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 Bäcklund transformation is given for the $N=2, a=1$ supersymmetric KdV equation.
Casorati determinant solution to the non-autonomous discrete KdV equation is constructed by using the bilinear formalism. We present three different bilinear formulations which have different origins.
Hirota bilinear form and multisoliton solution for semidiscrete and fully discrete (difference-difference) versions of supersymmetric KdV equation found by Xue, Levi and Liu [1] is presented. The solitonic interaction term displays a fermionic dressing factor as in the continuous supersymmetric case. Using bilinear equations it is shown also that there can be constructed a new integrable semidiscrete (and fully discrete) version of supersymmetric KdV which has simpler bilinear form but more complicated interaction dressing. Its continuum limit is also computed.
Hirota bilinear form and soliton solutions for super-KdV of Kuperschmidt (Kuper-KdV) are given. It is shown that even though the collision of supersolitons is more complicated than in the case of supersymmetric KdV 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, goes to the Kuper-KdV equation in a multiple-scale approach.
In this paper, an efficient algorithm of logarithmic transformation to Hirota bilinear form of the KdV-type bilinear equation is established. In the algorithm, some properties of Hirota operator and logarithmic transformation are successfully applied, which helps to prove that the linear terms of the nonlinear partial differential equation play a crucial role in finding the Hirota bilinear form. Experimented with various integro-differential equations, our algorithm is proven to be more efficient than the algorithm referred by Zhou, Fu, and Li in getting the Hirota bilinear form, especially in…
Two different types of N=1 modified KdV equations are shown to possess $N$ soliton solutions. The soliton solutions of these equations are obtained by casting the equations in the bilinear forms using the supersymmetric extension of the Hirota method. The distinguishing features of the soliton solutions of N=1 mKdV and N=1 mKdV B equations are discussed.
Various solutions to the discrete Schwarzian KdV equation are discussed. We first derive the bilinear difference equations of Hirota type of the discrete Schwarzian KP equation, which is decomposed into three discrete two-dimensional Toda lattice equations. We then construct two kinds of solutions in terms of the Casorati determinant. We derive the discrete Schwarzian KdV equation on an inhomogeneous lattice and its solutions by a reduction process. We finally discuss the solutions in terms of the $τ$ functions of some Painlevé systems.
Extending the gauge-invariance principle for τfunctions of the standard bilinear formalism to the supersymmetric case, we define N=1 supersymmetric Hirota operators. Using them, we bilinearize SUSY KdV-type equations (KdV, Sawada-Kotera, Hirota-Satsuma). The solutions for multiple collisions of super-solitons and extension to SUSY sine-Gordon are also discussed.
In the paper we discuss the Bäcklund transformation of the KdV equation between solitons and solitons, between negatons and negatons, between positons and positons, between rational solution and rational solution, and between complexitons and complexitons. We investigate the conditions that Wronskian entries satisfy for the bilinear Bäcklund transformation of the KdV equation. By choosing suitable Wronskian entries and the parameter in the bilinear Bäcklund transformation, we obtain transformations between many kinds of solutions.
A detailed description is given for the construction of the deformation of the N=2 supersymmetric $α=1$ KdV-equation, leading to the recursion operator for symmetries and the zero-th Hamiltonian structure; the solution to a longstanding problem.
In this paper, we derive a Bäcklund transformation for the supersymmetric Kortweg-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 N=2 supersymmetric KdV equation of Inami and Kanno is bilinearized employing the Hirota method and the existence of $N$ soliton solutions is demonstrated. The exact form of the solutions are explicitly obtained and an interesting symmetry of the equations of motion is observed.
Ultradiscrete soliton equations and Bäcklund transformation for a generalized soliton solution are presented. The equations include the ultradiscrete KdV equation or the ultradiscrete Toda equation in a special case. We also express the solution by the ultradiscrete permanent, which is defined by ultradiscretizing the signature-free determinant, that is, the permanent. Moreover, we discuss a relation between Bäcklund transformations for discrete and ultradiscrete KdV equations.
Writing the Hirota-Satsuma (HS) system of equations in a symmetrical form we find its local and new nonlocal reductions. It turns out that all reductions of the HS system are Korteweg-de Vries (KdV), complex KdV, and new nonlocal KdV equations. We obtain one-soliton solutions of these KdV equations by using the method of Hirota bilinearization.
The $N=2 \;a=-2$ 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.
We show that the supersymmetric KdV and KP equations, related to the non-trivial flows, can be cast in the Hirota bilinear form. The existence of one, two and subsequently $N$-soliton solutions is explicitly demonstrated.
We consider the bi-Hamiltonian representation of the two-component coupled KdV equations discovered by Drinfel'd and Sokolov and rediscovered by Sakovich and Foursov. Connection of this equation with the supersymmetric Kadomtsev-Petviashvilli-Radul-Manin hierarchy is presented. For this new supersymmetric equation the Lax representation and odd Hamiltonian structure is given.
For the Harry Dym hierarchy, a non-standard Lax formulation is deduced from that of 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 superconformal transformation. The bosonic limit of this new N=2 supersymmetric Harry Dym equation is related to a coupled system of KdV-MKdV equations.
We preesent a new supersymmetric integrable extensions of the a=4,N=2 KdV hierarchy. The root of the supersymmetric Lax operator of the KdV equation is generalized, by including additional fields. This generalized root generate new hierarchy of integrable equations, for which we investigate the hamiltonian structure. In special case our system describes the interaction of the KdV equation with the two MKdV equations.
A bilinear formulation for the supersymmetric two-boson equation is derived. As applications, some solutions are calculated for it. We also construct a bilinear Backlund transformation.
We prove that P.Mathieu's Open problem on constructing Gardner's deformation for the N=2 supersymmetric a=4-Korteweg-de Vries equation has no supersymmetry invariant solutions, whenever it is assumed that they retract to Gardner's deformation of the scalar KdV equation under the component reduction. At the same time, we propose a two-step scheme for the recursive production of the integrals of motion for the N=2, a=4-SKdV. First, we find a new Gardner's deformation of the Kaup-Boussinesq equation, which is contained in the bosonic limit of the super-hierarchy. This yields the recurrence relati…
We present an alternative integrable discretization of differential-difference KdV equation based on Hirota bilinear formalism. It is shown that using two tau functions the direct discretisation of the bilinear equations gives immediately the well known discrete KdV equation. We comment also on integrability and relation with the classical bilinear form involving only one tau function.
We propose the Lax operators for N=2 supersymmetric matrix generalization of the bosonic (1,s)-KdV hierarchies. The simplest examples - the N=2 supersymmetric a=4 KdV and a=5/2 Boussinesq hierarchies - are discussed in detail.
A proper bilinear form is proposed for the N=1 supersymmetric modified Korteweg-de Vries equation. The bilinear Bäcklund transformation of this system is constructed. As applications, some solutions are presented for it.
We propose a hamiltonian formulation of the $N=2$ supersymmetric KP type hierarchy recently studied by Krivonos and Sorin. We obtain a quadratic hamiltonian structure which allows for several reductions of the KP type hierarchy. In particular, the third family of $N=2$ KdV hierarchies is recovered. We also give an easy construction of Wronskian solutions of the KP and KdV type equations.
Darboux transformation is reconsidered for the supersymmetric KdV system. By iterating the Darboux transformation, a supersymmetric extension of the Crum transformation is obtained for the Manin-Radul SKdV equation, in doing so one gets Wronskian superdeterminant representations for the solutions. Particular examples provide us explicit supersymmetric extensions, super solitons, of the standard soliton of the KdV equation. The KdV soliton appears as the body of the super soliton.
Wronski determinant (Wronskian) provides a compact form for $τ$-functions that play roles in a large range of mathematical physics. In 1979 Matveev and Satsuma, independently, obtained solutions in Wronskian form for the Kadomtsev-Petviashvili equation. Later, in 1981 these solutions were constructed from Sato's approach. Then in 1983, Freeman and Nimmo invented the so-called Wronskian technique, which allows directly verifying bilinear equations when their solutions are given in terms of Wronskians. In this technique the considered bilinear equation is usually reduced to the Plücker relation …
We present a systematic procedure for obtaining the dispersionless limit of a class of N=1 supersymmetric systems starting from the Lax description of their dispersive counterparts. This is achieved by starting with an N=2 supersymmetric system and scaling the fields in an alternative manner so as to maintain N=1 supersymmetry. We illustrate our method by working out explicitly the examples of the dispersionless supersymmetric two boson hierarchy and the dispersionless supersymmetric Boussinesq hierarchy.
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 Kortweg-de Vries equations, thus are connected with two N=2 supersymmetric Kortweg-de Vries equations.
In this paper, based on the regular KdV system, we study negative order KdV (NKdV) equations about their Hamiltonian structures, Lax pairs, infinitely many conservation laws, and explicit multi-soliton and multi-kink wave solutions thorough bilinear Bäcklund transformations. The NKdV equations studied in our paper are differential and actually derived from the first member in the negative order KdV hierarchy. The NKdV equations are not only gauge-equivalent to the Camassa-Holm equation through some hodograph transformations, but also closely related to the Ermakov-Pinney systems, and the Kuper…
This paper can be an overview on solutions in Wronskian/Casoratian form to soliton equations with KdV-type bilinear forms. We first investigate properties of matrices commuting with a Jordan block, by which we derive explicit general solutions to equations satisfied by Wronskian/Casoratian entry vectors, which we call condition equations. These solutions are given according to the coefficient matrix in the condition equations taking diagonal or Jordan block form. Limit relations between these different solutions are described. We take the KdV equation and the Toda lattice to serve as two examp…
We present the Lax operator for the N=3 KdV hierarchy and consider its extensions. We also construct a new infinite family of N=2 supersymmetric hierarchies by exhibiting the corresponding super Lax operators. The new realization of N=4 supersymmetry on the two general N=2 superfields, bosonic spin 1 and fermionic spin 1/2, is discussed.
The general method of the supersymmetrization of the soliton equations with the odd (bi) hamiltonian structure is established. New version of the supersymmetric N=2,4 (Modified) Korteweg de Vries equation is given, as an example. The second odd Hamiltonian operator of the SUSY KdV equation generates the odd N=2,4 SUSY Virasoro - like algebra.
A manifestly N=2 supersymmetric coset formalism is applied to analyse the "fermionic" extensions of N=2 $a=4$ and $a=-2$ KdV hierarchies. Both these hierarchies can be obtained from a manifest N=2 coset construction. This coset is defined as the quotient of some local but non-linear superalgebra by a $\hat{U(1)}$ subalgebra. Three superextensions of N=2 KdV hierarchy are proposed, among which one seems to be entirely new.
The N=2 supersymmetric α=1 KdV hierarchy in N=2 superspace is considered and its rich symmetry structure is uncovered. New nonpolynomial and nonlocal, bosonic and fermionic symmetries and Hamiltonians, bi-Hamiltonian structure as well as a recursion operator connecting all symmetries and Hamiltonian structures of the N=2 α=1 KdV hierarchy are constructed in explicit form. It is observed that the algebra of symmetries of the N=2 supersymmetric α=1 KdV hierarchy possesses two different subalgebras of N=2 supersymmetry.
For the supersymmetric KdV equation, a proper Darboux transformation is presented. This Darboux transformation leads to the Bäcklund transformation found early by Liu and Xie \cite{liu2}. 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.
In this paper we present a vectorial Darboux transformation, in terms of ordinary determinants, for the supersymmetric extension of the Korteweg-de Vries equation proposed by Manin and Radul. It is shown how this transformation reduces to the Korteweg-de Vries equation. Soliton type solutions are constructed by dressing the vacuum and we present some relevant plots.
The Poisson structure of a coupled system arising from a supersymmetric breaking of N=1 Super KdV equations is obtained. The supersymmetric breaking is implemented by introducing a Clifford algebra instead of a Grassmann algebra. The Poisson structure follows from the Dirac brackets obtained by the constraint analysis of the hamiltonian of the system. The coupled system has multisolitonic solutions. We show that the one soliton solutions are Liapunov stable.
We devise a lucid and straightforward way for explicitly constructing quasi-periodic wave solutions (also called multi-periodic wave solutions) of supersymmetric equations in superspace $\mathbb{R}_Λ^{2,1}$ over two-dimensional Grassmann algebra $G_1(σ)$. Once a nonlinear equation is written in a bilinear form, its quasi-periodic wave solutions can be directly obtained by using a formula. Moreover, properties of these solutions are investigated in detail by analyzing their structures, plots and asymptotic behaviors. The relations between the quasi-periodic wave solutions and soliton solutions …
We establish a bilinear framework for elliptic soliton solutions which are composed by the Lamé-type plane wave factors. $τ$ functions in Hirota's form are derived and vertex operators that generate such $τ$ functions are presented. Bilinear identities are constructed and an algorithm to calculate residues and bilinear equations is formulated. These are investigated in detail for the KdV equation and sketched for the KP hierarchy. Degenerations by the periods of elliptic functions are investigated, giving rise to the bilinear framework associated with trigonometric/hyperbolic and rational func…
We study the distribution of singularities for partial difference equations, in particular, the bilinear and nonlinear form of the discrete version of the Korteweg-de Vries (dKdV) equation. By the Laurent property, the irreducibility, and the co-primeness of the terms of the bilinear dKdV equation, we clarify the relationship of these properties with the appearance of zeros in the time evolution. The results are applied to the nonlinear dKdV equation and we formulate the famous integrability criterion (singularity confinement test) for nonlinear partial difference equations with respect to the…
We reduce the vectorial binary Darboux transformation for the Manin-Radul supersymmetric KdV system in such a way that it preserves the Manin-Radul-Mathieu supersymmetric KdV equation reduction. Expressions in terms of bosonic Pfaffians are provided for transformed solutions and wave functions. We also consider the implications of these results for the supersymmetric sine-Gordon equation.
Bosonization approach to the classical supersymmetric systems is presented. By introducing the multi-fermionic parameters in the expansions of the superfields, the $\mathcal {N}=1$ supersymmetric KdV (sKdV) equations are transformed to a system of coupled bosonic equations. The method can be applied to any fermionic systems. By solving the coupled bosonic equations, some novel types of exact solutions can be explicitly obtained. Especially, the richness of the localized excitations of the supersymmetric integrable system are discovered. The rich multi-soliton solutions obtained here have not y…
A study of Hamiltonian structures associated with supersymmetric Lax operators is presented. Following a constructive approach, the Hamiltonian structures of Inami-Kanno super KdV hierarchy and constrained modified super KP hierarchy are investigated from the reduced supersymmetric Gelfand-Dickey brackets. By applying a gauge transformation on the Hamiltonian structures associated with these two nonstandard super Lax hierarchies, we obtain the Hamiltonian structures of generalized Manin-Radul super KdV and constrained super KP hierarchies. We also work out a few examples and compare them with …
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 Painlevé property of the BSKdV system is proved and a Bäcklund transformation (BT…
In this paper, we introduce a class of new generalized super Bell polynomials on a superspace, explore their properties, and show that they are a natural and effective tool to systematically investigate integrability of supersymmetric equations. The connections between the super Bell polynomials and super bilinear representation, bilinear Bäcklund transformation, Lax pair and infinite conservation laws of supersymmetric equations are established. We take supersymmetric KdV equation and supersymmetric sine-Gordon equation to illustrate this procedure.
A new integrable nonautonomous nonlinear ordinary difference equation is presented which can be considered to be a discrete analogue of the Painleve V equation. Its derivation is based on the similarity reduction on the two-dimensional lattice of integrable partial difference equations of KdV type. The new equation which is referred to as GDP (generalised discrete Painleve equation) contains various ``discrete Painleve equations'' as subcases for special values/limits of the parameters, some of which were already given before in the literature. The general solution of the GDP can be expressed …
We construct a new variety of $N=2$ supersymmetric integrable systems by junction of pseudo-differential superspace Lax operators for $a=4$, $N=2$ KdV and multi-component $N=2$ NLS hierarchies. As an important particular case, we obtain Lax operator for $N=4$ super KdV system. A similar extension of one of $N=2$ super Boussinesq hierarchies is given. We also present a minimal $N=4$ supersymmetric extension of the second flow of $N=4$ KdV hierarchy and comment on its possible integrability.
Using methods of math.DG/0304245 and [I.S.Krasil'shchik and P.H.M.Kersten, Symmetries and recursion operators for classical and supersymmetric differential equations, Kluwer, 2000], we accomplish an extensive study of the N=1 supersymmetric Korteweg-de Vries 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…
Quantization procedure of the Gardner-Zakharov-Faddeev and Magri brackets by means of the fermionic representation for the KdV field is considered. It is shown that in both cases the corresponding Hamiltonians are given as sums of two well defined operators. Each of them is bilinear and diagonal with respect to either fermion, or boson (current) creation-annihilation operators. As a result the quantization procedure needs no any space cut-off and can be performed on the whole axis. Existence of the solitonic states in the Hilbert space as well as quantization of soliton parameters are shown to…
Based on a Riemann theta function and the super-Hirota bilinear form, we propose a key formula for explicitly constructing quasi-periodic wave solutions of the supersymmetric Ito's equation in superspace $\mathbb{C}_Λ^{2,1}$. Once a nonlinear equation is written in bilinear forms, then the quasi-periodic wave solutions can be directly obtained from our formula. The relations between the periodic wave solutions and the well-known soliton solutions are rigorously established. It is shown that the quasi-periodic wave solutions tends to the soliton solutions under small amplitude limits.
Extending the gauge-invariance principle for $τ$ functions of the standard bilinear formalism to the supersymmetric case, we define ${\cal N}=1$ supersymmetric Hirota bilinear operators. Using them we bilinearize supersymmetric nonlinear evolution equations. The super-soliton solutions are discussed. As a quite strange paradox it is shown that the Lax integrable supersymmetric KdV of Manin-Radul-Mathieu equation does not possesses N super-soliton solution for $N\geq 3$ for arbitrary parameters. Only for a particular choice of them the N super-soliton solution exists.
