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Admissible transformations and the group classification of Schrödinger equations
Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, Faculty of Science & Engineering.
2017 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

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.

The aim of the thesis is twofold. The first is the construction of the new theory of uniform seminormalized 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 equivalence group, with respect to a proper subgroup of the equivalence group as well as 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.

The class of linear Schrödinger equations with complex potentials is of this type 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 seminormalized 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 generalized nonlinear Schrödinger equations which are not uniformly semi-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.

Place, publisher, year, edition, pages
Linköping: Linköping University Electronic Press, 2017. , 7 p.
Series
Linköping Studies in Science and Technology. Dissertations, ISSN 0345-7524 ; 1846
National Category
Mathematical Analysis Algebra and Logic Geometry Computational Mathematics Discrete Mathematics
Identifiers
URN: urn:nbn:se:liu:diva-137424DOI: 10.3384/diss.diva-137424ISBN: 978-91-7685-540-9 (print)OAI: oai:DiVA.org:liu-137424DiVA: diva2:1095590
Public defence
2017-06-01, Nobel BL32, B-huset, Campus Valla, Linköping, 10:00 (English)
Opponent
Supervisors
Available from: 2017-05-15 Created: 2017-05-15 Last updated: 2017-05-17Bibliographically approved
List of papers
1. Equivalence groupoid for (1+2)-dimensional linear Schrodinger equations with complex potentials
Open this publication in new window or tab >>Equivalence groupoid for (1+2)-dimensional linear Schrodinger equations with complex potentials
2015 (English)In: SEVENTH INTERNATIONAL WORKSHOP: GROUP ANALYSIS OF DIFFERENTIAL EQUATIONS AND INTEGRABLE SYSTEMS (GADEISVII), IOP Publishing: Conference Series / Institute of Physics (IoP) , 2015, Vol. 621, no UNSP 012008, UNSP 012008- p.Conference paper, Published paper (Refereed)
Abstract [en]

We describe admissible point transformations in the class of (1+2)-dimensional linear Schrodinger equations with complex potentials. We prove that any point transformation connecting two equations from this class is the composition of a linear superposition transformation of the corresponding initial equation and an equivalence transformation of the class. This shows that the class under study is semi-normalized.

Place, publisher, year, edition, pages
IOP Publishing: Conference Series / Institute of Physics (IoP), 2015
Series
Journal of Physics Conference Series, ISSN 1742-6588 ; 621
National Category
Mathematics
Identifiers
urn:nbn:se:liu:diva-120668 (URN)10.1088/1742-6596/621/1/012008 (DOI)000357939100008 ()
Conference
7th International Workshop on Group Analysis of Differential Equations and Integrable Systems (GADEIS)
Available from: 2015-08-20 Created: 2015-08-20 Last updated: 2017-05-15

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