On preliminary symmetry classification of nonlinear Schrödinger equations with some applications to Doebner-Goldin models

Zhdanov, Renat; Roman, Olena

doi:10.1016/s0034-4877(00)89037-0

Cited by 20 publications

(22 citation statements)

References 15 publications

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“…The corresponding values of the arbitrary elements are connected in a local way, i.e., the transformation T belongs to the equivalence group G ∼ F of the class F . This allows us to reformulate and to strengthen the results of [79] on the equivalence group G ∼ F .…”

Section: Lemma 1 If a Point Transformation T Connects Two Equations Fmentioning

confidence: 96%

“…All the equations from the class F , which are invariant with respect to subalgebras of the Lie symmetry algebra of the (1 + 1)-dimensional free Schrödinger equation were constructed in [13]. Later the more general problem of the description of the equations from the class F , possessing at most three-dimensional Lie invariance algebras, was solved in [79]. It was observed [13,24] that (1 + n)-dimensional NSchEs with nonlinearities of the form F = f (|ψ|)ψ are notable for their symmetry properties because any such equation is invariant with respect to a representation of the (1 + n)-dimensional Galilean group.…”

Section: Lie Symmetries Of Nonlinear Schrödinger Equations: Known Resmentioning

confidence: 99%

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Admissible Transformations and Normalized Classes of Nonlinear Schrödinger Equations

2008

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The theory of group classification of differential equations is analyzed, substantially extended and enhanced based on the new notions of conditional equivalence group and normalized class of differential equations. Effective new techniques are proposed. Using these, we exhaustively describe admissible point transformations in classes of nonlinear (1 + 1)-dimensional Schrödinger equations, in particular, in the class of nonlinear (1 + 1)-dimensional Schrödinger equations with modular nonlinearities and potentials and some subclasses thereof. We then carry out a complete group classification in this class, representing it as a union of disjoint normalized subclasses and applying a combination of algebraic and compatibility methods. Moreover, we introduce the complete classification of (1 + 2)-dimensional cubic Schrödinger equations with potentials. The proposed approach can be applied to studying symmetry properties of a wide range of differential equations.Keywords Group classification of differential equations · Group analysis of differential equations · Equivalence group · Admissible transformations · Normalized classes of differential equations · Lie symmetry · Nonlinear Schrödinger equations R.O. Popovych ( ) · M. Kunzinger

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Section: Lemma 1 If a Point Transformation T Connects Two Equations Fmentioning

confidence: 96%

Section: Lie Symmetries Of Nonlinear Schrödinger Equations: Known Resmentioning

confidence: 99%

Admissible Transformations and Normalized Classes of Nonlinear Schrödinger Equations

2008

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“…Note also that group classification of the nonlinear wave and Schrödinger equations in the same spirit has been done in [15,16].…”

Section: Introductionmentioning

confidence: 99%

Symmetry classification of KdV-type nonlinear evolution equations

Güngör

Lahno

Zhdanov

2004

Journal of Mathematical Physics

109

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Group classification of a class of third-order nonlinear evolution equations generalizing KdV and mKdV equations is performed. It is shown that there are two equations admitting simple Lie algebras of dimension three. Next, we prove that there exist only four equations invariant with respect to Lie algebras having nontrivial Levi factors of dimension four and six. Our analysis shows that there are no equations invariant under algebras which are semi-direct sums of Levi factor and radical. Making use of these results we prove that there are three, nine, thirty-eight, fifty-two inequivalent KdV-type nonlinear evolution equations admitting one-, two-, three-, and four-dimensional solvable Lie algebras, respectively. Finally, we perform a complete group classification of the most general linear third-order evolution equation. *

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“…Note also that group classification of the nonlinear wave and the Schrödinger equations in the same spirit has been done in Refs. [30][31].…”

Section: Summary Of the Group Classification Algorithmmentioning

confidence: 99%

Preliminary group classification of quasilinear third-order evolution equations

Huang

Zhang

2009

Appl. Math. Mech.-Engl. Ed.

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Group classification of quasilinear third-order evolution equations is given by using the classical infinitesimal Lie method, the technique of equivalence transformations, and the theory of classification of abstract low-dimensional Lie algebras. We show that there are three equations admitting simple Lie algebras of dimension three. All non-equivalent equations admitting simple Lie algebras are nothing but these three. Furthermore, we also show that there exist two, five, twenty-nine and twenty-six nonequivalent third-order nonlinear evolution equations admitting one-, two-, three-, and four-dimensional solvable Lie algebras, respectively.

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On preliminary symmetry classification of nonlinear Schrödinger equations with some applications to Doebner-Goldin models

Cited by 20 publications

References 15 publications

Admissible Transformations and Normalized Classes of Nonlinear Schrödinger Equations

Admissible Transformations and Normalized Classes of Nonlinear Schrödinger Equations

Symmetry classification of KdV-type nonlinear evolution equations

Preliminary group classification of quasilinear third-order evolution equations

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