Exactly Solvable And Integrable Systems

Two properties of Calogero wave functions for rational Calogero models are studied: (i) the representation of the wave functions in terms of the exponential of Lassalle operators, (ii) the $sL(2,\rr)$ structure of the Calogero--Moser wave functions.

In this work we develop a general procedure for constructing the recursion operators fro non-linear integrable equations admitting Lax representation. Svereal new examples are given. In particular we find the recursion operators for some KdV-type of integrable equations.

We study Darboux-Bäcklund transformations (DBTs) for the q -deformed Korteweg-de Vries hierarchy by using the q -deformed pseudodifferential operators. We identify the elementary DBTs which are triggered by the gauge operators constructed from the (adjoint) wave functions of the hierarchy. Iterating these elementary DBTs we obtain not only q -deformed Wronskian-type but also binary-type representations of the tau-function to the hierarchy.

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.

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 in terms of Painleve VI (PVI) transcendents. In fact, continuous PVI emerges as the equation obeyed by the solutions of the discrete equation in terms of the lattice parameters rather than the lattice variables that label the lattice sites. We show that the bilinear form of PVI is embedded naturally in the lattice systems leading to the GDP. Further results include the establishment of Baecklund and Schlesinger transformations for the GDP, the corresponding isomonodromic deformation problem, and the self-duality of its bilinear scheme.

Two discretizations of the vector nonlinear Schrodinger (NLS) equation are studied. One of these discretizations, referred to as the symmetric system, is a natural vector extension of the scalar integrable discrete NLS equation. The other discretization, referred to as the asymmetric system, has an associated linear scattering pair. General formulae for soliton solutions of the asymmetric system are presented. Formulae for a constrained class of solutions of the symmetric system may be obtained. Numerical studies support the hypothesis that the symmetric system has general soliton solutions.

The system of N classical particles on the line with the Weierstrass ℘ function as potential is known to be completely integrable. Recently D'Hoker and Phong found a beautiful parameterization by the polynomial of degree N of the space of Riemann surfaces associated with this system. In the trigonometric limit of the elliptic potential these Riemann surfaces degenerate into rational curves. The D'Hoker-Phong polynomial in the limit describes the intersection points of the rational curves. We found an explicit determinant representation of the polynomial in the trigonometric case. We consider applications of this result to the theory of Toeplitz determinants and to geometry of the spectral curves. We also prove our earlier conjecture on the asymptotic behavior of the ratio of two symplectic volumes when the number of particles tends to infinity.

This note develops an explicit construction of the constrained KP hierarchy within the Sato Grassmannian framework. Useful relations are established between the kernel elements of the underlying ordinary differential operator and the eigenfunctions of the associated KP hierarchy as well as between the related bilinear concomitant and the squared eigenfunction potential.

The scalar nonlinear Schrodinger (NLS) equation and a suitable discretization are well known integrable systems which exhibit the phenomena of ``effective'' chaos. Vector generalizations of both the continuous and discrete system are discussed. Some attention is directed upon the issue of the integrability of a discrete version of the vector NLS equation.

We apply the exchange operator formalism in polar coordinates to a one-parameter family of three-body problems in one dimension and prove the integrability of the model both with and without the oscillator potential. We also present exact scattering solution of a new family of three-body problems in one dimension.