| 1. |
- Kurujyibwami, Celestin
(författare)
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Admissible transformations and the group classification of Schrödinger equations
- 2017
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Doktorsavhandling (övrigt vetenskapligt/konstnärligt)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.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.
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| 2. |
- Kurujyibwami, Celestin, et al.
(författare)
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Algebraic Method for Group Classification of (1+1)-Dimensional Linear Schrodinger Equations
- 2018
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Ingår i: Acta Applicandae Mathematicae - An International Survey Journal on Applying Mathematics and Mathematical Applications. - : SPRINGER. - 0167-8019 .- 1572-9036. ; 157:1, s. 171-203
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Tidskriftsartikel (refereegranskat)abstract
- We carry out the complete group classification of the class of (1+1)-dimensional linear Schrodinger 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.
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| 3. |
- Kurujyibwami, Célestin, et al.
(författare)
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Algebraic method for group classification of (1+1)-dimensional linear Schrödinger equations
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Annan publikation (övrigt vetenskapligt/konstnärligt)abstract
- 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.
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| 4. |
- Kurujyibwami, Celestin
(författare)
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Equivalence groupoid for (1+2)-dimensional linear Schrodinger equations with complex potentials
- 2015
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Ingår i: SEVENTH INTERNATIONAL WORKSHOP: GROUP ANALYSIS OF DIFFERENTIAL EQUATIONS AND INTEGRABLE SYSTEMS (GADEISVII). - : IOP Publishing: Conference Series / Institute of Physics (IoP). ; , s. UNSP 012008-
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Konferensbidrag (refereegranskat)abstract
- 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.
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| 5. |
- Kurujyibwami, Célestin
(författare)
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Group classification of linear Schrödinger equations by the algebraic method
- 2016
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Licentiatavhandling (övrigt vetenskapligt/konstnärligt)abstract
- This thesis is devoted to the group classification of linear Schrödinger equations. The study of Lie symmetries of such equations was initiated more than 40 years ago using the classical Lie infinitesimal method for specific types of real-valued potentials. In first papers on this subject, most attention was paid to dynamical transformations, which necessarily change the time and space variables. This is why phase translations were missed. Later, the study of Lie symmetries was extended to nonlinear Schrödinger equations. At the same time, the group classification problem for the class of linear Schrödinger equations with complex potentials remains unsolved.The aim of the present thesis is to carry out the group classification for the class of linear Schrödinger equations with complex potentials. These potentials are nowadays important in quantum mechanics, scattering theory, condensed matter physics, quantum field theory, optics, electromagnetics and so forth. We exhaustively solve the group classification problem for space dimensions one and two.The thesis comprises two parts. The first part is a brief review of Lie symmetries and group classification of differential equations. Next, we outline the equivalence transformations in a class of differential equations, normalization properties of such class and the algebraic method for group classification of differential equations.The second part consists of two research papers. In the first paper, the algebraic method is applied to solve the group classification problem for (1+1)-dimensional linear Schrödinger equations with complex potentials. With this technique, the problem of the group classification of the class under study is reduced to the classification of certain subalgebras of its equivalence algebra. As a result, we find that the inequivalent cases are exhausted by eight families of potentials and we give the corresponding maximal Lie invariance algebras.In the second paper we carry out the preliminary symmetry analysis of the class of linear Schrödinger equations with complex potentials in the multi-dimensional case. Using the direct method, we find the equivalence groupoid and the equivalence group of this class. Due to the multi-dimensionality, the results of the computations are quite different from the ones presented in Paper I. We obtain the complete group classification of (1+2)-dimensional linear Schrödinger equations with complex potentials.
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| 6. |
- Kurujyibwami, Célestin, et al.
(författare)
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Group classification of multidimensional linear Schrödinger equations with algebraic method
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Annan publikation (övrigt vetenskapligt/konstnärligt)abstract
- We consider the group classification problem for multidimensional linear Schrödinger equations with complex-valued potentials. Using the algebraic approach, we compute the equivalence groupoid of the class and thus show that this class is uniformly semi-normalized. More specifically, any point transformation connecting two equations from the class is the composition of a linear superposition transformation of the corresponding initial equation and an equivalence transformation of this class. This is why the algebraic method of group classification is applied, which reduces the group classification to the classification of specific low-dimensional subalgebras of the associated equivalence algebra. Inequivalent Lie symmetry extensions are listed for dimension 1+2. Splitting into different classification cases is based on several integer parameters that are invariant with respect to the adjoint action of equivalence transformations. These parameters characterize the dimensions of parts of the corresponding Lie symmetry algebra that are related to generalized scalings, rotations and generalized Galilean boosts, respectively. As expected, the computation for dimension 1+2 is much trickier and cumbersome than for dimension 1+1 due to two reasons, appearing a new kind of transformations rotations and increasing the range for dimensions of essential subalgebras of Lie symmetry algebras, whose least upper bound is n(n+3)/2+5.
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| 7. |
- Maniraguha, Jean de Dieu, et al.
(författare)
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Transforming Stäckel Hamiltonians of Benenti type to polynomial form
- 2022
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Ingår i: Advances in Theoretical and Mathematical Physics. - : International Press. - 1095-0761 .- 1095-0753. ; 26:3, s. 711-734
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Tidskriftsartikel (refereegranskat)abstract
- In this paper we discuss two canonical transformations that turn Stäckel separable Hamiltonians of Benenti type into polynomial form: transformation to Viète coordinates and transformation to Newton coordinates. Transformation to Newton coordinates has been applied to these systems only very recently and in this paper we present a new proof that this transformation indeed leads to polynomial form of Stäckel Hamiltonians of Benenti type. Moreover we present all geometric ingredients of these Hamiltonians in both Viète and Newton coordinates.
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