Supersymmetric Korteweg-de Vries Equation Research Articles

The fully supersymmetric AKNS problem: Darboux transformations and discrete systems

The fully supersymmetric AKNS spectral problem with four superfields proposed by Morosi and Pizzocchero is considered. This is a hybrid problem of the classical AKNS and supersymmetric AKNS spectral problems. Under the assumption that Darboux matrices are linear with respect to the spectral parameter, three Darboux transformations and the corresponding Bäcklund transformations are constructed. Through proper reductions, we obtain Darboux transformations for the coupled supersymmetric nonlinear Schrödinger and coupled supersymmetric modified Korteweg–de Vries equations, and recover the Darboux transformations for Manin-Radul supersymmetric Korteweg–de Vries equation and supersymmetric AKNS hierarchy, respectively. The first Darboux-Bäcklund transformation is further applied to construct integrable discrete super systems, and both semi-discrete and fully discrete systems are obtained. The associated continuum limits are studied as well.

On a fermionic extension of [formula omitted] equation

A fermionic extension of K(−1,−2) equation, alias a super K(−1,−2) equation, proposed by Tempesta et al. (2003), is converted to the N=1 supersymmetric Korteweg–de Vries equation via invertible transformations involving both independent and dependent variables. As implementation of this intimate connection, some integrable properties are established for the fermionic extension of K(−1,−2) equation, including a linear spectral problem in terms of 3 × 3 matrices, a Bäcklund transformation, a nonlinear superposition formula, infinitely many conservation laws, as well as its bi-Hamiltonian formulation.

Bilinearization and Soliton Solutions of the Supersymmetric Coupled KdV Equation

We propose an N=1 supersymmetric generalization of the coupled Korteweg-de Vries (KdV) equation and use the Hirota superoperator to obtain a superfield bilinear form of the supersymmetric coupled KdV equation. Using the Hirota method, we obtain explicit expressions for superfield soliton solutions of the supersymmetric coupled KdV equation. We also find superfield one- and two-soliton solutions.

N = 2 Supercomplexification of the Korteweg-de Vries, Sawada-Kotera and Kaup-Kupershmidt Equations

The supercomplexification is a special method of N = 2 supersymmetrization of the integrable equations in which the bosonic sector can be reduced to the complex version of these equations. The N = 2 supercomplex Korteweg-de Vries, Sawada-Kotera and Kaup-Kupershmidt equations are defined and investigated. The common attribute of the supercomplex equations is appearance of the odd Hamiltonian structures and superfermionic conservation laws. The odd bi-Hamiltonian structure, Lax representation and superfermionic conserved currents for new N = 2 supersymmetric Korteweg-de Vries equation and for Sawada-Kotera one, are given.

Inverse scattering transform for a supersymmetric Korteweg-de Vries equation

In this paper, the inverse scattering transform is extended to a super Korteweg-de Vries equation with an arbitrary variable coefficient by using Kulish and Zeitlin?s approach. As a result, exact solutions of the super Korteweg-de Vries equation are obtained. In the case of reflectionless potentials, the obtained exact solutions are reduced to soliton solutions. More importantly, based on the obtained results, an approach to extending the scattering transform is proposed for the supersymmetric Korteweg-de Vries equation in the 1-D Grassmann algebra. It is shown the the approach can be applied to some other supersymmetric non-linear evolution equations in fluids.

Constructing soliton solutions and super-bilinear form of lattice supersymmetric KdV equation

The Hirota bilinear form and multisoliton solution for semidiscrete and fully discrete (difference–difference) versions of the supersymmetric Korteweg–de Vries (KdV) equation found by Xue et al (2013 J. Phys. A: Math. Theor 46 502001) are presented. The solitonic interaction term displays a fermionic dressing factor as in the continuous supersymmetric case. Using bilinear equations it is also shown that a new integrable semidiscrete (and fully discrete) version of supersymmetric KdV can be constructed with a simpler bilinear form but a more complicated interaction dressing. Its continuum limit is also computed.

Exact Solutions of Supersymmetric KdV-a System via Bosonization Approach

Bosonization approach is applied in solving the most general N = 1 supersymmetric Korteweg de-Vries equation with an arbitrary parameter a (sKdV-a) equation. By introducing some fermionic parameters in the expansion of the superfield, the sKdV-a equation is transformed to a new coupled bosonic system. The Lie point symmetries of this model are considered and similarity reductions of it are conducted. Several types of similarity reduction solutions of the coupled bosonic equations are simply obtained for all values of a. Some kinds of exact solutions of the sKdV-a equation are discussed which was not considered integrable previously.

The transformations between N = 2 supersymmetric Korteweg-de Vries and Harry Dym equations

The N = 2 supercomformal transformations are employed to study supersymmetric integrable systems. It is proved that two known N = 2 supersymmetric Harry Dym equations are transformed into two N = 2 supersymmetric modified Korteweg-de Vries equations, thus are connected with two N = 2 supersymmetric Korteweg-de Vries equations.

A non-standard Lax formulation of the Harry Dym hierarchy and its supersymmetric extension

For the Harry Dym hierarchy, a non-standard Lax formulation is deduced from that of the Korteweg–de Vries (KdV) equation through a reciprocal transformation. By supersymmetrizing this Lax operator, a new N = 2 supersymmetric extension of the Harry Dym hierarchy is constructed, and is further shown to be linked to one of the N = 2 supersymmetric KdV equations through the superconformal transformation. The bosonic limit of this new N = 2 supersymmetric Harry Dym equation is related to a coupled system of KdV–MKdV equations.

On the non-abelian superalgebra spanned by the conserved quantities of [formula omitted] supersymmetric Korteweg–de Vries equation

We obtain an infinite sequence of bosonic non-local conserved quantities for the N=1 supersymmetric Korteweg–de Vries equation. It is generated from a bosonic non-local conserved quantity of Super Gardner equation. In distinction to the already known one with odd parity and dimension 12, it has even parity and dimension 1. It fits exactly in the supersymmetric cohomology in the space of conserved quantities that we also introduce here. Using results from this cohomology we obtain the Poisson bracket of several non-local conserved quantities, including the already known odd ones and the new even ones. The algebra closes in terms of polynomials of local and non-local conserved quantities. We prove that the bosonic non-local conserved quantities cannot be expressed as functions of the already known local and non-local conserved quantities of Super KdV equation.

Hirota’s virtual multisoliton solutions of N=2 supersymmetric Korteweg-de Vries equations

We prove that Mathieu's N=2 supersymmetric Korteweg-de Vries equations with a=1 or a=4 admit Hirota's n-supersoliton solutions, whose nonlinear interaction does not produce any phase shifts. For initial profiles that can not be distinguished from a one-soliton solution at times t<<0, we reveal the possibility of a spontaneous decay and, within a finite time, transformation into a solitonic solution with a different wave number. This paradoxal effect is realized by the completely integrable N=2 super-KdV systems, whenever the initial soliton is loaded with other solitons that are virtual and become manifest through the tau-function as the time grows. Key words and phrases: Hirota's solitons, N=2 supersymmetric KdV, Krasil'shchik-Kersten system, phase shift, spontaneous decay.

Group-Theoretical Structure and the Inverse Scattering Method for the Super-KdV Equation

Using the group-theoretical approach to the inverse scattering method the supersymmetric Korteweg-de Vries equation is obtained by application of the Drinfeld-Sokolov reduction to osp(1|2) loop superalgebra. The direct and inverse scattering problems are discussed for the corresponding Lax pair.

(Non)local Hamiltonian and symplectic structures, recursions and hierarchies: a new approach and applications to theN= 1 supersymmetric KdV equation

Using methods of Kersten et al (2004 J. Geom. Phys. 50 273–302) and Krasil'shchik and Kersten (2000 Symmetries and Recursion Operators for Classical and Supersymmetric Differential Equations (Dordrecht: Kluwer)), we accomplish an extensive study of the N = 1 supersymmetric Korteweg–de Vries (KdV) equation. The results include a description of local and nonlocal Hamiltonian and symplectic structures, five hierarchies of symmetries, the corresponding hierarchies of conservation laws, recursion operators for symmetries and generating functions of conservation laws. We stress that the main point of the paper is not just the results on super-KdV equation itself, but merely exposition of the efficiency of the geometrical approach and of the computational algorithms based on it.

Nonlinear superposition formula for N=1 supersymmetric KdV equation

In this Letter, we derive a Bäcklund transformation for the supersymmetric Korteweg–de Vries equation. We also construct a nonlinear superposition formula, which allows us to rebuild systematically for the supersymmetric KdV equation the soliton solutions of Carstea, Ramani and Grammaticos.

The Painlevé analysis for N=2 super Korteweg–de Vries equations

The Painlevé analysis of a generic multiparameter N=2 extension of the Korteweg–de Vries (KdV) equation is presented. Unusual aspects of the analysis, pertaining to the presence of two fermionic fields, are emphasized. For the general class of models considered, we find that the only ones which manifestly pass the test are precisely the four known integrable supersymmetric KdV equations, including the SKdV1 case.

Group invariant solutions for the N=2 super Korteweg–de Vries equation

The method of symmetry reduction is used to solve Grassmann-valued differential equations. The (N=2) supersymmetric Korteweg–de Vries equation is considered. It admits a Lie superalgebra of symmetries of dimension 5. A two-dimensional subsuperalgebra is chosen to reduce the number of independent variables in this equation. We are then able to give different types of exact solutions, in particular soliton solutions.

THE QUANTUM SKdV1,4 EQUATION AT c=3

At c=3, two of the three integrable quantum N=2 supersymmetric Korteweg-de Vries equations become identical ( SKdV 1 and SKdV 4). Quite remarkably, all their conservation laws can be written in closed form, which provides thus a simple constructive integrability proof.

Self-dual supersymmetric Yang-Mills theory generates Witten's topological field theory

We show that the recently constructed N=4 supersymmetric self-dual Yang-Mills theory as the consistent background of the N=2 open superstring will generate Witten's topological field theory in two dimensions as a descendant theory after appropriate twisted dimensional reduction/truncations. We also show that this topological field theory further generates supersymmetric Korteweg de Vries equations, SL ( n)-Toda theory and W ∞-gravity in the n→∞ limit. Considering also that this topological field theory is to generate other integrable and topological systems, our resutls give supporting evidence for the conjecture that the N=2 open superstring theory is the underlying “master theory” of integrable and topological systems, via the intermediate N=4 supersymmetric self-dual Yang-Mills theory as its consistent target space-time background.

Supersymmetric integrable systems embedded in supersymmetric self-dual Yang-Mills theory

We perform dimensional reductions of recently constructed self-dual N=2 supersymmetric Yang-Mills theory in 2+2 dimensions into two dimensions. We show that the universal equations obtained in these dimensional reductions can embed supersymmetric integrable systems, such as N=1 and N=2 supersymmetric Korteweg-de Vries equations, N=1 supersymmetric Liouville theory or supersymmetric Toda theory. This is the first supporting evidence for the conjecture that the (2+2)-dimensional self-dual supersymmetric Yang-Mills theory generates supersymmetric integrable systems in lower dimensions.

Integrability of perturbed superconformal minimal models

It is argued that the perturbation of a superconformal minimal model by a particular relevant operator yields a theory which admits an infinite number of commuting conservation laws, which turn out to be exactly the conservation laws of the quantum supersymmetric Korteweg-de Vries equation.