Admissible Transformations and Normalized Classes of Nonlinear Schrödinger Equations

Popovych, Roman O.; Kunzinger, Michael; Eshraghi, Homayoon

doi:10.1007/s10440-008-9321-4

Cited by 129 publications

(315 citation statements)

References 60 publications

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“…They constitute a group called the equivalence group. Another notion which complements the equivalence transformations was introduced by Kingston and Sophocleous [11,12] under the name form preserving transformations, which are referred to as admissible transformations in [24]. The set of admissible transformations forms a groupoid and we refer to it as the equivalence groupoid and its computation is based on the direct method.…”

Section: Background and Motivationmentioning

confidence: 99%

“…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%

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Group classification of linear Schrödinger equations by the algebraic method

Kurujyibwami

2016

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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...

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Section: Background and Motivationmentioning

confidence: 99%

Section: Background and Motivationmentioning

confidence: 99%

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Group classification of linear Schrödinger equations by the algebraic method

Kurujyibwami

2016

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“…The group of point symmetries of the radial gNLS equation (1.1) for n = 1 (m = 0) is well-known [21,22] Note the inversion (3.4) is called a pseudo-conformal transformation, and the special power p = 4/n for which it exists is commonly called the critical power. On solutions u = f (t, r) of the radial gNLS equation (1.1), the one-dimensional symmetry transformation groups arising from the separate generators (3.1)-(3.4) are given by…”

Section: Symmetries and Group Foliationsmentioning

confidence: 99%

Exact solutions of semilinear radial Schrödinger equations by separation of group foliation variables

Anco

Feng

Wolf

2015

Journal of Mathematical Analysis and Applications

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Abstract. Explicit solutions are obtained for a class of semilinear radial Schrödinger equations with power nonlinearities in multi-dimensions. These solutions include new similarity solutions and other new group-invariant solutions, as well as new solutions that are not invariant under any symmetries of this class of equations. Many of the solutions have interesting analytical behavior connected with blow-up and dispersion. Several interesting nonlinearity powers arise in these solutions, including the case of the critical (pseudo-conformal) power. In contrast, standard symmetry reduction methods lead to nonlinear ordinary differential equations for which few if any explicit solutions can be derived by standard integration methods.

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“…Recent years, this method has been extended by many authors, in which they proposed a numbers of novel techniques, such as algebraic methods based on subgroup analysis of the equivalence group [45][46][47][48] and their generalizations [49][50][51][52][53][54], local transformations and form-preserving transformations [44,[55][56][57][58], to solve group classification problem for numerous nonlinear partial differential equations. In this paper we extend the classical Lie-Ovsiannikov method based on equivalence transformations to the generalized nonlinear beam equation.…”

Section: Introductionmentioning

confidence: 99%

Lie Symmetry Classification of the Generalized Nonlinear Beam Equation

Huang

2017

Symmetry

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Abstract:In this paper we make a Lie symmetry analysis of a generalized nonlinear beam equation with both second-order and fourth-order wave terms, which is extended from the classical beam equation arising in the historical events of travelling wave behavior in the Golden Gate Bridge in San Francisco. We perform a complete Lie symmetry group classification by using the equivalence transformation group theory for the equation under consideration. Lie symmetry reductions of a nonlinear beam-like equation which are singled out from the classification results are investigated. Some classes of exact solutions, including solitary wave solutions, triangular periodic wave solutions and rational solutions of the nonlinear beam-like equations are constructed by means of the reductions and symbolic computation.

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Admissible Transformations and Normalized Classes of Nonlinear Schrödinger Equations

Cited by 129 publications

References 60 publications

Group classification of linear Schrödinger equations by the algebraic method

Group classification of linear Schrödinger equations by the algebraic method

Exact solutions of semilinear radial Schrödinger equations by separation of group foliation variables

Lie Symmetry Classification of the Generalized Nonlinear Beam Equation

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