We carry out the complete group classification of the class of (1+1)-dimensional linear Schrödinger equations with complex-valued potentials. After introducing the notion of uniformly semi-normalized classes of differential equations, we compute the equivalence groupoid of the class under study and show that it is uniformly semi-normalized. More specifically, each admissible transformation in the class is the composition of a linear superposition transformation of the corresponding initial equation and an equivalence transformation of this class. This allows us to apply the new version of the algebraic method based on uniform semi-normalization and reduce the group classification of the class under study to the classification of low-dimensional appropriate subalgebras of the associated equivalence algebra. The partition into classification cases involves two integers that characterize Lie symmetry extensions and are invariant with respect to equivalence transformations.
We suggest a new approach to the problem of dimensional reduction of initial/boundary value problems for evolution equations in one spatial variable. The approach is based on higher-order (generalized) conditional symmetries of the equations involved. It is shown that reducibility of an initial value problem for an evolution equation to a Cauchy problem for a system of ordinary di erential equations can be fully characterized in terms of conditional symmetries which leave invariant the equation in question. We also give some examples of the solution of initial value problems for second-and third-order nonlinear di erential equations by reduction by their conditional symmetries. We give a systematic classi cation of general second-order partial di erential equations admitting second-order conditional symmetries, based on Lie's classi cation of invariant secondorder ordinary di erential equations. This yields ve classes of principally new initial value problems for nonlinear evolution equations which admit no Lie symmetries and are reducible via second-order conditional symmetries.
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