KdV Hierarchy Research Articles

A note on the gauge equivalence between the Manin-Radul and Laberge-Mathieu super KdV hierarchies

The gauge equivalence between the Manin-Radul and Laberge-Mathieu super KdV hierarchies is revisited. Apart from the Inami-Kanno transformation, we show that there is another gauge transformation which also possess the canonical property. We explore the relationship of these two gauge transformations from the Kupershmidt-Wilson theorem viewpoint and, as a by-product, obtain the Darboux-Backlund transformation for the Manin-Radul super KdV hierarchy. The geometrical intepretation of these transformations is also briefly discussed.

On density of state of quantized Willmore surface - a way to a quantized extrinsic string in

Recently I quantized an elastica with Bernoulli-Euler functional in two-dimensional space using the modified KdV hierarchy. In this article, I will quantize a Willmore surface, or equivalently a surface with the Polyakov extrinsic curvature action, using the modified Novikov-Veselov (MNV) equation. In other words, I show that the density of state of the partition function for the quantized Willmore surface is expressed by volume of a subspace of the moduli of the MNV equation.

Negative Kadomtsev–Petviashvili Hierarchy

The negative KdV hierarchy is reobtained from the conformalinvariance of its Schwartz form. By means of the Schrödingeroperator, the negative KdV hierarchy can be rewritten to an equivalentdifferential equation system. The new method used for the KdV equationcan be extended to get more symmetries from known ones for arbitrarypartial differential equations without using a recursion operator. Thegeneral theory can also be used as a new criterion to verify whether amodel is integrable or not. Using the new method on the KP equation, aset of infinitely many nonlocal symmetries and then a negative KPhierarchy is obtained. The negative KP hierarchy can also be writtenas an equivalent differential equation system by using two“time (y)” dependentSchrödinger operators. The only difference between the negativeKdV hierarchy and the negative KP hierarchy is that one timeindependent Schrödinger operator in the KdV case is replaced bytwo time dependent Schrödinger operators in the KP case. A(2+1)-dimensional sinh-Gordon hierarchy is obtained from the negativeKP hierarchy by using the (2+1)-dimensional Miuratransformation.

Elliptic algebro-geometric solutions of the KdV and AKNS hierarchies - an analytic approach

We provide an overview of elliptic algebro-geometric solutions of the KdV and AKNS hierarchies, with special emphasis on Floquet theoretic and spectral theoretic methods. Our treatment includes an effective characterization of all stationary elliptic KdV and AKNS solutions based on a theory developed by Hermite and Picard.

A SIMPLE AND DIRECT PERIODIC SOLUTION OF THE KORTEWEG-DE VRIES EQUATIONS

The solutionq(x, t) of one of the KdV hierarchy is assumed to be a potential in the Schrodinger equation as usual. We differentiate this equation with respect to the time variable and solve it with the aid of the Green function. The obtained equation relatesw t (x, t, λ)=φ t (x + c, x, t, λ) withq t (x, t). The functionφ(x, x 0,t, λ) obeys the Schrodinger equation and the boundary conditionsφ(x 0,x 0,t, λ)=0,φ x (x 0,x 0;t, λ)=1. The shiftingc is equal to the period. We differentiatew t (x, t, λ) three times with respect to thex coordinate and obtain the time derivative of the Milne equation. The integration of this equation with respect tox allows to solve simply the inverse problem. The reconstructed periodic potential is given by means of the well known formula for the root functions ofw(x, t, λ). The time behaviour of this function, i.e. the solution of the KdV equation, is obtained when one replacesq t (x, t) by an expression of the KdV hiearchy in the relation betweenq t (x, t) andw t (x, t, λ) and transforms it. We estimated also the limit, whenc → ∞, i.e. the possible relation of the periodic solutions with the soliton ones.

R-Matrix for the Restricted KdV Flows with the Neumann Constraints

Under the Neumann constraints, each equation of the KdV hierarchy is decomposed into two finite dimensional systems, including the well-known Neumann model. Like in the case of the Bargmann constraint, the explicit Lax representations are deduced from the adjoint representation of the auxiliary spectral problem. It is shown that the Lax operator satisfies the r-matrix relation in the Dirac bracket. Thus, the integrabilities of these resulting systems with the Neumann constraints are obtained.

Systems of PDEs obtained from factorization in loop groups

We propose a generalization of a Drinfeld–Sokolov scheme of attaching integrable systems of PDEs to affine Kac–Moody algebras. With every affine Kac–Moody algebra \(\mathfrak{g} \) and a parabolic subalgebra \($$\) , we associate two hierarchies of PDEs. One, called positive, is a generalization of the KdV hierarchy, the other, called negative, generalizes the Toda hierarchy. We prove a coordinatization theorem which establishes that the number of functions needed to express all PDEs of the the total hierarchy equals the rank of\(\mathfrak{g} \) . The choice of functions, however, is shown to depend in a noncanonical way on \($$\). We employ a version of the Birkhoff decomposition and a ‘2-loop’ formulation which allows us to incorporate geometrically meaningful solutions to those hierarchies. We illustrate our formalism for positive hierarchies with a generalization of the Boussinesq system and for the negative hierarchies with the stationary Bogoyavlenskii equation.

Hamiltonian Structure and Coset Construction of the Supersymmetric Extensions of N=2 KdV Hierarchy

A manifestly N=2 supersymmetric coset formalism is applied to analyze the "fermionic" extensions of N=2, a=4 and a=-2 KdV hierarchies. Both hierarchies can be obtained from a manifest N=2 coset construction. This coset is defined as the quotient of some local but nonlinear superalgebra by a Û (1) subalgebra. Three superextensions of N=2 KdV hierarchy are proposed, among which one seems to be entirely new.

On the Korteweg–de Vries equation: Frequencies and initial value problem

The Korteweg-de Vries equation (KdV) @tv(x;t )+ @ 3 xv (x;t) 3@xv(x;t) 2 =0 ( x2 S 1 ;t2R ) is a completely integrable system with phase space L 2 (S 1 ). Although the Hamiltonian H(q ): = R S 1 1 @ x q ( x ) 2 + q ( x ) 3 dx is dened only on the dense subspace H 1 (S 1 ), we prove that the frequencies !j = @H @Jj can be dened on the whole space L 2 (S 1 ), where (Jj)j1 denote the action variables which are globally dened on L 2 (S 1 ). These frequencies are real analytic functionals and can be used to analyze Bourgain’s weak solutions of KdV with initial data in L 2 (S 1 ). The same method can be used for any equation in the KdV hierarchy.

On the integrable hierarchies associated with the N = 2 super Wn algebra

A new Lax operator is proposed from the viewpoint of constructing the integrable hierarchies related with the N = 2 super W n algebra. It is shown that the Poisson algebra associated to the second Hamiltonian structure for the resulting hierarchy contains the N = 2 super Virasoro algebra as a proper subalgebra. The simplest cases are discussed in detail. In particular, it is proved that the supersymmetric two-boson hierarchy is one of the N = 2 supersymmetric KdV hierarchies. Also, a Lax operator is supplied for one of the N = 2 supersymmetric Boussinesq hierarchies.

The bispectral property of aq-deformation of the Schur polynomials and theq-KdV hierarchy

We show that appropriate q-analogues of the Schur polynomials provide rational solutions of a q-deformation of the Nth KdV hierarchy. This allows us to construct explicit examples of bispectral commutative rings of q-difference operators.

Form factors, KdV and deformed hyperelliptic curves

We review and summarize recent works on the relation between form factors in integrable quantum field theory and deformation of geometrical data associated to hyper-elliptic curves. This relation, which is based on a deformation of the Riemann bilinear identity, in particular leads to the notion of null vectors in integrable field theory and to a new description of the KdV hierarchy.

A note on fractional KdV hierarchies

One of the cornerstones of the theory of integrable systems of KdV type has been the remark that the n-GD (Gel’fand–Dickey) equations are reductions of the full Kadomtsev–Petviashvilij (KP) theory. In this paper we address the analogous problem for the fractional KdV theories, by suggesting candidates of the “KP theories” lying behind them. These equations are called “KP(m) hierarchies,” and are obtained as reductions of a bigger dynamical system, which we call the “central system.” The procedure allowing passage from the central system to the KP(m) equations, and then to the fractional KdVnm equations, is discussed in detail in the paper. The case of KdV32 is given as a paradigmatic example.

N = 4 super NLS-mKdV hierarchies

N=2 extension of affine algebra $\hat{sl(2)\oplus u(1)}$ possesses a hidden global N=4 supersymmetry and provides a second hamiltonian structure for a new N=4 supersymmetric integrable hierarchy defined on N=2 affine supercurrents. This system is an N=4 extension of at once two hierarchies, N=2 NLS and N=2 mKdV ones. It is related to N=4 KdV hierarchy via a generalized Sugawara-Feigin-Fuks construction which relates N=2 $\hat{sl(2)\oplus u(1)}$ algebra to ``small'' N=4 SCA. We also find the underlying affine hierarchy for another integrable system with the N=4 SCA second hamiltonian structure, ``quasi'' N=4 KdV hierarchy. It respects only N=2 supersymmetry. For both new hierarchies we construct scalar Lax formulations. We speculate that any N=2 affine algebra admitting a quaternionic structure possesses N=4 supersymmetry and so can be used to produce N=4 supersymmetric hierarchies. This suggests a way of classifying all such hierarchies.

New integrable extensions of N = 2 KdV and Boussinesq hierarchies

We construct a new variety of N = 2 supersymmetric integrable systems by the junction of pseudo-differential superspace Lax operators for a = 4, N = 2 KdV and multi-component N = 2 NLS hierarchies. As an important special case, we obtain the Lax operator for the N = 4 super KdV system. A similar extension of one of the N = 2 super Boussinesq hierarchies is given. We also present a minimal N = 4 supersymmetric extension of the second flow of the N = 4 KdV hierarchy, and comment on its possible integrability.

Nonlinear dynamics of solitary waves in an extensible rod

In 1991, Goldstein and Petrich showed that the dynamics of a curve in a plane is governed geometrically by the modified KdV (mKdV) hierarchy. In this paper, we show a way of utilizing their new tool of the soliton approach to analyse the nonlinear dynamics of elastic rods. Considering the stretching effect of a narrow rod, nonlinear dynamics is studied by using exact soliton solutions of the mKdV equation, such as one soliton and breather soliton solutions. The metric of the rod which represents the stretching effect is explicitly given by perturbation analyses, and it is shown that an initial tension of the rod plays an important role in forming solitons in the rod.

Constrained KP Hierarchies: Additional Symmetries, Darboux–Bäcklund Solutions and Relations to Multi-Matrix Models

This paper provides a systematic description of the interplay between a specific class of reductions denoted as cKP r,m(r,m ≥ 1) of the primary continuum integrable system — the Kadomtsev–Petviashvili (KP) hierarchy and discrete multi-matrix models. The relevant integrable cKP r,m structure is a generalization of the familiar r-reduction of the full KP hierarchy to the SL (r) generalized KdV hierarchy cKP r,0. The important feature of cKP r,m hierarchies is the presence of a discrete symmetry structure generated by successive Darboux–Bäcklund (DB) transformations. This symmetry allows for expressing the relevant tau-functions as Wronskians within a formalism which realizes the tau-functions as DB orbits of simple initial solutions. In particular, it is shown that any DB orbit of a cKP r,1 defines a generalized two-dimensional Toda lattice structure. Furthermore, we consider the class of truncated KP hierarchies (i.e. those defined via Wilson–Sato dressing operator with a finite truncated pseudo-differential series) and establish explicitly their close relationship with DB orbits of cKP r,m hierarchies. This construction is relevant for finding partition functions of the discrete multi-matrix models. The next important step involves the reformulation of the familiar nonisospectral additional symmetries of the full KP hierarchy so that their action on cKP r,m hierarchies becomes consistent with the constraints of the reduction. Moreover, we show that the correct modified additional symmetries are compatible with the discrete DB symmetry on the cKP r,m DB orbits.

New super KdV system with the N = 4 SCA as the hamiltonian structure

We present a new integrable extension of the a = −2, N = 2 SKdV hierarchy, with the “small” N = 4 superconformal algebra (SCA) as the second hamiltonian structure. As distinct from the previously known N = 4 supersymmetric KdV hierarchy associated with the same N = 4 SCA, the new system respects only N = 2 rigid supersymmetry. We give for it both matrix and scalar Lax formulations and consider its various integrable reductions which complete the list of known SKdV systems with the N = 2 SCA as the second hamiltonian structure. We construct a generalized Miura transformation which relates our system to the α = −2, N = 2 super Boussinesq hierarchy and, respectively, the “small” N = 4 SCA to the N = 2 W 3 superalgebra.

Null-Vectors in Integrable Field Theory

The form factor bootstrap approach allows to construct the space of local fields in the massive restricted sine-Gordon model. This space has to be isomorphic to that of the corresponding minimal model of conformal field theory. We describe the subspaces which correspond to the Verma modules of primary fields in terms of the commutative algebra of local integrals of motion and of a fermion (Neveu–Schwarz or Ramond depending on the particular primary field). The description of null-vectors relies on the relation between form factors and deformed hyper-elliptic integrals. The null-vectors correspond to the deformed exact forms and to the deformed Riemann bilinear identity. In the operator language, the null-vectors are created by the action of two operators ? (linear in the fermion) and ? (quadratic in the fermion). We show that by factorizing out the null-vectors one gets the space of operators with the correct character. In the classical limit, using the operators ? and ? we obtain a new, very compact, description of the KdV hierarchy. We also discuss a beautiful relation with the method of Whitham.

Tau-functions and dressing transformations for zero-curvature affine integrable equations

The solutions of a large class of hierarchies of zero-curvature equations that includes Toda- and KdV-type hierarchies are investigated. All these hierarchies are constructed from affine (twisted or untwisted) Kac–Moody algebras 𝔤. Their common feature is that they have some special “vacuum solutions” corresponding to Lax operators lying in some Abelian (up to the central term) subalgebra of 𝔤; in some interesting cases such subalgebras are of the Heisenberg type. Using the dressing transformation method, the solutions in the orbit of those vacuum solutions are constructed in a uniform way. Then, the generalized tau-functions for those hierarchies are defined as an alternative set of variables corresponding to certain matrix elements evaluated in the integrable highest-weight representations of 𝔤. Such definition of tau-functions applies for any level of the representation, and it is independent of its realization (vertex operator or not). The particular important cases of generalized mKdV and KdV hierarchies as well as the Abelian and non-Abelian affine Toda theories are discussed in detail.