Ivanova N. Classification problems for diffusion-convection equations and Schrodinger equations

The new approach for construction of conservation laws is proposed. The notion of equivalence of conservation laws with respect to local transformation group is introduced. We introduce the notion of local dependence of potentials. The local conservation laws of diffusion-convection equations are classifyed, complete lists of inequivalent potential conservation laws and inequivalent potential systems are constructed. Potential equivalence transformations and potential symmetries are found. Connection between potential and Lie symmetries via potential equivalence transformations is investigated. The complete group classification in the class of nonlinear diffusion-convection equations with variable coefficients is performed. Exact invariant solutions are constructed for some subclasses. The problems of group classification for classes of nonlinear Schrodinger equations with potentials are solved. Sufficient conditions of existence and uniqueness of solutions of Cauchy problem for some subclasses of nonlinear Schrodinger equations with potentials are found.