A supersymmetric Sawada–Kotera equation

Tian, Kai; Liu, Q. P.

doi:10.1016/j.physleta.2009.03.039

Cited by 39 publications

(43 citation statements)

References 19 publications

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“…A number of supersymmetric extensions have been formulated for both classical and quantum mechanical systems. In particular, such supersymmetric generalizations have been constructed for hydrodynamic-type systems (e.g., the Korteweg-de Vries equation [2,3], the Sawada-Kotera equation [4], polytropic gas dynamics [5,6] and a Gaussian irrotational compressible fluid [7]) as well as other nonlinear wave equations, e.g., the Schrödinger equation [8] and the sine/sinh-Gordon equation [9][10][11]. Parameterizations of strings and Nambu-Goto membranes have been used to supersymmetrize the Chaplygin gas in (1 + 1) and (2 + 1) dimensions respectively [12].…”

Section: Introductionmentioning

confidence: 99%

Algebraic Aspects of the Supersymmetric Minimal Surface Equation

Grundland

Hariton

2017

Symmetry

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Abstract:In this paper, a supersymmetric extension of the minimal surface equation is formulated. Based on this formulation, a Lie superalgebra of infinitesimal symmetries of this equation is determined. A classification of the one-dimensional subalgebras is performed, which results in a list of 143 conjugacy classes with respect to action by the supergroup generated by the Lie superalgebra. The symmetry reduction method is used to obtain invariant solutions of the supersymmetric minimal surface equation. The classical minimal surface equation is also examined and its group-theoretical properties are compared with those of the supersymmetric version.

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Section: Introductionmentioning

confidence: 99%

Algebraic Aspects of the Supersymmetric Minimal Surface Equation

Grundland

Hariton

2017

Symmetry

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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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“… Supersymmetric Sawada–Kotera equation (

λ = \frac{3}{2}

)

{normalΦ}_{t} = {normalΦ}_{5} + 5 {normalΦ}_{3} false(scriptD normalΦ false) + 5 {normalΦ}_{2} false(scriptD Φ_{1} false) + 5 {normalΦ}_{1} {(D Φ)}^{2}

and in components it is of the form

({, \begin{matrix} v_{t} = v_{5} & + 5 v_{3} v + 5 v_{2} v_{1} + 5 v_{1} v^{2} - 5 ξ_{3} ξ_{1} \\ ξ_{t} = ξ_{5} & + 5 ξ_{3} v + 5 ξ_{2} v_{1} + 5 ξ_{1} v^{2} . \end{matrix})

The recursion operator of the supersymmetric Sawada–Kotera equation was deduced from its Lax equation , and later factorized by Popowicz into two “odd” Hamiltonian operators, which allow us to rewrite the recursion operator in a neater form as

\begin{matrix} true \\ right ℜ & left = & right (() \partial_{x}^{2} D + 2 Φ \partial_{x} + 2 \partial_{x} Φ + D Φ D) \partial_{x}^{- 1} false(\partial_{x}^{2} scriptD + 2 normalΦ \partial_{x} \\ right + 2 \partial_{x} normalΦ + scriptD normalΦ scriptD false) {scriptD}^{- 1} {false(\partial_{x} scriptD + normalΦ false)}^{2} {scriptD}^{- 1} . \end{matrix}

…”

Section: A Complete List Of Scalar λ‐Homogeneous (λ>0) N=1 Supersymmementioning

confidence: 99%

“…Supersymmetric Sawada–Kotera equation (

λ = \frac{3}{2}

)

{normalΦ}_{t} = {normalΦ}_{5} + 5 {normalΦ}_{3} false(scriptD normalΦ false) + 5 {normalΦ}_{2} false(scriptD Φ_{1} false) + 5 {normalΦ}_{1} {(D Φ)}^{2}

and in components it is of the form

({, \begin{matrix} v_{t} = v_{5} & + 5 v_{3} v + 5 v_{2} v_{1} + 5 v_{1} v^{2} - 5 ξ_{3} ξ_{1} \\ ξ_{t} = ξ_{5} & + 5 ξ_{3} v + 5 ξ_{2} v_{1} + 5 ξ_{1} v^{2} . \end{matrix})

…”

Section: A Complete List Of Scalar λ‐Homogeneous (λ>0) N=1 Supersymmementioning

confidence: 99%

Symbolic Representation and Classification of Supersymmetric Evolutionary Equations

Tian

Wang

2017

Stud Appl Math

Self Cite

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We extend the symbolic representation to the ring of N=1 supersymmetric differential polynomials, and demonstrate that operations on the ring, such as the super derivative, Fréchet derivative, and super commutator, can be carried out in the symbolic way. Using the symbolic representation, we classify scalar λ‐homogeneous N=1 supersymmetric evolutionary equations with nonzero linear term when λ>0 for arbitrary order and give a comprehensive description of all such integrable equations.

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A supersymmetric Sawada–Kotera equation

Abstract: A new supersymmetric equation is proposed for the Sawada-Kotera equation. The integrability of this equation is shown by the existence of Lax representation and infinite conserved quantities and a recursion operator.

Cited by 39 publications

References 19 publications

Algebraic Aspects of the Supersymmetric Minimal Surface Equation

Algebraic Aspects of the Supersymmetric Minimal Surface Equation

Supersymmetric versions of the equations of conformally parametrized surfaces

Symbolic Representation and Classification of Supersymmetric Evolutionary Equations

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