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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)
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)
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.
This paper completes investigation of symmetry properties of nonlinear 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 which was started in [9–11]. Potential symmetries of equations from the considered class are found and the connection of them with Lie symmetries of diffusion-type equations is shown.… (More)
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)
We show that the so-called hidden potential symmetries considered in a recent paper  are ordinary potential symmetries that can be obtained using the method introduced by Bluman and collaborators [7, 8]. In fact, these are simplest potential symmetries associated with potential systems which are constructed with single conservation laws having no… (More)