Grassmannians, Nonlinear Wave Equations and Generalized Schur Functions

We show how solutions to a large class of partial differential equations with nonlocal Riccati-type nonlinearities can be generated from the corresponding linearized equations, from arbitrary initial data. It is well known that evolutionary matrix Riccati equations can be generated by projecting linear evolutionary flows on a Stiefel manifold onto a coordinate chart of the underlying Grassmann manifold. Our method relies on extending this idea to the infinite dimensional case...

Given a point A in the real Grassmannian, it is well-known that one can construct a soliton solution u_A(x,y,t) to the KP equation. The contour plot of such a solution provides a tropical approximation to the solution when the variables x, y, and t are considered on a large scale and the time t is fixed. In this paper we use several decompositions of the Grassmannian in order to gain an understanding of the contour plots of the corresponding soliton solutions. First we use th...

The goal of this paper is to define the Grassmann integral in terms of a limit of a sum around a well-defined contour so that Grassmann numbers gain geometric meaning rather than symbols. The unusual rescaling properties of the integration of an exponential is due to the fact that the integral attains the known values only over a specific set of contours and not over their rescaled versions. Such contours live in infinite dimensional space and their sides are infinitesimal, a...

In this paper we give a geometric description in terms of the Grassmann manifold of Segal and Wilson, of the reduction of the KP hierarchy known as the vector $k$-constrained KP hierarchy. We also show in a geometric way that these hierarchies are equivalent to Krichever's general rational reductions of the KP hierarchy.

We review Aomoto's generalized hypergeometric functions on Grassmannian spaces Gr(k +1, n+1). Particularly, we clarify integral representations of the generalized hypergeometric functions in terms of twisted homology and cohomology. With an example of the Gr(2, 4) case, we consider in detail Gauss' original hypergeometric functions in Aomoto's framework. This leads us to present a new systematic description of Gauss' hypergeometric differential equation in a form of a first o...

In this paper, we introduce a family of symmetric polynomials by specializing the factorial Schur polynomials. These polynomials represent the weighted Schubert classes of the cohomology of the weighted Grassmannian introduced by Corti-Reid, and we regard these polynomials as analogue of the Schur polynomials. We show that those twisted Schur polynomials are the characters of certain representations. Thus we give an interpretation of the Schubert structure constants of the we...

We show that all (n-component) KP tau-functions, which are related to the twisted loop group of $GL_n$, give solutions of the Darboux-Egoroff system of PDE's. Using the Geometry of the Grassmannian we construct from the corresponding wave function the deformed flat coordinates of the Egoroff metric and from this the corresponding solution of the Witten-Dijkgraaf-E. Verlinde-H. Verlinde equations.

For a simple Lie algebra $\mathfrak{g}$ and an irreducible faithful representation $\pi$ of $\mathfrak{g}$, we introduce the Schur polynomials of $(\mathfrak{g},\pi)$-type. We then derive the Sato-Zhou type formula for tau functions of the Drinfeld-Sokolov (DS) hierarchy of $\mathfrak{g}$-type. Namely, we show that the tau functions are linear combinations of the Schur polynomials of $(\mathfrak{g},\pi)$-type with the coefficients being the Pl\"ucker coordinates. As an applic...

This paper is concerned with the construction of the polynomial tau-functions of the symplectic KP (SKP), orthogonal KP (OKP) hierarchies and universal character hierarchy of B-type (BUC hierarchy), which are proved as zero modes of certain combinations of the generating functions. By applying the strategy of carrying out the action of the quantum fields on vacuum vector, the generating functions for symplectic Schur function, orthogonal Schur function and generalized Q-funct...

The solitons solution of BKP equation can be constructed by the Pfaffian structure. Then one investigates the real line solitons structure of BKP equation using the totally non-negative Grassmannian. Especially, the N-soliton solution is studied and its self-dual Tau function is obtained. Also, one can construct the totally non-negative Grassmannian of the Sawada-Kotera equation for its real line solitons.