riemann–Liouville derivatives, namely, the space–time fractional generalized reaction duffing equation and the space–time fractional diffusion reaction equation with cubic nonlinearity. The proposed method is effective and convenient for solving nonlinear evolution equations with fractional order. Copyright © 2014 John Wiley & Sons, Ltd.