After discussing the classical statement of group classification problem and some its extensions in the general case, 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… (More)
This is the second part of the series of papers on symmetry properties of a class of variable coefficient (1+1)-dimensional nonlinear diffusion–convection equations of general form f (x)u t = (g(x)A(u)u x) x + h(x)B(u)u x. At first, we review the results of  on equivalence transformations and group classification of the class under consideration.… (More)
We study local conservation laws of variable coefficient diffusion–convection equations of the form f (x)u t = (g(x)A(u)u x) x + h(x)B(u)u x. The main tool of our investigation is the notion of equivalence of conservation laws with respect to the equivalence groups. That is why, for the class under consideration we first construct the usual equivalence… (More)
A class of variable coefficient (1+1)-dimensional nonlinear reaction–diffusion equations of the general form f (x)u t = (g(x)u n u x) x + h(x)u m is investigated. Different kinds of equivalence groups are constructed including ones with transformations which are nonlocal with respect to arbitrary elements. For the class under consideration the complete… (More)
The notions of generating sets of conservation laws of systems of differential equations with respect to symmetry groups and equivalence groups are introduced and applied. This allows us to generalize essentially the procedure of finding potential symmetries for the systems with multidimensional spaces of conservation laws. A class of variable coefficient… (More)
A new approach to group classification problems and more general investigations on transfor-mational 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… (More)
We prove that the classical, non–periodic Toda lattice is super–integrable. In other words, we show that it possesses 2N − 1 independent constants of motion, where N is the number of degrees of freedom. The main ingredient of the proof is the use of some special action–angle coordinates introduced by Moser to solve the equations of motion.
Group classification of a class of nonlinear fin equations is carried out exhaustively. Additional equivalence transformations and conditional equivalence groups are also found. These allow to simplify results of classification and further applications of them. The derived Lie symmetries are used to construct exact solutions of truly nonlinear equations for… (More)