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.     

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.     

Invariant subspaces and conditional Lie-Bäcklund symmetries of inhomogeneous nonlinear diffusion equations

The inhomogeneous nonlinear difusion equation is studied by invariant subspace and conditional Lie-Bcklund symmetry methods.It is shown that the equations admit a class of invariant subspaces governed by the nonlinear ordinary diferential equations,which is equivalent to a kind of higher-order conditional Lie-Bcklund symmetries of the equations.As a consequence,a number of new solutions to the inhomogeneous nonlinear difusion equations are constructed explicitly or reduced to solving fnite-dimensional dynamical systems.     

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 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.     

Cartan’s Structure of Symmetry Pseudo-Group and Coverings for the r-th Modified Dispersionless Kadomtsev–Petviashvili Equation

We derive two non-equivalent coverings for the r-th dKP equation from Maurer–Cartan forms of its symmetry pseudo-group. Also we find Bäcklund transformations between the obtained covering equations.     

Cartan’s Structure Theory of Symmetry Pseudo-Groups, Coverings and Multi-Valued Solutions for the Khokhlov–Zabolotskaya Equation

We derive two non-equivalent coverings for the modified Khokhlov–Zabolotskaya equation from Maurer–Cartan forms of its symmetry pseudo-group. Also we find Bäcklund transformations between the obtained covering equations. We apply these results to constructing multi-valued solutions for the Khokhlov–Zabolotskaya equation.     

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.     

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.     

Abel’s theorem and Bäcklund transformations for the Hamilton-Jacobi equations

We consider an algorithm for constructing auto-Bäcklund transformations for finitedimensional Hamiltonian systems whose integration reduces to the inversion of the Abel map. In this case, using equations of motion, one can construct Abel differential equations and identify the sought Bäcklund transformation with the well-known equivalence relation between the roots of the Abel polynomial. As examples, we construct Bäcklund transformations for the Lagrange top, Kowalevski top, and Goryachev–Chaplygin top, which are related to hyperelliptic curves of genera 1 and 2, as well as for the Goryachev and Dullin–Matveev systems, which are related to trigonal curves in the plane.     

Integrable seepage equations

Well-known results of the classification of parabolic-type differential equations, possessing an infinite Lie–Bäcklund algebra, are used to describe seepage models, which, using (in general) differential substitutions, can be reduced to the heat conduction equation. Relations between the functional parameters, characterizing the properties of the liquid and gas phases and the porous medium, are obtained that ensure the existence of such substitutions.     

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.     

A NEW APPROACH TO SOLVE PERTURBED NONLINEAR EVOLUTION EQUATIONS THROUGH LIE-BACKLUND SYMMETRY METHOD

A new method based on Lie-Backlund symmetry method to solve the perturbed nonlinear evolution equations is presented. New approximate solutions of perturbed nonlinear evolution equations stemming from the exact solutions of unperturbed equations are obtained. This method is a generalization of Burde＇s Lie point symmetry technique.     

On a homoclinic manifold of a coupled long-wave–short-wave system

A Bäcklund transformation is obtained for linearly unstable spatially independent plane-wave solutions of a system of coupled long-wave–short-wave resonance equations. Explicit expressions are constructed for the periodic orbits lying on a homoclinic manifold of a torus of planewaves by evaluating the Bäcklund transformation at double points of an irreducible factor of the Floquet spectral curve of the associated scattering problem.     

Nonlinear self-adjointness of the Krichever–Novikov equation

It is known that the classification of third-order evolutionary equations with the constant separant possessing a nontrivial Lie–Bäcklund algebra (in other words, integrable equations) results in the linear equation, the KdV equation and the Krichever–Novikov equation. The first two of these equations are nonlinearly self-adjoint. This property allows to associate conservation laws of the equations in question with their symmetries. The problem on nonlinear self-adjointness of the Krichever–Novikov equation has not been solved yet. In the present paper we solve this problem and find the explicit form of the differential substitution providing the nonlinear self-adjointness.     

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.     

System of equations for stimulated combination scattering and the related double periodic <Emphasis Type="Italic">A</Emphasis><Stack><Subscript><Emphasis Type="Italic">n</Emphasis></Subscript><Superscript>(1)</Superscript></Stack> Toda chains

We consider a system of equations describing stimulated combination scattering of light. We show that solutions of this system are expressed in terms of two solutions of the sine-Gordon equation that are related to each other by a Bäcklund transformation. We also show that this system is integrable and admits a Zakharov-Shabat pair. In the general case, the system of equations for the Bäcklund transformation of periodic A _n ⁽¹⁾ Toda chains is also shown to be integrable and to have a Zakharov-Shabat pair.     

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.     

Bianchi Permutability for the Anti‐Self‐Dual Yang‐Mills Equations

The anti‐self‐dual Yang‐Mills equations are known to have reductions to many integrable differential equations. A general Bäcklund transformation (BT) for the anti‐self‐dual Yang‐Mills (ASDYM) equations generated by a Darboux matrix with an affine dependence on the spectral parameter is obtained, together with its Bianchi permutability equation. We give examples in which we obtain BTs of symmetry reductions of the ASDYM equations by reducing this ASDYM BT. Some discrete integrable systems are obtained directly from reductions of the ASDYM Bianchi system.