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Journal ArticleDOI

Generating symmetry operators for the KdV equation and seeking their supersymmetric generalizations

20 Jul 1998-International Journal of Modern Physics A (World Scientific Publishing Co. Pte Ltd)-Vol. 13, Iss: 18, pp 3203-3213
TL;DR: In this article, the Lax form of the Korteweg-de Vries equation is exploited to determine the whole set of its symmetry operators and relate them to conserved quantities.
Abstract: Exploiting the Lax form of the Korteweg-de Vries equation, we determine the whole set of its symmetry operators and relate them to conserved quantities. We also study a KdV supersymmetric version and put in evidence its even (bosonic) and odd (fermionic) symmetries. We then get the Lie superalgebra characterizing this supersymmetric nonlinear equation.
Citations
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Journal ArticleDOI
TL;DR: In this paper, the supersymmetric Korteweg-de-vries equation is formulated in terms of superspace-valued superfields containing bosonic and fermionic fields.
Abstract: We discuss several PT-symmetric deformations of superderivatives. Based on these various possibilities, we propose new families of complex PT-symmetric deformations of the supersymmetric Korteweg–de Vries equation. Some of these new models are mere fermionic extensions of the former in the sense that they are formulated in terms of superspace-valued superfields containing bosonic and fermionic fields, breaking however the supersymmetry invariance. Nonetheless, we also find extensions, which may be viewed as new supersymmetric Korteweg–de Vries equation. Moreover, we show that these deformations allow for a non-Hermitian Hamiltonian formulation.

24 citations

Journal ArticleDOI
TL;DR: In this article, an extended elliptic function method is proposed and applied to the generalized shallow water wave equation and the derived new solutions include rational, periodic, singular and solitary wave solutions.
Abstract: In this work, an extended elliptic function method is proposed and applied to the generalized shallow water wave equation. We systematically investigate on how to classify new exact travelling wave solutions expressible in terms of quasi-periodic elliptic integral functions and doubly periodic Jacobian elliptic functions. The derived new solutions include rational, periodic, singular and solitary wave solutions. An interesting comparison with the canonical procedure is provided. In some cases the obtained elliptic solution has singularity at a certain region in the whole space. For such solutions we have computed the effective region where the obtained solution is free from such a singularity.

15 citations

Journal ArticleDOI
TL;DR: In this article, an extended elliptic function method is proposed and applied to the generalized shallow water wave equation and the derived new solutions include rational, periodic, singular and solitary wave solutions.
Abstract: In this work an extended elliptic function method is proposed and applied to the generalized shallow water wave equation. We systematically investigate to classify new exact travelling wave solutions expressible in terms of quasi-periodic elliptic integral function and doubly-periodic Jacobian elliptic functions. The derived new solutions include rational, periodic, singular and solitary wave solutions. An interesting comparison with the canonical procedure is provided. In some cases the obtained elliptic solution has singularity at certain region in the whole space. For such solutions we have computed the effective region where the obtained solution is free from such a singularity.

3 citations

Journal ArticleDOI
TL;DR: In this article, the supersymmetric Korteweg-de-vries equation has been studied in terms of superspace valued superfields containing bosonic and fermionic fields.
Abstract: We discuss several PT-symmetric deformations of superderivatives. Based on these various possibilities, we propose new families of complex PT-symmetric deformations of the supersymmetric Korteweg-de Vries equation. Some of these new models are mere fermionic extensions of the former in the sense that they are formulated in terms of superspace valued superfields containing bosonic and fermionic fields, breaking however the supersymmetry invariance. Nonetheless, we also find extensions, which may be viewed as new supersymmetric Korteweg-de Vries equation. Moreover, we show that these deformations allow for a non-Hermitian Hamiltonian formulation and construct three charges associated to the corresponding flow.

1 citations

References
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Book
01 Jan 1984
TL;DR: The role of Lie Algebras and Lie Group Representation Theory is discussed in this paper, where the Homogeneous Lorentz Group and the Poincare Group are discussed.
Abstract: OF VOLUME 2: The Role of Lie Algebras. Relationships between Lie Groups and Lie Algebras. The Three-Dimensional Rotation Groups. The Structure of Semi-Simple Lie Algebras. Semi-Simple Real Lie Algebras. Representations of Semi-Simple Lie Algebras and Groups. Developments of the Representation Theory. The Homogeneous Lorentz Groups and the Poincare Groups. Global Internal Symmetries of Elementary Particles.

731 citations

Journal ArticleDOI
TL;DR: In this paper, it was shown that among a family of supersymmetric extensions of the Kortewegde Vries equation, there is a special system that has an infinite number of conservation laws, which can be formulated in the second Hamiltonian structure, and which has a nontrivial Lax representation.
Abstract: It is shown that among a one‐parameter family of supersymmetric extensions of the Korteweg–de Vries equation, there is a special system that has an infinite number of conservation laws, which can be formulated in the second Hamiltonian structure, and which has a nontrivial Lax representation. Its modified version is also discussed.

344 citations

Journal Article
TL;DR: In this paper, the largest group of coordinate transformations leaving invariant the Schroedinger equation of the n-dimensional harmonic oscillator is determined and shown to be isomorphic to the corresponding group of the free-particle equation.
Abstract: The largest group of coordinate transformations leaving invariant the Schroedinger equation of the n-dimensional harmonic oscillator is determined and shown to be isomorphic to the corresponding group of the free-particle equation. It can be described as a Galilei group in which the time translations have been replaced by the group SL(2,R) of projective transformations. The relation between the oscillator group and the spectrumgenenating algebra of the harmonic oscillator is investigated. The relevance of the oscillator group and the group SL(2,R) for general quantum systems is discussed. (auth)

236 citations

Journal ArticleDOI
TL;DR: The invariance groups of nonlinear time evolution equations have been studied from the perspective of a generalized Lie transformation as mentioned in this paper, and the doublet solution of the KdV equation is characterized as the invariant solution of one of the groups.
Abstract: Group theoretic properties of nonlinear time evolution equations have been studied from the standpoint of a generalized Lie transformation. It has been found that with each constant of motion of the KdV type equation fxxx+a (f) fx+ft=0 and of the coupled nonlinear Schrodinger equation fxx +a (f,g)+ift=0, gxx+a (g,f) −igt=0 one invariance group of the equations is always associated. The well‐known series of constants of motion of the KdV equation and the cubic Schrodinger equation will be recovered from the invariance groups of the equations. The doublet solution of the KdV equation will be characterized as the invariant solution of one of the groups. In a more general context, it will be shown that the well‐known equation of quantum mechanics (d/dt) 〈U〉=〈[iH,U] +∂U/∂t〉 can be generalized to a class of nonlinear time evolution equations and that if U is a generator of an invariance group of the equation then (d/dt) 〈U〉=0. The class includes equations such as the KdV, the cubic Schrodinger, and the Hirota e...

33 citations