Supersymmetric KdV Equation Research Articles

[formula omitted]-periodic wave solutions of the [formula omitted] supersymmetric KdV equation

In this paper, the N-periodic wave solutions of the N=2 supersymmetric KdV equation are studied by combining the super Hirota bilinear form with the super Riemann-theta function, which can be used to describe new phenomena on super quasi-periodic waves with the fermionic field. With the aid of the Gauss–Newton method, the three-periodic and four-periodic wave solutions are obtained. In particular, these quasi-periodic waves can produce parallel, crossed and degenerated patterns. The analytical method related to the characteristic lines is used to analyze the dynamic characteristics of the three-periodic and four-periodic waves. In addition, it has been indicated that N-periodic waves can exist in the supersymmetric integrable systems.

Darboux transformation for the supersymmetric nonisospectral KdV equation

Darboux transformation for the supersymmetric nonisospectral KdV equation is investigated. Based on the Lax pair, we successively construct the one-fold, two-fold and three-fold Darboux tansformations for the supersymmetric nonisospectral KdV equation. Moreover, we present the [Formula: see text]-fold Darboux transformation in the form of superdeterminant.

The quasi-periodic solutions for the supersymmetric variable-coefficient KdV equation

The supersymmetric variable-coefficient KdV equation is presented and it admits Painlevé property by the standard singularity analysis. Based on Hirota bilinear method and Riemann theta function, one and two quasi-periodic wave solutions for the supersymmetric variable-coefficient KdV equation are studied. In addition, we give the asymptotic relations between quasi-periodic wave solutions and soliton solutions.

Bilinear approach to Kuperschmidt super-KdV type equations

Hirota bilinear form and soliton solutions for the super-KdV (Korteweg–de Vries) equation of Kuperschmidt (Kuper–KdV) are given. It is shown that even though the collision of supersolitons is more complicated than in the case of the supersymmetric KdV equation of Manin–Radul, the asymptotic effect of the interaction is simpler. As a physical application it is shown that the well-known FPU problem, having a phonon-mediated interaction of some internal degrees of freedom expressed through Grassmann fields, transforms to the Kuper–KdV equation in a multiple-scale approach.

Bäcklund–Darboux transformations and discretizations of N = 2 a = −2 supersymmetric KdV equation

The N=2a=−2 supersymmetric KdV equation is studied. A Darboux transformation and the corresponding Bäcklund transformation are constructed for this equation. Also, a nonlinear superposition formula is worked out for the associated Bäcklund transformation. The Bäcklund transformation and the related nonlinear superposition formula are used to construct integrable super semi-discrete and full discrete systems. The continuum limits of these discrete systems are also considered.

A N = 2 extension of the Hirota bilinear formalism and the supersymmetric KdV equation

We present a bilinear Hirota representation of the N = 2 supersymmetric extension of the Korteweg-de Vries equation. This representation is deduced using binary Bell polynomials, hierarchies, and fermionic limits. We, also, propose a new approach for the generalisation of the Hirota bilinear formalism in the N = 2 supersymmetric context.

Decay Mode Solutions for the Supersymmetric Cylindrical KdV Equation

Abstract Supersymmetric cylindrical KdV equation is presented. Decay mode solutions for the supersymmetric KdV equation are derived by supersymmetric Hirota operator.

Nonholonomic deformation of coupled and supersymmetric KdV equations and Euler–Poincaré–Suslov method

Recently, Kupershmidt [38] presented a Lie algebraic derivation of a new sixth-order wave equation, which was proposed by Karasu-Kalkani et al. [31]. In this paper, we demonstrate that Kupershmidt's method can be interpreted as an infinite-dimensional analogue of the Euler–Poincaré–Suslov (EPS) formulation. In a finite-dimensional case, we modify Kupershmidt's deformation of the Euler top equation to obtain the standard EPS construction on SO(3). We extend Kupershmidt's infinite-dimensional construction to construct a nonholonomic deformation of a wide class of coupled KdV equations, where all these equations follow from the Euler–Poincaré–Suslov flows of the right invariant L2 metric on the semidirect product group [Formula: see text], where Diff (S1) is the group of orientation preserving diffeomorphisms on a circle. We generalize our construction to the two-component Camassa–Holm equation. We also give a derivation of a nonholonomic deformation of the N = 1 supersymmetric KdV equation, dubbed as sKdV6 equation and this method can be interpreted as an infinite-dimensional supersymmetric analogue of the Euler–Poincaré–Suslov (EPS) method.

Exact Solutions for the N = 2 Supersymmetric KdV Equation with the Bosonization Approach

The bosonization approach is applied to the N=2 supersymmetric KdV (SKdV) equation with a goal of building a system of coupled bosonic equations. It avoids the difficulties caused by intractable anticommuting fermionic fields. By solving the coupled bosonic equations, the exact solutions for the N=2 SKdV equation are demonstrated with the mapping and deformation method and the Lie point symmetry theory.

Supersymmetric KdV equation: Darboux transformation and discrete systems

For the supersymmetric KdV equation, a proper Darboux transformation is presented. This Darboux transformation leads to the Bäcklund transformation found earlier by Liu and Xie (2004 Phys. Lett. A 325 139–43). The Darboux transformation and the related Bäcklund transformation are used to construct integrable super differential–difference and difference–difference systems. The continuum limits of these discrete systems and of their Lax pairs are also considered.

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.

New solution of the \U0001d4a9 = 2 supersymmetric KdV equation via Hirota methods

We consider the resolution of the \U0001d4a9 = 2 supersymmetric KdV equation with a = −2 (SKdVa = −2) from the Hirota formalism. For the first time, a bilinear form of the SKdVa = −2 equation is constructed. We construct multisoliton solutions and rational similarity solutions.

SOLUTIONS AND LAX PAIRS BASED ON BILINEAR BÄCKLUND TRANSFORMATIONS OF SOME SUPERSYMMETRIC EQUATIONS

The paper investigates solutions and Lax pairs through bilinear Bäcklund transformations for some supersymmetric equations. We derive variety of solutions from the known bilinear Bäcklund transformations. Besides, using the gauge invariance of (super) Hirota bilinear derivatives we may get deformed bilinear Bäcklund transformations and consequently deformed Lax pairs with fermionic parameters. Examples are the supersymmetric KdV equation and the supersymmetric KdV1 equation.

Super extension of Bell polynomials with applications to supersymmetric equations

In this paper, we generalize classical Bell polynomials into super version, which are found to be effective in systematically constructing super bilinear representation, bilinear Bäcklund transformation, Lax pair, and infinite conservation laws of supersymmetric equations. We take \documentclass[12pt]{minimal}\begin{document}$\mathcal{N}=1$\end{document}N=1 supersymmetric KdV equation and \documentclass[12pt]{minimal}\begin{document}$\mathcal{N}=2$\end{document}N=2 supersymmetric sine-Gordon equation to illustrate this procedure.

Integrable fermionic extensions of the Garnier system and the anharmonic oscillator

We first derive a fermionic extension of the Garnier system by applying the method of binary nonlinearization of spectral problem to the supersymmetric KdV equation of Kupershmidt and then a fermionic extension of the anharmonic oscillator by a simple reduction. The integrable properties of resulting systems such as Lax representations, corresponding r-matrices and conversed integrals of motion are established.

Super quasiperiodic wave solutions and asymptotic analysis for % MathType!MTEF!2!1!+- % feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKf % MBHbqeduuDJXwAKbYu51MyVXgaruWqVvNCPvMCG4uz3bqefqvATv2C % G4uz3bIuV1wyUbqeeuuDJXwAKbsr4rNCHbGeaGqiVu0Je9sqqrpepC % 0xbbL8F4rqqrFfpeea0xe9Lq-Jc9vqaqpepm0xbba9pwe9Q8fs0-yq % aqpepae9pg0FirpepeKkFr0xfr-xfr-xb9adbaqaaeGaciGaaiaabe % qaamaaeaqbaaGcbaWefv3ySLgznfgDOfdaryqr1ngBPrginfgDObYt % UvgaiyaacqWFneVtrmWu51MyVXgaiCaacqGF9aqptCvAUfKttLearC % qr1ngBPrgaiGaacqqFXaqmaaa!51B5! $$ \mathcal{N} = 1 $$ supersymmetric KdV-type equations

Based on a general multidimensional Riemann theta function and the super Hirota bilinear form, we extend the Hirota method to construct explicit super quasiperiodic (multiperiodic) wave solutions of \( \mathcal{N} = 1 \) supersymmetric KdV-type equations in superspace. We show that the supersymmetric KdV equation does not have an N-periodic wave solution with arbitrary parameters for N ≥ 2. In addition, an interesting influencing band occurs among the super quasiperiodic waves under the presence of a Grassmann variable. We also observe that the super quasiperiodic waves are symmetric about this band but collapse along with it. We present a limit procedure for analyzing the asymptotic properties of the super quasiperiodic waves and rigorously show that the super periodic wave solutions tend to super soliton solutions under some “small amplitude” limits.

Quasiperiodic Wave Solutions of Supersymmetric KdV Equation in Superspace

A direct and unifying scheme for explicitly constructing quasiperiodic wave solutions (multiperiodic wave solutions) of supersymmetric KdV equation in a superspace is proposed. The scheme is based on the concept of super Hirota forms and on the use of super Riemann theta functions. In contrast to ordinary KdV equation with purely bosonic field, some new phenomena on super quasiperiodic waves occur in the supersymmetric KdV equation with the fermionic field. For instance, it is shown that the supersymmetric KdV equation does not possess an N-periodic wave solution for N≥ 2 for arbitrary parameters. It is further observed that there is an influencing band occurred among the quasiperiodic waves under the presence of the Grassmann variable. The quasiperiodic waves are symmetric about the band but collapse along with the band. In addition, the relations between the quasiperiodic wave solutions and soliton solutions are rigorously established. It is shown that quasiperiodic wave solution convergence to the soliton solutions under certain conditions and small amplitude limit.

Bilinear approach to N = 2 supersymmetric KdV equations

The N=2 supersymmetric KdV equations are studied within the framework of Hirota's bilinear method. For two such equations, namely $N=2, a=4$ and $N=2, a=1$ supersymmetric KdV equations, we obtain the corresponding bilinear formulations. Using them, we construct particular solutions for both cases. In particular, a bilinear B\"{a}cklund transformation is given for the $N=2, a=1$ supersymmetric KdV equation.

SUPERSYMMETRIC KUPER CAMASSA–HOLM EQUATION AND GEODESIC FLOW: A NOVEL APPROACH

We use the logarithmic 2-cocycle and the action of V ect (S1) on the space of pseudodifferential symbols to derive one particular type of supersymmetric KdV equation, known as Kuper-KdV equation. This equation was formulated by Kupershmidt and it is different from the Manin–Radul–Mathieu type equation. The two Super KdV equations behave differently under a supersymmetric transformation and Kupershmidt version does not preserve SUSY transformation. In this paper we study the second type of supersymmetric generalization of the Camassa–Holm equation correspoding to Kuper-KdV equation via standard embedding of super vector fields into the Lie algebra of graded pseudodifferential symbols. The natural lift of the action of superconformal group SDiff yields SDiff module. This method is particularly useful to construct Moyal quantized systems.

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.