The theory of group classification of differential equations is analyzed, substantially extended and enhanced based on the new notions of conditional equivalence group and normalized class of differential equations. Effective new techniques are proposed. Using these, we exhaustively describe admissible point transformations in classes of nonlinear (1 + 1)-dimensional Schrödinger equations, in particular, in the class of nonlinear (1 + 1)-dimensional Schrödinger equations with modular nonlinearities and potentials and some subclasses thereof. We then carry out a complete group classification in this class, representing it as a union of disjoint normalized subclasses and applying a combination of algebraic and compatibility methods. Moreover, we introduce the complete classification of (1 + 2)-dimensional cubic Schrödinger equations with potentials. The proposed approach can be applied to studying symmetry properties of a wide range of differential equations.Keywords Group classification of differential equations · Group analysis of differential equations · Equivalence group · Admissible transformations · Normalized classes of differential equations · Lie symmetry · Nonlinear Schrödinger equations R.O. Popovych ( ) · M. Kunzinger

Using a new powerful technique based on the notion of megaideal, we construct a complete set of inequivalent realizations of real Lie algebras of dimension no greater than four in vector fields on a space of an arbitrary (finite) number of variables. Our classification amends and essentially generalizes earlier works on the subject.Known results on classification of low-dimensional real Lie algebras, their automorphisms, differentiations, ideals, subalgebras and realizations are reviewed. On classification of Lie algebrasThe necessary step to classify realizations of Lie algebras is classification of these algebras, i.e. classification of possible commutative relations between basis elements. By the Levi-Maltsev theorem any finite-dimensional Lie algebra over a field of characteristic 0 is a semi-direct sum (the Levi-Maltsev decomposition) of the radical (its maximal solvable ideal) and a semi-simple subalgebra (called the Levi factor) (see, e.g., [25]). This result reduces the task of classifying all Lie algebras to the following problems:1) classification of all semi-simple Lie algebras; 2) classification of all solvable Lie algebras;3) classification of all algebras that are semi-direct sums of semi-simple Lie algebras and solvable Lie algebras.Of the problems listed above, only that of classifying all semi-simple Lie algebras is completely solved in the well-known Cartan theorem: any semi-simple complex or real Lie algebra can be decomposed into a direct sum of ideals which are simple subalgebras being mutually orthogonal with respect to the Cartan-Killing form. Thus, the problem of classifying semi-simple Lie algebras is equivalent to that of classifying all non-isomorphic simple Lie algebras. This classification is known (see, e.g., [14,5]).At the best of our knowledge, the problem of classifying solvable Lie algebras is completely solved only for Lie algebras of dimension up to and including six (see, for example, [48,49,50,51,80,81]). Below we shortly list some results on classifying of low-dimensional Lie algebras.All the possible complex Lie algebras of dimension ≤ 4 were listed by S. Lie himself [37]. In 1918 L. Bianchi investigated three-dimensional real Lie algebras [7]. Considerably later this problem was again considered by H.C. Lee [32] and G. Vranceanu [85], and their classifications are equivalent to Bianchi's one. Using Lie's results on complex structures, G.I. Kruchkovich [27,28,29] classified four-dimensional real Lie algebras which do not contain three-dimensional abelian subalgebras.Complete, correct and easy to use classification of real Lie algebras of dimension ≤ 4 was first carried out by G.M. Mubarakzyanov [49] (see also citation of these results as well as description of subalgebras and invariants of real low-dimensional Lie algebras in [58,59]). At the same year a variant of such classification was obtained by J. Dozias [13] and then adduced in [84]. Analogous results are given in [62]. Namely, after citing classifications of L. Bianchi [7] and G.I. Kruchkovich [27], A.Z. Petrov clas...

Using a new method and additional (conditional and partial) equivalence transformations, we performed group classification in a class of variable coefficientWe obtain new interesting cases of such equations with the density f localized in space, which have non-trivial invariance algebra. Exact solutions of these equations are constructed. We also consider the problem of investigation of the possible local transformations for an arbitrary pair of equations from the class under consideration, i.e. of describing all the possible partial equivalence transformations in this class.

This paper completes investigation of symmetry properties of nonlinear variable coefficient -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. Exact solutions of the Fujita-Storm equation u t = (u −2 u x ) x are constructed.

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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 group classification is performed with respect to convenient equivalence groups (generalized extended and conditional ones) and with respect to the set of all local transformations. Usage of different equivalences and coefficient gauges plays the major role for simple and clear formulation of the final results. The corresponding set of admissible transformations is described exhaustively. Then, using the most direct method, we classify local conservation laws. Some exact solutions are constructed by the classical Lie method.

A new approach to group classification problems and more general investigations on transformational 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 coefficientis studied from the symmetry point of view in the framework of the approach proposed. The singular subclass of the equations with m = 2 is singled out. The group classifications of the entire class, the singular subclass and their images are performed with respect to both the corresponding (generalized extended) equivalence groups and all point transformations. The set of admissible transformations of the imaged class is exhaustively described in the general case m = 2. The procedure of classification of nonclassical symmetries, which involves mappings between classes of differential equations, is discussed. Wide families of new exact solutions are also constructed for equations from the classes under consideration by the classical method of Lie reductions and by generation of new solutions from known ones for other equations with point transformations of different kinds (such as additional equivalence transformations and mappings between classes of equations).

We give a comprehensive analysis of interrelations between the basic concepts of Ž . the modern theory of symmetry classical and non-classical reductions of partial differential equations. Using the introduced definition of reduction of differential Ž . equations we establish equivalence of the non-classical conditional symmetry and Ž . direct Ansatz approaches to reduction of partial differential equations. As an illustration we give an example of non-classical reduction of the nonlinear wave equation in 1 q 3 dimensions. The conditional symmetry approach when applied to the equation in question yields a number of non-Lie reductions which are far-reaching generalizations of the well-known symmetry reductions of the nonlinear wave equations.