Equivalence groupoid for (1+2)-dimensional linear Schrödinger equations with complex potentials

Kurujyibwami, Célestin

doi:10.1088/1742-6596/621/1/012008

Cited by 3 publications

(5 citation statements)

References 18 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…That is, if for each triple (θ,θ, ϕ) ∈ G ∼ (L| S ) there exists a transformation Φ ∈ G ∼ and a transformationφ ∈ G θ of the system L θ such thatθ = Φθ and ϕ = Φ| (x,u) •φ. (see [13]). …”

Section: Normalization Properties Of Classes Of Differential Equationsmentioning

confidence: 96%

“…The relation between the equivalence groupoid and the equivalence group for a given class influences the choice of the appropriate technique for group classification and this leads to an efficient way of presenting the results [24,13]. It simplifies the process of classifying non trivial Lie symmetries within this class.…”

Section: Background and Motivationmentioning

confidence: 99%

“…In this section we look at equivalence transformations of classes of differential equations and some of their properties. More details can be found in [3,13,24].…”

Section: Equivalence Transformations and Normalization Propertiesmentioning

confidence: 99%

See 2 more Smart Citations

Group classification of linear Schrödinger equations by the algebraic method

Kurujyibwami

2016

Self Cite

View full text Add to dashboard Cite

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.iii Populärvetenskaplig sammanfattningAvhandlingen behandlar gruppklassificeringen av linjära Schrödingerekvationer. Studiet av sådana ekvationers Lie symmetrier påbörjades för mer än 40 år sedan. Man använde då Lies klassiska infinitesimala metod och begränsade sig till speciella typer av reellvärd potential. De första arbetena ägnades huvudsakligen åt dynamiska transformationer. Eftersom dessa nödvändigtvis ändrar både tidsoch rumsvariablerna behandlades inte fastranslationer. Senare utvidgades studierna till icke-linjära Schrödingerekvationer, men gruppklassificeringsproblemet för klassen av linjä...

show abstract

Section: Normalization Properties Of Classes Of Differential Equationsmentioning

confidence: 96%

Section: Background and Motivationmentioning

confidence: 99%

See 1 more Smart Citation

Group classification of linear Schrödinger equations by the algebraic method

Kurujyibwami

2016

Self Cite

View full text Add to dashboard Cite

show abstract

“…Some recent studies on admissible transformations (equivalence groupoid) of physically interesting classes of PDEs, such as variable coefficient Burgers, Kawahara and Schrödinger equations, can be found in particular in [20,21,24,25].…”

Section: Introductionmentioning

confidence: 99%

Equivalence groupoid of a class of variable coefficient Korteweg–de Vries equations

Vaneeva

Pošta

2017

Journal of Mathematical Physics

View full text Add to dashboard Cite

We classify the admissible transformations in a class of variable coefficient Korteweg-de Vries equations. As a result, full description of the structure of the equivalence groupoid of the class is given. The class under study is partitioned into six disjoint normalized subclasses. The widest possible equivalence group for each subclass is found which appears to be generalized extended in five cases. Ways for improvement of transformational properties of the subclasses are proposed using gaugings of arbitrary elements and mapping between classes. The group classification of one of the subclasses is carried out as an illustrative example.

show abstract

“…We note (see [23]) that the admissible transformations for the class L| S have the structure of a groupoid : the identity transformation is quite obviously an admissible transformation; if ϕ ∈ T(θ 1 , θ 2 ) and ψ ∈ T(θ 3 , θ 4 ) then the composition ψ • ϕ is defined only if θ 2 = θ 3 and then ψ • ϕ ∈ T(θ 1 , θ 4 ). The associativity of composition is inherited from the associativity of point transformations : ϕ • (ψ • ρ) = (ϕ • ψ) • ρ whenever the composition is defined.…”

Section: Definition 32 We Define the Set Of Admissible Transformatimentioning

confidence: 99%

Admissible transformations and the group classification of Schrödinger equations

Kurujyibwami¹

View full text Add to dashboard Cite

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 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 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 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 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)-di...

show abstract

Equivalence groupoid for (1+2)-dimensional linear Schrödinger equations with complex potentials

Cited by 3 publications

References 18 publications

Group classification of linear Schrödinger equations by the algebraic method

Group classification of linear Schrödinger equations by the algebraic method

Equivalence groupoid of a class of variable coefficient Korteweg–de Vries equations

Admissible transformations and the group classification of Schrödinger equations

Contact Info

Product

Resources

About