Stationary mKdV hierarchy and integrability of the Dirac equations by quadratures

Using the Lie's infinitesimal method we establish that the Dirac equation in one variable is integrable by quadratures if the potential V(x) is a solution of one of the equations of the stationary mKdV hierarchy.Comment: 6 pages, LaTe

A precise definition of reduction of partial differential equations

We give a comprehensive analysis of interrelations between the basic concepts of the modern theory of symmetry (classical and non-classical) reductions of partial differential equations. Using the introduced definition of reduction of differential equations we establish equivalence of the non-classical (conditional symmetry) and direct (Ansatz) approaches to reduction of partial differential equations. As an illustration we give an example of non-classical reduction of the nonlinear wave equation in (1+3) dimensions. The conditional symmetry approach when applied to the equation in question yields a number of non-Lie reductions which are far-reaching generalization of the well-known symmetry reductions of the nonlinear wave equations.Comment: LaTeX, 21 page

Symmetry classification of third-order nonlinear evolution equations. Part I: Semi-simple algebras

We give a complete point-symmetry classification of all third-order evolution equations of the form $u_t=F(t,x,u,u_x, u_{xx})u_{xxx}+G(t,x,u,u_x, u_{xx})$ which admit semi-simple symmetry algebras and extensions of these semi-simple Lie algebras by solvable Lie algebras. The methods we employ are extensions and refinements of previous techniques which have been used in such classifications.Comment: 53 page

Enhanced Group Analysis and Exact Solutions of Variable Coefficient Semilinear Diffusion Equations with a Power Source

A new approach to group classification problems and more general investigations on transformational properties of classes of differential equations is proposed. It is based on mappings between classes of differential equations, generated by families of point transformations. A class of variable coefficient (1+1)-dimensional semilinear reaction-diffusion equations of the general form $f(x)u_t=(g(x)u_x)_x+h(x)u^m$ ($m\ne0,1$) is studied from the symmetry point of view in the framework of the approach proposed. The singular subclass of the equations with $m=2$ is singled out. The group classifications of the entire class, the singular subclass and their images are performed with respect to both the corresponding (generalized extended) equivalence groups and all point transformations. The set of admissible transformations of the imaged class is exhaustively described in the general case $m\ne2$. The procedure of classification of nonclassical symmetries, which involves mappings between classes of differential equations, is discussed. Wide families of new exact solutions are also constructed for equations from the classes under consideration by the classical method of Lie reductions and by generation of new solutions from known ones for other equations with point transformations of different kinds (such as additional equivalence transformations and mappings between classes of equations).Comment: 40 pages, this is version published in Acta Applicanda Mathematica

Infinitely many local higher symmetries without recursion operator or master symmetry: integrability of the Foursov--Burgers system revisited

We consider the Burgers-type system studied by Foursov, w_t &=& w_{xx} + 8 w w_x + (2-4\alpha)z z_x, z_t &=& (1-2\alpha)z_{xx} - 4\alpha z w_x + (4-8\alpha)w z_x - (4+8\alpha)w^2 z + (-2+4\alpha)z^3, (*) for which no recursion operator or master symmetry was known so far, and prove that the system (*) admits infinitely many local generalized symmetries that are constructed using a nonlocal {\em two-term} recursion relation rather than from a recursion operator.Comment: 10 pages, LaTeX; minor changes in terminology; some references and definitions adde

Group Analysis of Variable Coefficient Diffusion-Convection Equations. I. Enhanced Group Classification

We discuss the classical statement of group classification problem and some its extensions in the general case. After that, we carry out the complete extended group classification for a class of (1+1)-dimensional nonlinear diffusion--convection equations with coefficients depending on the space variable. At first, we construct the usual equivalence group and the extended one including transformations which are nonlocal with respect to arbitrary elements. The extended equivalence group has interesting structure since it contains a non-trivial subgroup of non-local gauge equivalence transformations. The complete group classification of the class under consideration is carried out with respect to the extended equivalence group and with respect to the set of all point transformations. Usage of extended equivalence and correct choice of gauges of arbitrary elements play the major role for simple and clear formulation of the final results. The set of admissible transformations of this class is preliminary investigated.Comment: 25 page

Conditional Lie-Bäcklund symmetry and reduction of evolution equations

We suggest a generalization of the notion of invariance of a given partial differential equation with respect to Lie-Backlund vector field. Such generalization proves to be effective and enables us to construct principally new Ansatze reducing evolution-type equations to several ordinary differential equations. In the framework of the said generalization we obtain principally new reductions of a number of nonlinear heat conductivity equations u t = u xx + F (u; u x ) with poor Lie symmetry and obtain their exact solutions. It is shown that these solutions can not be constructed by means of the symmetry reduction procedure

Separation of variables in the two-dimensional wave equation with potential

The paper is devoted to solution of a problem of separation of variables in the wave equation utt−uxx+V(x)u=0. We give a complete classification of potentials V(x) for which this equation admits a nontrivial separation of variables. Furthermore, we obtain all coordinate systems that provide separability of the equation considered

Reduction of the self-dual Yang-Mills equations. I. The Poincaré group

For the vector potential of the Yang-Mills field, we give a complete description of ansatzes invariant under three-parameterP (1, 3) -inequivalent subgroups of the Poincaré group. By using these ansatzes, we reduce the self-dual Yang-Mills equations to a system of ordinary differential equations.Для вектор-потенціалу поля Янга - Міллса побудовано повний набір інваріантних відносно Р(1,3)- нееквівалентних підгруп групи Пуанкаре анзаців, з використанням яких проведено редукцію самодуальних рівнянь Янга - Мілса до систем звичайних диференціальних рівнянь