A generalized derivative nonlinear Schrödinger equation,is studied by means of Hirota's bilinear formalism. Soliton solutions are constructed as quotients of Wronski-type determinants. A relationship between the bilinear structure and gauge transformation is also discussed. *

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Abstract. We present a systematic construction of the discrete KP hierarchy in terms of Sato-Wilson-type shift operators. Reductions of the equations in this hierarchy to 1+1-dimensional integrable lattice systems are considered, and the problems that arise with regard to the symmetry algebra underlying the reduced systems as well as the ultradiscretizability of these systems are discussed. A scheme for constructing ultradiscretizable reductions that give rise to Yang-Baxter maps is explained in two explicit examples.2000 Mathematics Subject Classification. 39A10, 39A12. Introduction.A problem widely recognized as one of the most fundamental open problems in the theory of integrable systems is that of the relation between classical and quantum integrable systems with infinitely many degrees of freedom. Recently a possible route emerged through which a firm connection between the classical and quantum settings might be established: the ultradiscretization of (classical) discrete integrable systems [8,22]. Crystal bases, arising in the zero-temperature limit of quantum enveloping algebras, have been shown to play an important rôle in the description of the dynamical properties of so-called box and ball systems (BBSs) [20], which in turn can be obtained from 1+1-dimensional discrete integrable systems through a special limiting procedure: the ultradiscrete limit [22]. On the other hand, geometric crystals [1] are classical analogues of such crystals. These are such that (quantum) crystal bases can be obtained from them through a limiting procedure very much like the ultradiscrete limit. Geometric crystals have been studied extensively in connection with the combinatorial properties of BBSs [10, 11], but, most importantly, they also offer prime examples of Yang-Baxter (YB) maps [7], i.e. of set-theoretical solutions to the YB equation [6,26]. In particular, it can be shown that the R-matrix associated to such a YB map (or with the tropical version thereof, obtained from the geometric crystal) in the ultradiscrete limit turns into the combinatorial R-matrix that governs the time evolution of the BBS [8]. This R-matrix, in turn, is directly related to the crystal associated with that BBS. The question that remains however,

Orthogonal and symplectic matrix integrals are investigated. It is shown that the matrix integrals can be considered as a τ -function of the coupled KP hierarchy, whose solution can be expressed in terms of pfaffians.Over the past decade, the intimate relationships between matrix integrals and nonlinear integrable systems have been clarified, particularly in the context of string theory. In such cases, nonperturbative properties of physical quantities can be evaluated by the use of the integrable structures of the models. (For review, see refs. 1-4.)Here, we consider a matrix integral over an ensemble of Hermitian matriceswhere β = 1, 2 or 4, for which the ensemble is, respectively, orthogonal, unitary or symplectic and η(x, t) = ∞ m=1 x m t m . An interesting observation is that, when β = 2, the quantity Z (β=2) N {t} is a τ -function of the Kadomtsev-Petviashvili (KP) hierarchy [2,3,4,5]. In fact, the integral (1) can be identified as a continuum limit of the multisoliton solution of the hierarchy. This fact may be easily observed if we rewrite Z (β=2) N in the following form:It may be also clear that the quantity Z (β=2) N satisfies the bilinear Toda equationHere, we give other examples of such kinds of relationship. The orthogonal case β = 1 and the symplectic case β = 4 lead to τ -functions of the coupled KP hierarchy proposed

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.One dimensional quantum integrable models with long-range interaction have attracted much interest, because of not only their physical significance, but also their beautiful mathematical structure. One of such models is the Sutherland (trigonometric) model, which describes interacting particles on a circle [1]. The total momentum and Hamiltonian of the model are respectively given by

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The coupled KP hierarchy, introduced by Hirota and Ohta, are investigated by using the dressing method. It is shown that the coupled KP hierarchy can be reformulated as a reduced case of the 2-component KP hierarchy.

The hierarchy structure associated with a (2 + 1)-dimensional Nonlinear Schrödinger equation is discussed as an extension of the theory of the KP hierarchy. Several methods to construct special solutions are given. The relation between the hierarchy and a representation of toroidal Lie algebras are established by using the language of free fermions. A relation to the self-dual Yang-Mills equation is also discussed. 818, x = ix 1 , y = y 0 , t = −y 1 . In this sense, the evolution equations (2.5) and 822