Exactly Solvable And Integrable Systems

We use the singular manifold method to obtain the Lax pair, Darboux transformations and soliton solutions for a (2+1) dimensional integrable equation.

For the n-dimensional integrable system with a twisted so(p,q) reduction, Darboux transformations given by Darboux matrices of degree 2 are constructed explicitly. These Darboux transformations are applied to the local isometric immersion of space forms with flat normal bundle and linearly independent curvature normals to give the explicit expression of the position vector. Some examples are given from the trivial solutions and standard imbedding T^n\to R^{2n}.

We systematically study Darboux-type transformations for the KdV and AKNS hierarchies and provide a complete account of their effects on hyperelliptic curves associated with algebro-geometric solutions of these hierarchies.

The bi-Hamiltonian structure of certain multi-component integrable systems, generalizations of the dispersionless Toda hierarchy, is studies for systems derived from a rational Lax function. One consequence of having a rational rather than a polynomial Lax function is that the corresponding bi-Hamiltonian structures are degenerate, i.e. the metric which defines the Hamiltonian structure has vanishing determinant. Frobenius manifolds provide a natural setting in which to study the bi-Hamiltonian structure of certain classes of hydrodynamic systems. Some ideas on how this structure may be extanded to include degenerate bi-Hamiltonian structures, such as those given in the first part of the paper, are given.

Determinant formulas for the general solutions of the Toda and discrete Toda equations are presented. Application to the τ functions for the Painlevé equations is also discussed.

Rational solutions for the Painlevé IV equation are investigated by Hirota bilinear formalism. It is shown that the solutions in one hierarchy are expressed by 3-reduced Schur functions, and those in another two hierarchies by Casorati determinant of the Hermite polynomials, or by special case of the Schur polynomials.

Using the Quantum Inverse Scattering Method for the XXZ model with open boundary conditions, we obtained the determinant formula for the six vertex model with reflecting end.

Functional relations play a key role in the study of integrable models. We argue in this paper that for massless field theories at zero temperature, these relations can in fact be interpreted as monodromy relations. Combined with a recently discovered duality, this gives a way to bypass the Bethe ansatz, and compute directly physical quantities as solutions of a linear differential equation, or as integrals over a hyperelliptic curve. We illustrate these ideas in details in the case of the c=1 theory, and the associated boundary sine-Gordon model.

In the previous report (J. Phys. A (1997) 30 4019-4029), I showed that the Dirac operator defined over a conformal surface immersed in R^3 is identified with the Dirac operator which is generalized the Weierstrass- Enneper equation and Lax operator of the modified Novikov-Veselov (MNV) equation. In this article, I determine the Dirac operator defined over a conformal surface immersed in R^4, which is reduced to the Lax operators of the nonlinear Schrodinger and the MNV equations by taking appropriate limits. Thus the Dirac operator might be the Lax operator of (2+1)- dimensional soliton equation.

A universal system of difference equations associated with a hyperelliptic curve is derived constituting the discrete analogue of the Dubrovin equations arising in the theory of finite-gap integration. The parametrisation of the solutions in terms of Abelian functions of Kleinian type (i.e. the higher-genus analogues of the Weierstrass elliptic functions) is discussed as well as the connections with the method of separation of variables.