Related papers: KdV type hierarchies, the string equation and $W_{…

An action is constructed that gives an arbitrary equation in the KdV or MKdV hierarchies as equation of motion; the second Hamiltonian structure of the KdV equation and the Hamiltonian structure of the MKdV equation appear as Poisson…

It is well known that tau functions of the KP hierarchy satisfy addition formulas. We consider the general addition formula in the determinant form and take a certain limit of it. It expresses certain shifts of a tau function in terms of…

Starting from free charged fermions we give equivalent definitions of the $n\/$-component KP hierarchy, in terms of $\tau\/$-functions $\tau_\alpha\/$ (where $\alpha \in M =\/$ root lattice of $sl_n\/$), in terms of $n \times n\/$ matrix…

The structure of Hamiltonian reductions of the Wess-Zumino-Novikov-Witten (WZNW) theory by first class Kac-Moody constraints is analyzed in detail. Lie algebraic conditions are given for ensuring the presence of exact integrability,…

We give the formulation in extended superspace of an $N=2$ supersymmetric KP hierarchy using chirality preserving pseudo-differential operators. We obtain two quadratic hamiltonian structures, which lead to different reductions of the KP…

Near an arbitrary finite gap potential we construct real analytic, canonical coordinates for the KdV equation on the torus having the following two main properties: (1) up to a remainder term, which is smoothing to any given order, the…

For the mass critical generalized KdV equation $\partial_t u + \partial_x (\partial_x^2 u + u^5)=0$ on $\mathbb R$, we construct a full family of flattening solitary wave solutions. Let $Q$ be the unique even positive solution of…

We use the connection between the Frobrenius manifold and the Douglas string equation to further investigate Minimal Liouville gravity. We search a solution of the Douglas string equation and simultaneously a proper transformation from the…

We study $\hbar$ expansion of the KP hierarchy following Takasaki-Takebe arXiv:hep-th/9405096 considering several examples of matrix model $\tau$-functions with natural genus expansion. Among the examples there are solutions of KP equations…

Inspired by a recent work of Dubrovin [7], for each simple Lie algebra $\mathfrak{g}$, we introduce an infinite family of pairwise commuting ODEs and define their $\tau$-functions. We show that these $\tau$-functions can be identified with…

The exhaustive group classification of the class of KdV-like equations with time-dependent coefficients $u_t+uu_x+g(t)u_{xxx}+h(t)u=0$ is carried out using equivalence based approach. A simple way for the construction of exact solutions of…

In this thesis, we consider heterotic string vacua based on a warped product of a four-dimensional domain wall and a six-dimensional internal manifold preserving only two supercharges. Thus, they correspond to half-BPS states of heterotic…

We examine the conformal property of the second Hamiltonian structure of constrained KP hierarchy derived by Oevel and Strampp. We find that it naturallygives a family of nonlocal extended conformal algebras. We give two examples of such…

The elliptic Korteweg-de Vries (KdV) system is a multi-component generalization of the lattice potential KdV equation, whose soliton solutions are associated with an elliptic Cauchy kernel (i.e., a Cauchy kernel on the torus). In this paper…

We describe the Sato-Wilson type formulation of the KP hierarchy within the framework of closed string theory. A matrix generalization of this formalism is shown to allow natural interpretation of coincident D-branes as a sourse of…

The Lam\'e function can be used to construct plane wave factors and solutions to the Korteweg-de Vries (KdV) and Kadomtsev-Petviashvili (KP) hierarchy. The solutions are usually called elliptic solitons. In this chapter, first, we review…

Using relations from random matrix theory, we derive exact expressions for all $n$-point spectral correlation functions of Dirac operator eigenvalues in terms of finite-volume partition functions. This is done for both chiral symplectic and…

We sharpen the duality between open and closed topological string partition functions for topological gravity coupled to matter. The closed string partition function is a generalised Kontsevich matrix model in the large dimension limit. We…

The Toda lattice hierarchy is discussed in connection with the topological description of the $c=1$ string theory compactified at the self-dual radius. It is shown that when special constraints are imposed on the Toda hierarchy, it…

We derive an explicit generating function of correlations functions of an arbitrary tau-function of the KdV hierarchy. In particular applications, our formulation gives closed formul\ae\ of a new type for the generating series of…