Nonlinear Maxwell Equations

A new relativistic invariant version of nonlinear Maxwell equations is offerred. Some properties of these equations are considered.Comment: 6 pages, LaTe

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

Graded Symmetry Algebras of Time-Dependent Evolution Equations and Application to the Modified KP equations

By starting from known graded Lie algebras, including Virasoro algebras, new kinds of time-dependent evolution equations are found possessing graded symmetry algebras. The modified KP equations are taken as an illustrative example: new modified KP equations with $m$ arbitrary time-dependent coefficients are obtained possessing symmetries involving $m$ arbitrary functions of time. A particular graded symmetry algebra for the modified KP equations is derived in this connection homomorphic to the Virasoro algebras.Comment: 19 pages, latex, to appear in J. Nonlinear Math. Phy

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

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