On KP generators and the geometry of the HBDE

Abstract

Sato theory provides a correspondence between solutions to the KP hierarchy and points in an infinite dimensional Grassmannian. In this correspondence, flows generated infinitesimally by powers of the “shift” operator give time dependence to the first coordinate of an arbitrarily selected point, making it a tau-function. These tau-functions satisfy a number of integrable equations, including the Hirota bilinear difference equation (HBDE). Here, we rederive the HBDE as a statement about linear maps between Grassmannians. In addition to illustrating the fundamental nature of this equation in the standard theory, we make use of this geometric interpretation of the HBDE to answer the question of what other infinitesimal generators could be used for similarly creating tau-functions. The answer to this question involves a “rank one condition”, tying this investigation to the existing results on integrable systems involving such conditions and providing an interpretation for their significance in terms of the relationship between the HBDE and the geometry of Grassmannians.

Introduction

It was the seminal work of Sato [29] which related the geometry of the Grassmannian to the solution of soliton equations. That relationship is analogous to the relationship of the functions sine and cosine and the geometry of the unit circle in the plane. These trigonometric functions, of course, arise as the dependence of the x- and y-coordinates on the time parameter of a uniform flow around the circle. In the case of Sato theory, it is the tau-functions of the KP hierarchy which arise as the dependence of the “first” Plücker coordinate upon the time variables $\text{t}=(t_1\text{,}{t}_2\text{,}{t}_3\text{,}\dots )$, where the flow corresponding to the variable $t_i$ is generated infinitesimally by the operator which takes the basis element $e_j$ of the underlying vector space to $e_{j+i}$[29], [30], [34]. The remainder of this introduction will briefly review this construction and motivate the following question: What other choice of infinitesimal generator could have been made that similarly generate KP tau-functions? In other words, we are looking for other flows, in both finite and infinite dimensional Grassmannians, which have this property of creating tau-functions through the projection onto the first coordinate.

Our approach to this question will be algebro-geometric in nature, rather than analytic. In Section 2, we will reinterpret the Hirota bilinear difference equation (HBDE), which characterizes KP tau-functions, as a linear map between Grassmann cones with certain geometric properties. It will be precisely the existence of such a map that characterizes the alternate KP generators.

The main result appears in Section 3, where we identify those operators S that can serve as generators of the KP flow in a Grassmannian. As it turns out, this property is characterized only by a restriction on the rank of one block of the operator. This result is applied and discussed in Sections 4 and 5, with special emphasis on its relationship to the rank one conditions that have appeared elsewhere in the literature on integrable systems.

The KP hierarchy is usually considered as an infinite set of compatible dynamical systems on the space of monic pseudo-differential operators of order one. A solution of the KP hierarchy is any pseudo-differential operator of the form

$$L=\partial +w_1(\text{t}){\partial }^{-1}+w_2(\text{t}){\partial }^{-2}+\cdots \text{,}\text{t}=(t_1\text{,}{t}_2\text{,}{t}_3\text{,}\dots )\text{,}$$

satisfying the evolution equations

$$\frac{\partial }{\partial t_i}L=[L\text{,}{(L^i)}_{+}]\text{,}i=1\text{,}2\text{,}3\text{,}\dots \text{,}$$

where the “+” subscript indicates projection onto the differential operators by simply eliminating all negative powers of $\partial$, and $[A\text{,}B]=A\circ B-B\circ A$.

Remarkably, there exists a convenient way to encode all information about the KP solution $L$ in a single function $\tau (\text{t})$ satisfying certain bilinear differential equations. Specifically, each of the coefficients $w_i$ of $L$ can be written as a certain rational function of $\tau (t_1\text{,}{t}_2\text{,}\dots )$ and its derivatives [30]. Alternatively, one can construct $L$ from $\tau$ by letting W be the pseudo-differential operator

$$W=\frac{1}{\tau }\tau \left(t_1-{\partial }^{-1}\text{,}{t}_2-\frac{1}{2}{\partial }^{-2}\text{,}\dots \right)\text{,}$$

and then $L≔W\circ \partial \circ W^{-1}$ is a solution to the KP hierarchy [2]. Every solution to the KP hierarchy can be written this way in terms of a tau-function, though the choice of tau-function is not unique. For example, note that one may always multiply W on the right by any constant coefficient series $1+\text{O}({\partial }^{-1})$ without affecting the corresponding solution.

If $L$ is a solution to the KP hierarchy, then the function

$$u(x\text{,}y\text{,}t)=-2\frac{\partial }{\partial x}{w}_1(x\text{,}y\text{,}t\text{,}\dots )=2\frac{{\partial }^2}{\partial x^2}\log \tau$$

is a solution of the KP equation which is used to model ocean waves. Moreover, many of the other equations that show up as particular reductions of the KP hierarchy have also been previously studied as physically relevant wave equations. The KP hierarchy also arises in theories of quantum gravity [21], the probability distributions of the eigenvalues of random matrices [3], [32], and has applications to questions of classical differential geometry [7].

Certainly one of the most significant observations regarding these equations, which is a consequence of the form (2), is that all of these equations are completely integrable. Among the many ways to solve the equations of the KP hierarchy are several with connections to the algebraic geometry of “spectral curves” [4], [13], [22], [25], [24], [31], [30]. However, more relevant to the subject of this note is the observation of M. Sato that the geometry of an infinite dimensional Grassmannian underlies the solutions to the KP hierarchy [29].

Let k and n be two positive integers with $k<n$. For later convenience, we will choose a non-standard notation for the basis of $C^n$, denoting it by

$$C^n=〈e_{k-n}\text{,}{e}_{k-n+1}\text{,}\dots \text{,}{e}_{-1}\text{,}{e}_0\text{,}{e}_1\text{,}\dots \text{,}{e}_{k-1}〉\text{.}$$

Then, for instance, an arbitrary element of “wedge space” ${\bigwedge }^k{C}^n$ can be written in the form

$$\omega =\sum _{I\in I_{k\text{,}n}}{\pi }_I{e}_I\text{,}$$

where ${\pi }_I\in C$ are coefficients, $I_{k\text{,}n}$ denotes the set

$$I_{k\text{,}n}=\{I=(i_0\text{,}{i}_1\text{,}\dots \text{,}{i}_{k-1})|k-n\le i_0<i_1<i_2<\cdots <i_{k-1}\le k-1\}\text{,}$$

and $e_I=e_{i_0}\wedge e_{i_1}\wedge \cdots \wedge e_{i_{k-1}}$.

A linear operator $M:C^n\to C^n$ naturally extends to an operator $\hat{M}:{\bigwedge }^k\to {\bigwedge }^k$, where we consider the action to be applied to each term of the wedge product

$$\hat{M}{e}_I=M(e_{i_1})\wedge M(e_{i_2})\wedge \cdots \wedge M(e_{i_k})\text{,}$$

and extend it linearly across sums.

We denote by ${\Gamma }^{k\text{,}n}\subset {\bigwedge }^k{C}^n$ the set of decomposable k-wedges in the exterior algebra of $C^n$

$${\Gamma }^{k\text{,}n}=\{v_1\wedge v_2\wedge \cdots \wedge v_k|v_i\in C^n\}\text{.}$$

This Grassmann cone is in fact an affine variety in the $\left(\genfrac{}{}{0ex}{}{n}{k}\right)$-dimensional vector space ${\bigwedge }^k{C}^n$ because $\omega$ is in ${\Gamma }^{k\text{,}n}$ if and only if the coefficients ${\pi }_I$ satisfy a collection of quadratic polynomial relations known as the Plücker relations [12]. Specifically, we consider the coefficients ${\pi }_I$ to be skew-symmetric in the ordering of their subindices and select any two sets I and J of integers between $k-n$ and n of cardinality $k-1$ and $k+1$, respectively

$$k-n\le i_1<i_2<\cdots <i_{k-1}\le n\text{,}$$

$$k-n\le j_1<j_2<\cdots <\le j_{k+1}\le n\text{.}$$

It follows that $\omega$ is decomposable if and only if

$$\sum _{l=1}^{l+1}{(-1)}^l{\pi }_{i_1\text{,}{i}_2\text{,}\dots \text{,}{i}_{k-1}\text{,}{j}_l}{\pi }_{j_1\text{,}{j}_2\text{,}\dots \text{,}{j}_{l-1}\text{,}{j}_{l+1}\text{,}\dots \text{,}{j}_{k+1}}=0$$

for all such selections of subsets I and J.

In general, therefore, the Grassmann cone ${\Gamma }^{k\text{,}n}$ is defined by a collection of quadratic equations involving up to $k+1$ terms. In the special case $k=2$ and $n=4$, only a single three-term relation is required. Specifically, $\omega \in {\bigwedge }^2{C}^4$ is decomposable if and only if the coefficients satisfy the equation

$${\pi }_{-2\text{,}-1}{\pi }_{0\text{,}1}-{\pi }_{-2\text{,}0}{\pi }_{-1\text{,}1}+{\pi }_{-2\text{,}1}{\pi }_{-1\text{,}0}=0\text{.}$$

Later we will demonstrate a method through which the one relation (4) is sufficient to characterize the general case (cf. Section 4.5).

It is natural to associate a k-dimensional subspace $W_{\omega }\subset C^n$ to a non-zero element $\omega \in {\Gamma }^{k\text{,}n}$. If $\omega =v_1\wedge \cdots \wedge v_k$, then the $v_i$ are linearly independent and we associate to $\omega$ the subspace $W_{\omega }$ which they span. In fact, since $W_{\omega }=W_{{\omega }^{\prime }}$ if $\omega$ and ${\omega }^{\prime }$ are scalar multiples, it is more common to consider the Grassmannian $Gr(k\text{,}n)=P{\Gamma }^{k\text{,}n}$ as a projective variety whose points are in one-to-one correspondence with k-dimensional subspaces. This association of points in $P{\Gamma }^{k\text{,}n}$ to k-dimensional subspaces is the Plücker embedding of the Grassmannian in projective space. However, due to our interest in linear maps between these spaces – and our desire to avoid having to deal with the complications of viewing them as rational maps between the corresponding projective spaces – we choose to work with the cones instead.

Next, we briefly introduce the infinite dimensional Grassmannian of Sato theory and the notation which will be most useful in proving our main results. Additional information can be found by consulting [14], [20], [29], [30].

We formally consider the infinite dimensional Hilbert space $H$ over $C$ with basis $\{e_i|i\in Z\}$. It has the decomposition

$$H=H_{-}\oplus H_{+}\text{,}$$

where $H_{-}$ is spanned by $\{e_i|i<0\}$ and $H_{+}$ has the basis $\{e_i|i\ge 0\}$.

The wedge space $\bigwedge$ has the basis $e_I=e_{i_0}\wedge e_{i_1}\wedge \cdots$, where the (now infinite) multi-index $I=(i_0\text{,}{i}_1\text{,}{i}_2\text{,}\dots )$ is selected from the set $I$ whose elements are characterized by the properties $i_j<i_{j+1}$ and $i_j=j$ for j sufficiently large. (In other words, $I\in I$ can be constructed from the “ground state” $I_0=(0\text{,}1\text{,}2\text{,}3\text{,}4\text{,}\dots )$ by selecting a finite number of its elements and replacing them with distinct, negative integers.) A general element of $\bigwedge$ then is of the form

$$\omega =\sum _{I\in I}{\pi }_I{e}_I\text{.}$$

Since the multi-indices are of this form, it is notationally convenient to write only the first m elements of an element of $I\in I$ if it is true that $i_j=j$ for all $j\ge m$. For instance, we utilize the abbreviations

$${\pi }_{-2\text{,}-1}={\pi }_{-2\text{,}-1\text{,}2\text{,}3\text{,}4\text{,}5\text{,}\dots }\text{and}{e}_{-2\text{,}0\text{,}1}=e_{-2}\wedge e_0\wedge e_1\wedge e_3\wedge e_4\wedge \cdots \text{,}$$

and $e_{0\text{,}1}=e_0\wedge e_1\wedge e_2\wedge e_3\wedge \cdots$. Moreover, using this same abbreviation we are able to view the finite set $I_{k\text{,}n}$ introduced earlier as being a subset of the infinite $I$

$$I_{k\text{,}n}=\{I\in I:-k<i_0\text{,}{i}_j=j\text{for}j>n-1\}\text{.}$$

In this way, arbitrary finite dimensional Grassmann cones can be seen as being embedded in the infinite dimensional one in the form of points with only finitely many non-zero Plücker coordinates. Consequently, although we may not always emphasize this fact, the results we determine for $\bigwedge$ can all be stated in the finite dimensional case as well through this correspondence.

The Sato Grassmann cone $\Gamma \subset \bigwedge$ is precisely the set of those elements which can be written as

$$\omega =v_1\wedge v_2\wedge v_3\wedge \cdots \text{,}{v}_i\in H\text{.}$$

It can also be characterized by Plücker relations since $\omega \in \Gamma$ if and only if for every choice of k and n, the $\left(\genfrac{}{}{0ex}{}{n}{k}\right)$ Plücker coordinates ${\pi }_I$ for $I\in I_{k\text{,}n}$ satisfy the relations (3) for ${\Gamma }^{k\text{,}n}$. In order that the operations we are to utilize be well defined, we make the assumption that if $\omega$ is represented in this form, the vectors $\{v_i\}$ are chosen so that $v_i=e_i+{\sum }_{j=i+1}^{\infty }{c}_j{e}_j$ for i chosen to be sufficiently large.

As in the finite dimensional case, the Grassmannian $Gr=P\Gamma$ has an interpretation of being the set of subspaces of H meeting certain criteria. However, rather than being identified by their dimension, one can say that they are the subspaces for which the kernel and co-kernel of a certain projection map are finite dimensional and for which the index of that map is zero [29], [30]. Again, the subspace corresponding to $v_0\wedge v_1\wedge v_2\wedge \cdots \in \Gamma$ is the subspace spanned by the basis $\{v_i\}$.

Note. Those uncomfortable with the formal approach to this infinite dimensional object may choose to assume further restrictions on these definitions as specified in [30], where an analytic approach is used to ensure that all objects are well defined and that all infinite sums converge. Alternatively, one may consider the case that ${\pi }_I=0$ for $I⁄\in I_{k\text{,}n}$ in which case this reduces to the finite dimensional situation in which there are no questions of convergence.

The linear “shift” operator $\text{S}:H\to H$ is defined by the property that $\text{S}{e}_i=e_{i+1}$. (Written as a matrix, it would have ones on the sub-diagonal and zeros everywhere else.) The linear map

$$E(\text{t})=\exp \sum _{i=1}^{\infty }{t}_i{\text{S}}^i:H\to H\text{,}$$

induces a map $Ê(\text{t})$ on $\bigwedge$ for any fixed values of the parameters $\text{t}=(t_1\text{,}{t}_2\text{,}\dots )$. We use $Ê(\text{t})$ to introduce “time dependence” to each point $\omega \in \bigwedge$

$$\tilde{\omega }(t)=Ê(\text{t})\omega =\sum _{I\in I}{\tilde{\pi }}_I(\text{t})e_I\text{.}$$

The main object of Sato’s theory [29] is the function ${\tau }_{\omega }(\text{t})$ associated to any point $\omega \in \bigwedge$ and is defined as the first Plücker coordinate of the time-dependent point $\tilde{\omega }(\text{t})$ (cf. (7))

$${\tau }_{\omega }(\text{t})={\tilde{\pi }}_{0\text{,}1}(\text{t})\text{.}$$

There is very little that one can say about ${\tau }_{\omega }(\text{t})$ in general. In fact, since it can also be described as an infinite sum of Schur polynomials with the original coefficients ${\pi }_I$ of $\omega$ as coefficients [29], [30], one can select $\omega \in \bigwedge$ so that ${\tau }_{\omega }(\text{t})$ is any formal series in the variables $t_i$.

The main result of Sato theory is that ${\tau }_{\omega }(\text{t})$ is a KP tau-function precisely when $\omega \in \Gamma$. In fact, a function $\tau (\text{t})$ is a tau-function of the KP Hierarchy if and only if $\tau (\text{t})={\tau }_{\omega }(\text{t})$ for some $\omega \in \Gamma$[29].

Note. By virtue of the fact that we have chosen to work with Grassmann cones rather than projective Grassmannians, our correspondence between points and tau-functions necessarily involves the constant function ${\tau }_0(\text{t})\equiv 0$. The usual definition of “KP tau-function” specifically excludes this function, but here we will adopt the convention of referring to this function as a KP tau-function even though it does not correspond in the usual way to a Lax operator $L$.

The main question which we seek to address in this paper is the following: With what operator could you replace $\text{S}$ in (6) so that ${\tau }_{\omega }$(8) would still be a tau-function for any $\omega \in \Gamma$?

There is a sense in which this question seems uninteresting. After all, since Sato theory characterizes the totality of solutions of the KP hierarchy using only the shift operator $\text{S}$, it may not be clear why one would be interested in other choices. We therefore motivate the question with the following list:

• It is only by answering the question posed that we can recognize which of the many properties that characterize the operator $\text{S}$ are responsible for its role in generating KP tau-functions. For instance, it has the properties that it is a strictly lower triangular operator with respect to the basis $\{e_i\}$. Additionally, it has the property that for $v\in H_{-}$, $\text{S}v\in H_{-}\oplus C{e}_0$. It is not at first clear which, if any, of these properties is related to its role in generating KP flows.

• Although all solutions of the KP hierarchy can be generated using the operator $\text{S}$ and some point $\omega \in \Gamma$ through Sato’s construction, it is possible that solutions which are difficult to write or compute explicitly in that format can be derived in a simpler way using an alternative choice of generator for the flows. For instance, the simplest points in $\Gamma$ are those having only finitely many non-zero Plücker coordinates. (Equivalently, one may consider the case in which a finite dimensional Grassmannian is used in place of the infinite dimensional Sato Grassmannian.) Using powers of the shift operator $\text{S}$ to generate the KP flows, these correspond to tau-functions which are polynomials, depending only on a finite number of the variables $\{t_i\}$[30]. However, as we will show, using an alternative generator one gets a wider variety of interesting KP tau-functions using flows on finite dimensional Grassma- nnians.

• Finally, the answer to the question posed might provide an understanding of other phenomena in integrable systems which were not previously considered in the context of choice of KP generator in the Grassmannian at all. In particular, we suggestively point out that “rank one conditions” (the requirement that a certain matrix have rank of at most one) have arisen in the study of both finite and infinite dimensional integrable systems in a number of apparently unrelated contexts. We will argue that these are related and actually represent an unrecognized instance of the sort of alternative KP generator we investigate here.