Supersymmetric Korteweg-de Vries Research Articles

A novel Hirota bilinear approach to N = 2 supersymmetric equations

This article presents a novel application of the Hirota bilinear formalism to the N = 2 supersymmetric Korteweg–de Vries and Burgers equations. This new approach avoids splitting N = 2 equations into two N = 1 equations. We use the super Bell polynomials to obtain bilinear representations and present multi-soliton solutions.

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.

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.

On the formulation of a Bäcklund Wahlquist-Estabrook transformation for a supersymmetric Korteweg-de Vries equation

We present a local Bäcklund Wahlquist-Estabrook (WE) transformation for a supersymmetric Korteweg-de Vries (KdV) equation. As in the scalar case, such type of transformation generates infinite hierarchies of solutions and also implicitly gives the associated (local) conserved quantities. A nice property is that every of such hierarchies admits a nonlinear superposition principle, starting for an initial solution, including as a particular case the multisolitonic solutions of the system. We discuss the symmetries of the system and we present in an explicit way its local conserved quantities with the help of the associated Gardner transformation.

Critical Analyses of q-State Clock-Like Phase Transition

The N=1 supersymmetric modified Korteweg-de Vries (SmKdV) system is transformed to a system of coupled bosonic equations with the bosonization approach. The bosonized SmKdV (BSmKdV) passes the Painleve test and allows a set of Backlund transformation (BT) by truncating the series expansions. The traveling wave solutions of the BSmKdV system are obtained using the mapping and deformation method. Some special types of exact solutions for the BSmKdV system are found with the solutions and symmetries of the usual mKdV equation. In the meanwhile, the similarity reduction solutions of the system are investigated by using the Lie point symmetry theory. The generalized tanh function expansion method for the BSmKdV system leads to a nonauto-BT theorem. Using the nonauto-BT theorem, the novel exact solution of the BSmKdV system can be obtained. All these solutions obtained via the bosonization procedure are different from those obtained via other methods.

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.

Bosonization, singularity analysis, nonlocal symmetry reductions and exact solutions of supersymmetric KdV equation

Assuming that there exist at least two fermionic parameters, the classical N= 1 supersymmetric Korteweg-de Vries (SKdV) system can be transformed to some coupled bosonic systems. The boson fields in the bosonized SKdV (BSKdV) systems are defined on even Grassmann algebra. Due to the intrusion of other Grassmann parameters, the BSKdV systems are different from the usual non-supersymmetric integrable systems, and many more abundant solution structures can be unearthed. With the help of the singularity analysis, the Painlev\'e property of the BSKdV system is proved and a B\"acklund transformation (BT) is found. The BT related nonlocal symmetry, we call it as residual symmetry, is used to find symmetry reduction solutions of the BSKdV system. Hinted from the symmetry reduction solutions, a more generalized but much simpler method is established to find exact solutions of the BSKdV and then the SKdV systems, which actually can be applied to any fermionic systems.

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.

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.

Bilinearization ofN= 1 supersymmetric Korteweg–de Vries equation revisited

We consider the N = 1 supersymmetric Korteweg–de Vries (sKdV) equation within the framework of Hirota's bilinear method. We construct a Backlund transformation which may be interpreted as the modified sKdV equation. Also, we find a Lax representation and a nonlinear superposition formula. By direct applications of the nonlinear superposition formula, we calculate soliton solutions for the sKdV equation.

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.

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.

N=4 super KdV hierarchy in N=4 and N=2 superspaces

We present the results of further analysis of the integrability properties of the N=4 supersymmetric Korteweg–de Vries (KdV) equation deduced earlier by two of us [F. Delduc and E. Ivanov, Phys. Lett. B 309, 312 (1993)] as a Hamiltonian flow on N=4 SU(2) superconformal algebra in the harmonic N=4 superspace. To make this equation and the relevant Hamiltonian structures more tractable, we reformulate it in the ordinary N=4 and further in N=2 superspaces. In N=2 superspace it is represented by a coupled system of evolution equations for a general N=2 superfield and two chiral and antichiral superfields, and involves two independent real parameters, a and b. We construct a few first bosonic conserved charges in involution, of dimensions from 1 to 6, and show that they exist only for the following choices of the parameters: (i) a=4, b=0; (ii) a=−2, b=−6; (iii) a=−2, b=6. The same values are needed for the relevant evolution equations, including N=4 KdV itself, to be bi-Hamiltonian. We demonstrate that the above three options are related via SU(2) transformations and actually amount to the SU(2) covariant integrability condition found in the harmonic superspace approach. Our results provide a strong evidence that the unique N=4 SU(2) super KdV hierarchy exists. Upon reduction to N=2 KdV, the above three possibilities cease to be equivalent. They give rise to the a=4 and a=−2 N=2 KdV hierarchies, which thus prove to be different truncations of the single N=4 SU(2) KdV one.