Admissible transformations and the group classification of Schrödinger equations

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Fältnamn	Indikatorer	Metadata
000		06223nam a2200493 4500
001		oai:DiVA.org:liu-137424
003		SwePub
008		170515s2017 \| \|\|\|\|\|\|\|\|\|\|\|000 \|\|eng\|
020		a 9789176855409q print
024	7	a https://urn.kb.se/resolve?urn=urn:nbn:se:liu:diva-1374242 URI
024	7	a https://doi.org/10.3384/diss.diva-1374242 DOI
040		a (SwePub)liu
041		a engb eng
042		9 SwePub
072	7	a vet2 swepub-contenttype
072	7	a dok2 swepub-publicationtype
100	1	a Kurujyibwami, Celestinu Linköpings universitet,Matematik och tillämpad matematik,Tekniska fakulteten4 aut0 (Swepub:liu)celku57
245	1 0	a Admissible transformations and the group classification of Schrödinger equations
264	1	a Linköping :b Linköping University Electronic Press,c 2017
300		a 7 s.
338		a electronic2 rdacarrier
490	0	a Linköping Studies in Science and Technology. Dissertations,x 0345-7524 ;v 1846
520		a 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.
650	7	a NATURVETENSKAPx Matematikx Matematisk analys0 (SwePub)101012 hsv//swe
650	7	a NATURAL SCIENCESx Mathematicsx Mathematical Analysis0 (SwePub)101012 hsv//eng
650	7	a NATURVETENSKAPx Matematikx Algebra och logik0 (SwePub)101032 hsv//swe
650	7	a NATURAL SCIENCESx Mathematicsx Algebra and Logic0 (SwePub)101032 hsv//eng
650	7	a NATURVETENSKAPx Matematikx Geometri0 (SwePub)101022 hsv//swe
650	7	a NATURAL SCIENCESx Mathematicsx Geometry0 (SwePub)101022 hsv//eng
650	7	a NATURVETENSKAPx Matematikx Beräkningsmatematik0 (SwePub)101052 hsv//swe
650	7	a NATURAL SCIENCESx Mathematicsx Computational Mathematics0 (SwePub)101052 hsv//eng
650	7	a NATURVETENSKAPx Matematikx Diskret matematik0 (SwePub)101042 hsv//swe
650	7	a NATURAL SCIENCESx Mathematicsx Discrete Mathematics0 (SwePub)101042 hsv//eng
700	1	a Basarab-Horwath, Peter,c Professoru Linköpings universitet,Matematik och tillämpad matematik,Tekniska fakulteten4 ths0 (Swepub:liu)petba92
700	1	a Popovych, Roman,c Professoru Department of Mathematical Physics Institute of Mathematics National Academy of Sciences of Ukraine4 ths
700	1	a Damianou, Pontelis,c Professoru University of Cyprus, Cypern4 opn
710	2	a Linköpings universitetb Matematik och tillämpad matematik4 org
856	4	u https://doi.org/10.3384/diss.diva-137424y Fulltext
856	4	u https://liu.diva-portal.org/smash/get/diva2:1095590/COVER01.pdfy cover
856	4	u https://liu.diva-portal.org/smash/get/diva2:1095590/PREVIEW01.jpgx Previewy preview image
856	4	u https://liu.diva-portal.org/smash/get/diva2:1095590/FULLTEXT02.pdfx primaryx Raw objecty fulltext
856	4	u http://liu.diva-portal.org/smash/get/diva2:1095590/FULLTEXT02
856	4 8	u https://urn.kb.se/resolve?urn=urn:nbn:se:liu:diva-137424
856	4 8	u https://doi.org/10.3384/diss.diva-137424

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