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arXiv1998

Hamiltonian structure of real Monge-Ampère equations

The real homogeneous Monge-Ampère equation in one space and one time dimensions admits infinitely many Hamiltonian operators and is completely integrable by Magri's theorem. This remarkable property holds in arbitrary number of dimensions as well, so that among all integrable nonlinear evolution equations the real homogeneous Monge-Ampère equation is distinguished as one that retains its character as an integrable system in multi-dimensions. This property can be traced back to the appearance of arbitrary functions in the Lagrangian formulation of the real homogeneous Monge-Ampère equation which is degenerate and requires use of Dirac's theory of constraints for its Hamiltonian formulation. As in the case of most completely integrable systems the constraints are second class and Dirac brackets directly yield the Hamiltonian operators. The simplest Hamiltonian operator results in the Kac-Moody algebra of vector fields and functions on the unit circle.

solv-int↗
arXiv1996

Phase diagram of the non-hermitean asymmetric XXZ spin chain

The low-lying excitations of the asymmetric $XXZ$ spin chain are derived explicitly in the antiferromagnetic regime through the Bethe Ansatz. It is found that a massless and conformal invariant phase with central charge $c=1$ is separated from a massive phase by a line on which the low-lying excitations surprisingly scale with the lattice length as $ΔE \sim N^{-\frac{1}{2}}$. The mass gap vanishes with an exponent $\frac{1}{2}$ as one approaches the massless phase. The connection with the asymmetric $6$--vertex model and some physical consequences are discussed.

cond-mat↗
arXiv1995

Classical Poisson structures and r-matrices from constrained flows

We construct the classical Poisson structure and $r$-matrix for some finite dimensional integrable Hamiltonian systems obtained by constraining the flows of soliton equations in a certain way. This approach allows one to produce new kinds of classical, dynamical Yang-Baxter structures. To illustrate the method we present the $r$-matrices associated with the constrained flows of the Kaup-Newell, KdV, AKNS, WKI and TG hierarchies, all generated by a 2-dimensional eigenvalue problem. Some of the obtained $r$-matrices depend only on the spectral parameters, but others depend also on the dynamical variables. For consistency they have to obey a classical Yang-Baxter-type equation, possibly with dynamical extra terms.

solv-int↗
arXiv1996

Soliton equations and the zero curvature condition in noncommutative geometry

Familiar nonlinear and in particular soliton equations arise as zero curvature conditions for GL(1,R) connections with noncommutative differential calculi. The Burgers equation is formulated in this way and the Cole-Hopf transformation for it attains the interpretation of a transformation of the connection to a pure gauge in this mathematical framework. The KdV, modified KdV equation and the Miura transformation are obtained jointly in a similar setting and a rather straightforward generalization leads to the KP and a modified KP equation. Furthermore, a differential calculus associated with the Boussinesq equation is derived from the KP calculus.

hep-th↗
arXiv1996

Common Algebraic Structure for the Calogero-Sutherland Models

We investigate common algebraic structure for the rational and trigonometric Calogero-Sutherland models by using the exchange-operator formalism. We show that the set of the Jack polynomials whose arguments are Dunkl-type operators provides an orthogonal basis for the rational case.

solv-int↗
arXiv1996

Scattering of Plane Waves in Self-Dual Yang-Mills Theory

We consider the classical self-dual Yang-Mills equation in 3+1-dimensional Minkowski space. We have found an exact solution, which describes scattering of $n$ plane waves. In order to write the solution in a compact form, it is convenient to introduce a scattering operator $\hat{T}$. It acts in the direct product of three linear spaces: 1) universal enveloping of $su(N)$ Lie algebra, 2) $n$-dimensional vector space and 3) space of functions defined on the unit interval.

hep-th↗
arXiv1995

Negaton and Positon Solutions of the KDV Equation

We give a systematic classification and a detailed discussion of the structure, motion and scattering of the recently discovered negaton and positon solutions of the Korteweg-de Vries equation. There are two distinct types of negaton solutions which we label $[S^{n}]$ and $[C^{n}]$, where $(n+1)$ is the order of the Wronskian used in the derivation. For negatons, the number of singularities and zeros is finite and they show very interesting time dependence. The general motion is in the positive $x$ direction, except for certain negatons which exhibit one oscillation around the origin. In contrast, there is just one type of positon solution, which we label $[\tilde C^n]$. For positons, one gets a finite number of singularities for $n$ odd, but an infinite number for even values of $n$. The general motion of positons is in the negative $x$ direction with periodic oscillations. Negatons and positons retain their identities in a scattering process and their phase shifts are discussed. We obtain a simple explanation of all phase shifts by generalizing the notions of ``mass" and ``center of mass" to singular solutions. Finally, it is shown that negaton and positon solutions of the KdV equation can be used to obtain corresponding new solutions of the modified KdV equation.

hep-th↗
arXiv1997

Maxwell-Bloch equation and Correlation function for penetrable Bose gas

We consider the quantum nonlinear Schrödinger equation in one space and one time dimension. We are interested in the non-free-fermionic case. We consider static temperature-dependent correlation functions. The determinant representation for correlation functions simplifies in the small mass limit of the Bose particle. In this limit we describe the correlation functions by the vacuum expectation value of a boson-valued solution for Maxwell-Bloch differential equation. We evaluate long-distance asymptotics of correlation functions in the small mass limit.

hep-th↗
arXiv1998

On fusion algebra of chiral $SU(N)_{k}$ models

We discuss some algebraic setting of chiral $SU(N)_{k}$ models in terms of the statistical dimensions of their fields. In particular, the conformal dimensions and the central charge of the chiral $SU(N)_{k}$ models are calculated from their braid matrices. Futhermore, at level K=2, we present the characteristic polynomials of their fusion matrices in a factored form.

solv-int↗
arXiv1997

Time-Dependent Symmetries of Variable-Coefficient Evolution Equations and Graded Lie Algebras

Polynomial-in-time dependent symmetries are analysed for polynomial-in-time dependent evolution equations. Graded Lie algebras, especially Virasoro algebras, are used to construct nonlinear variable-coefficient evolution equations, both in 1+1 dimensions and in 2+1 dimensions, which possess higher-degree polynomial-in-time dependent symmetries. The theory also provides a kind of new realisation of graded Lie algebras. Some illustrative examples are given.

solv-int↗
arXiv1997

Simplified Calculation of Boundary S Matrices

The antiferromagnetic Heisenberg spin chain with N spins has a sector with N=odd, in which the number of excitations is odd. In particular, there is a state with a single one-particle excitation. We exploit this fact to give a simplified derivation of the boundary S matrix for the open antiferromagnetic spin-1/2 Heisenberg spin chain with diagonal boundary magnetic fields.

hep-th↗
arXiv1997

Extended matrix Gelfand-Dickey hierarchies: reduction to classical Lie algebras

The Drinfeld-Sokolov reduction method has been used to associate with $gl_n$ extensions of the matrix r-KdV system. Reductions of these systems to the fixed point sets of involutive Poisson maps, implementing reduction of $gl_n$ to classical Lie algebras of type $B, C, D$, are here presented. Modifications corresponding, in the first place to factorisation of the Lax operator, and then to Wakimoto realisations of the current algebra components of the factorisation, are also described.

solv-int↗
arXiv1997

Integrable su(3) spin chain combining different representations

The general expression for the local matrix $t(θ)$ of a quantum chain with the site space in any representation of su(3) is obtained. This is made by generalizing $t(θ)$ from the fundamental representation and imposing the fulfillment of the Yang-Baxter equation. Then, a non-homogeneous spin chain combining different representations of su(3) is solved by developing a method inspired in the nested Bethe ansatz. The solution for the eigenvalues of the trace of the monodromy matrix is given as two coupled Bethe equations. A conjecture about the solution of a chain with the site states in different representations of su(n) is presented. The thermodynamic limit of the ground state is calculated.

cond-mat↗
arXiv1997

Laplace transformations of hydrodynamic type systems in Riemann invariants: periodic sequences

The conserved densities of hydrodynamic type system in Riemann invariants satisfy a system of linear second order partial differential equations. For linear systems of this type Darboux introduced Laplace transformations, generalising the classical transformations in the scalar case. It is demonstrated that Laplace transformations can be pulled back to the transformations of the corresponding hydrodynamic type systems. We discuss periodic Laplace sequences of with the emphasize on the simplest nontrivial case of period 2. For 3-component systems in Riemann invariants a complete discription of closed quadruples is proposed. They turn to be related to a special quadratic reduction of the (2+1)-dimensional 3-wave system which can be reduced to a triple of pairwize commuting Monge-Ampere equations. In terms of the Lame and rotation coefficients Laplace transformations have a natural interpretation as the symmetries of the Dirac operator, associated with the (2+1)-dimensional n-wave system. The 2-component Laplace transformations can be interpreted also as the symmetries of the (2+1)-dimensional integrable equations of Davey-Stewartson type. Laplace transformations of hydrodynamic type systems originate from a canonical geometric correspondence between systems of conservation laws and line congruences in projective space.

solv-int↗
arXiv1997

Complex Analysis of a Piece of Toda Lattice

We study a small piece of two dimensional Toda lattice as a complex dynamical system. In particular the Julia set, which appears when the piece is deformed, is shown analytically how it disappears as the system approaches to the integrable limit.

solv-int↗
arXiv1997

The constrained modified KP hierarchy and the generalized Miura transformations

In this letter, we consider the second Hamiltonian structure of the constrained modified KP hierarchy. After mapping the Lax operator to a pure differential operator the second structure becomes the sum of the second and the third Gelfand-Dickey brackets defined by this differential operator. We simplify this Hamiltonian structure by factorizing the Lax operator into linear terms.

solv-int↗