Related papers: Exact Solutions of Integrable 2D Contour Dynamics
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~$\ggg$.…
I review the appearance of classical integrable systems as an effective tool for the description of non-perturbative exact results in quantum string and gauge theories. Various aspects of this relation: spectral curves, action-angle…
A sequence of solutions to the equation of symmetry for the continuous Toda chain in $1+2$ dimensions is represented in explicit form. This fact leads to the supposition that this equation is completely integrable.
We study how the solution of the two-dimensional Dirichlet boundary problem for smooth simply connected domains depends upon variations of the data of the problem. We show that the Hadamard formula for the variation of the Dirichlet Green…
We construct a certain reduction of the 2D Toda hierarchy and obtain a tau-symmetric Hamiltonian integrable hierarchy. This reduced integrable hierarchy controls the linear Hodge integrals in the way that one part of its flows yields the…
The negative integrable hierarchies of shallow water waves and dispersionless Toda lattice equations are considered. The integrability is shown by explicit construction of an infinite set of conservation laws.
The notion of non-degenerate solutions for the dispersionless Toda hierarchy is generalized to the universal Whitham hierarchy of genus zero with $M+1$ marked points. These solutions are characterized by a Riemann-Hilbert problem…
We construct a new class of exact string solutions with a four dimensional target space metric of signature ($-,+,+,+$) by gauging the independent left and right nilpotent subgroups with `null' generators of WZNW models for rank 2…
A tau function of the 2D Toda hierarchy can be obtained from a generating function of the two-partition cubic Hodge integrals. The associated Lax operators turn out to satisfy an algebraic relation. This algebraic relation can be used to…
We consider solutions of a Toda system for SU(N+1) and show that any solution with finite exponential integral cam be obtained from a rational curve in complex projective space of dimension N
We consider a generalization of the full symmetric Toda hierarchy where the matrix $\tilde {L}$ of the Lax pair is given by $\tilde {L}=LS$, with a full symmetric matrix $L$ and a nondegenerate diagonal matrix $S$. The key feature of the…
We study the dispersionless version of the recently introduced constrained Toda hierarchy. Like the Toda lattice itself, it admits three equivalent formulations: the formulation in terms of Lax equations, the formulation of the…
We derive a constraint (string equation) which together with the Toda Lattice hierarchy determines completely the integrable structure of the compactified 2D string theory. The form of the constraint depends on a continuous parameter, the…
An iterative algorithm for determining a class of solutions of the dispersionful 2-Toda hierarchy characterized by string equations is developed. This class includes the solution which underlies the large N-limit of the Hermitian matrix…
We define bi-infinite versions of four well-studied discrete integrable models, namely the ultra-discrete KdV equation, the discrete KdV equation, the ultra-discrete Toda equation, and the discrete Toda equation. For each equation, we show…
A systematic construction of a class of integrable hierarchy is discussed in terms of the twisted affine $A_{2r}^{(2)}$ Lie algebra. The zero curvature representation of the time evolution equations are shown to be classified according to…
We show how the two--matrix model and Toda lattice hierarchy presented in a previous paper can be solved exactly: we obtain compact formulas for correlators of pure tachyonic states at every genus. We then extend the model to incorporate a…
We discuss two classes of exact (in $\a'$) string solutions described by conformal sigma models. They can be viewed as two possibilities of constructing a conformal model out of the non-conformal one based on the metric of a $D$-dimensional…
We find several classes of exact classical solutions of critical bosonic string theory, constructed as twisted products of one Euclidean and one Minkowskian 2D black hole coset. One class of these solutions leads (after tensoring with free…
We study elliptic families of solutions to the recently introduced constrained Toda hierarchy, i.e., solutions which are elliptic functions of some linear combination of the hierarchical times. Equations of motion for poles of such…