Abstract:
We study admissible transformations and solve group classiﬁcation problems for various classes of linear and nonlinear Schrödinger equations with an arbitrary number n of space variables. Theaimofthethesisistwofold. Theﬁrstistheconstructionofthenewtheory of uniform semi-normalized classes of diﬀerential equations and its application to solving group classiﬁcation problems for these classes. Point transformations connecting two equations (source and target) from the class under study may have special properties of semi-normalization. This makes the group classiﬁcation of that class using the algebraic method more involved. To extend this method we introduce the new notion of uniformly semi-normalized classes. Various types of uniform semi-normalization are studied: with respect to the corresponding equivalencegroup,withrespecttoapropersubgroupoftheequivalencegroupaswellas the corresponding types of weak uniform semi-normalization. An important kind of uniform semi-normalization is given by classes of homogeneous linear diﬀerential equations, which we call uniform semi-normalization with respect to linear superposition of solutions. TheclassoflinearSchrödingerequationswithcomplexpotentialsisofthistype and its group classiﬁcation can be eﬀectively carried out within the framework of the uniform semi-normalization. Computing the equivalence groupoid and the equivalence group of this class, we show that it is uniformly semi-normalized with respect to linear superposition of solutions. This allow us to apply the version of the algebraic method for uniformly semi-normalized classes and to reduce the group classiﬁcation of this class to the classiﬁcation of appropriate subalgebras of its equivalence algebra. To single out the classiﬁcation cases, integers that are invariant under equivalence transformations are introduced. The complete group classiﬁcation of linear Schrödinger equations is carried out for the cases n =1 and n =2. The second aim is to study group classiﬁcation problem for classes of generalizednonlinearSchrödingerequationswhicharenotuniformlysemi-normalized. We ﬁnd their equivalence groupoids and their equivalence groups and then conclude whether these classes are normalized or not. The most appealing classes are the class of nonlinear Schrödinger equations with potentials and modular nonlinearities and the class of generalized Schrödinger equations with complex-valued and, in general, coeﬃcients of Laplacian term. Both these classes are not normalized. The ﬁrst is partitioned into an inﬁnite number of disjoint normalized subclasses of three kinds: logarithmic nonlinearity, power nonlinearity and general modular nonlinearity. The properties of the Lie invariance algebras of equations from each subclass are studied for arbitrary space dimension n, and the complete group classiﬁcation is carried out for each subclass in dimension (1+2). The second class is successively reduced into subclasses until we reach the subclass of (1+1)dimensional linear Schrödinger equations with variable mass, which also turns out to be non-normalized. We prove that this class is mapped by a family of point transformations to the class of (1+1)-dimensional linear Schrödinger equations with unique constant mass.