Pfaffian solutions for the Manin-Radul-Mathieu SUSY KdV and SUSY sine-Gordon equations

Liu, Q. P.; Mañas, Manuel

doi:10.1016/s0370-2693(98)00852-1

Cited by 28 publications

(45 citation statements)

References 11 publications

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“…A number of supersoliton and multi-supersoliton solutions were determined by a Crum-type transformation [19], [21], [25] and it was found that there exist infinitely many local conserved quantities. A connection was established between the super-Darboux transformations and super-Bäcklund transformations which allow for the construction of supersoliton solutions [11], [13], [16], [17], [20], [22], [24].…”

Section: Introductionmentioning

confidence: 99%

Supersymmetric versions of the equations of conformally parametrized surfaces

Bertrand¹,

Grundland²,

Hariton³

2015

J. Phys. A: Math. Theor.

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Abstract. The objective of this paper is to formulate two distinct supersymmetric (SUSY) extensions of the Gauss-Weingarten and Gauss-Codazzi (GC) equations for conformally parametrized surfaces immersed in a Grassmann superspace, one in terms of a bosonic superfield and the other in terms of a fermionic superfield. We perform this analysis using a superspace-superfield formalism together with a SUSY version of a moving frame on a surface. In constrast with the classical case, where we have three GC equations, we obtain six such equations in the bosonic SUSY case and four such equations in the fermionic SUSY case. In the fermionic case the GC equations resemble the form of the classical GC equations. We determine the Lie symmetry algebra of the classical GC equations to be infinite-dimensional and perform a subalgebra classification of the one-dimensional subalgebras of its largest finite-dimensional subalgebra. We then compute superalgebras of Lie point symmetries of the bosonic and fermionic SUSY GC equations respectively, and classify the onedimensional subalgebras of each superalgebra into conjugacy classes. We then use the symmetry reduction method to find invariants, orbits and reduced systems for two one-dimensional subalgebras for the classical case, two one-dimensional subalgebras for the bosonic SUSY case and two one-dimensional subalgebras for the fermionic SUSY case. We find explicit solutions of these reduced SUSY systems, which correspond to different surfaces immersed in a Grassmann superspace. Within this framework for the SUSY versions of the GC equations, a geometrical interpretation of the results is discussed.

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Section: Introductionmentioning

confidence: 99%

Supersymmetric versions of the equations of conformally parametrized surfaces

Bertrand¹,

Grundland²,

Hariton³

2015

J. Phys. A: Math. Theor.

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“…Recent development on SUSY integrable systems, especially on Bäcklund and Darboux transformations, allowed a better understanding and the construction of soliton solutions. For examples, Liu and Mañas [16,17] provided the N-Darboux transformation for the Manin-Radul(-Mathieu) SUSY Korteweg-de Vries (KdV) system, which can also be applied to the SUSY sine-Gordon equaiton, using Pfaffian solutions. In addition, Xue et al [18] investigated the Bäcklund transformation of the SUSY sine-Gordon equation as a nonlinear superposition formula and they were able to link it with the SUSY MKdV equations.…”

Section: Introductionmentioning

confidence: 99%

On geometric aspects of the supersymmetric Fokas–Gel’fand immersion formula

Bertrand¹

2017

J. Phys. A: Math. Theor.

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In this paper, we develop a new geometric characterization for the supersymmetric versions of the Fokas-Gel'fand formula for the immersion of soliton supermanifolds with two bosonic and two fermionic independent variables into Lie superalgebras. In order to do so, from a linear spectral problem of a supersymmetric integrable system using the covariant fermionic derivative, we provide a technique to obtain two additional linear spectral problems for that integrable system, one using the bosonic variable derivatives and the other using the fermionic variable derivatives. This allows us to investigate, through the first and second fundamental forms, the geometry of the (1 + 1|2)-supermanifolds immersed in Lie superalgebras. Whenever possible, the mean and Gaussian curvatures of the supermanifolds are calculated. These theoretical considerations are applied to the supersymmetric sine-Gordon equation.

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“…The proposed idea of constructing complexitons through special determinants, for example, the Wronskian and Casorati determinants, will also work for other integrable equations. In particular, super-complexitons can be similarly generated for super integrable equations, which will extend the theory of supersolitons [ 11,12].…”

Section: Wx Mamentioning

confidence: 99%

“…The Casoratian technique [ 17] is one of the ways to solve the bilinear Toda lattice equation (12). The corresponding solutions to the bilinear Toda lattice equation (12) are determined by the Casorati determinant:…”

Section: Toda Lattice Equationmentioning

confidence: 99%

Complexiton solutions to integrable equations

2005

Nonlinear Analysis: Theory, Methods & Applications

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Pfaffian solutions for the Manin-Radul-Mathieu SUSY KdV and SUSY sine-Gordon equations

Cited by 28 publications

References 11 publications

Supersymmetric versions of the equations of conformally parametrized surfaces

Supersymmetric versions of the equations of conformally parametrized surfaces

On geometric aspects of the supersymmetric Fokas–Gel’fand immersion formula

Complexiton solutions to integrable equations

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