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# KP solitons, total positivity, and cluster algebras

Edited by Percy A. Deift, New York University, New York, NY, and approved April 7, 2011 (received for review February 16, 2011)

## Abstract

Soliton solutions of the KP equation have been studied since 1970, when Kadomtsev and Petviashvili [Kadomtsev BB, Petviashvili VI (1970) *Sov Phys Dokl* 15:539–541] proposed a two-dimensional nonlinear dispersive wave equation now known as the KP equation. It is well-known that the Wronskian approach to the KP equation provides a method to construct soliton solutions. The *regular* soliton solutions that one obtains in this way come from points of the totally nonnegative part of the Grassmannian. In this paper we explain how the theory of total positivity and cluster algebras provides a framework for understanding these soliton solutions to the KP equation. We then use this framework to give an explicit construction of certain soliton contour graphs and solve the inverse problem for soliton solutions coming from the totally positive part of the Grassmannian.

The KP equation, introduced in 1970 (1), is considered to be a prototype of an integrable nonlinear dispersive wave equation with two spatial dimensions. Concretely, solutions to this equation provide a close approximation to the behavior of shallow water waves, such as beach waves. Given a point *A* in the real Grassmannian, one can construct a solution to the KP equation (2); this solution *u*_{A}(*x*,*y*,*t*) is written in terms of a *τ*-function, which is a sum of exponentials. More recently, several authors (3–6) have focused on understanding the *regular* soliton solutions that one obtains in this way: These come from points of the totally nonnegative part of the Grassmannian.

The classical theory of total positivity concerns square matrices in which all minors are positive. This theory was pioneered in the 1930s by Gantmacher, Krein, and Schoenberg (7, 8), and subsequently generalized in the 1990s by Lusztig (9, 10), who in particular introduced the totally positive and nonnegative parts of real partial flag varieties.

One of the most important partial flag varieties is the Grassmannian. Postnikov (11) investigated the totally nonnegative part of the Grassmannian (*Gr*_{kn})_{≥0}, which can be defined as the subset of the real Grassmannian where all Plücker coordinates are nonnegative. Specifying which minors are strictly positive and which are zero gives a decomposition into *positroid cells*. Postnikov introduced a variety of combinatorial objects, including *decorated permutations*, *Le diagrams*, *plabic graphs*, and *Grassmann necklaces*, in order to index the cells and describe their properties.

In this paper we develop a tight connection between the theory of total positivity for the Grassmannian and the behavior of the corresponding soliton solutions to the KP equation. To understand a soliton solution *u*_{A}(*x*,*y*,*t*), one fixes the time *t* and plots the points where *u*_{A}(*x*,*y*,*t*) has a local maximum. This gives rise to a *tropical curve* in the *xy* plane; concretely, this shows the positions in the plane where the corresponding wave has a peak. The *decorated permutation* indexing the cell containing *A* determines the asymptotic behavior of the soliton solution at *y* → ± ∞. When *t* is sufficiently small, we can predict the combinatorial structure of this tropical curve using the *Le diagram* indexing the cell containing *A*. When *A* comes from a totally positive Schubert cell, we show that generically this tropical curve is a realization of one of Postnikov’s *reduced plabic graphs*. Furthermore, if we label each region of the complement of the tropical curve with the dominant exponential in the *τ*-function, then the labels of the unbounded regions form the *Grassmann necklace* indexing the cell containing *A*. Finally, when *A* belongs to the totally positive Grassmannian, we show that the dominant exponentials labeling regions of the tropical curve form a *cluster* for the cluster algebra of the Grassmannian. Letting *t* vary, one may observe *cluster transformations*.

These previously undescribed connections between KP solitons, cluster algebras, and total positivity promise to be very powerful. For example, using some machinery from total positivity and cluster algebras, we solve the *inverse problem* for soliton solutions from the totally positive Grassmannian.

## Total Positivity for the Grassmannian

The real Grassmannian *Gr*_{kn} is the space of all *k*-dimensional subspaces of . An element of *Gr*_{kn} can be represented by a full-rank *k* × *n* matrix modulo left multiplication by nonsingular *k* × *k* matrices.

Let be the set of *k*-element subsets of [*n*]≔{1,…,*n*}. For , let Δ_{I}(*A*) denote the maximal minor of a *k* × *n* matrix *A* located in the column set *I*. The map *A*↦(Δ_{I}(*A*)), where *I* ranges over , induces the *Plücker embedding* , and the Δ_{I}(*A*) are called *Plücker coordinates*.

The totally nonnegative Grassmannian (*Gr*_{kn})_{≥0} (respectively, totally positive Grassmannian (*Gr*_{kn})_{>0}) is the subset of *Gr*_{kn} that can be represented by *k* × *n* matrices *A* with all Δ_{I}(*A*) nonnegative (respectively, positive).

Postnikov (11) gave a decomposition of (*Gr*_{kn})_{≥0} into *positroid cells*. For , the *positroid cell* is the set of elements of (*Gr*_{kn})_{≥0} represented by all *k* × *n* matrices *A* with the Δ_{I}(*A*) > 0 for and Δ_{J}(*A*) = 0 for .

Clearly (*Gr*_{kn})_{≥0} is a disjoint union of the positroid cells —in fact, it is a CW complex (12). Note that (*Gr*_{kn})_{>0} is a positroid cell; it is the unique positroid cell in (*Gr*_{kn})_{≥0} of top dimension *k*(*n* - *k*). Postnikov showed that the cells of (*Gr*_{kn})_{≥0} are naturally labeled by (and in bijection with) the following combinatorial objects (11):

Grassmann necklaces of type (

*k*,*n*)decorated permutations

*π*^{:}on*n*letters with*k*weak excedancesequivalence classes of

*reduced plabic graphs*of type (*k*,*n*)Le diagrams of type (

*k*,*n*).

For the purpose of studying solitons, we are interested only in the subset of positroid cells that are *irreducible*.

We say that a positroid cell is irreducible if the reduced-row echelon matrix *A* of any point in the cell has the following properties:

Each column of

*A*contains at least one nonzero element.Each row of

*A*contains at least one nonzero element in addition to the pivot.

The irreducible positroid cells are indexed by

irreducible Grassmann necklaces of type (

*k*,*n*)derangements

*π*on*n*letters with*k*excedancesequivalence classes of

*irreducible reduced plabic graphs*of type (*k*,*n*)irreducible Le diagrams of type (

*k*,*n*).

We now review the definitions of these objects and some of the bijections among them.

An irreducible Grassmann necklace of type (*k*,*n*) is a sequence of subsets *I*_{r} of [*n*] of size *k* such that, for *i*∈[*n*], *I*_{i+1} = (*I*_{i}∖{*i*})∪{*j*} for some *j* ≠ *i*. (Here indices *i* are taken modulo *n*.)

An example of a Grassmann necklace of type (4, 9) is (1257, 2357, 3457, 4567, 5678, 6789, 1789, 1289, 1259).

A derangement *π* = (*π*_{1},…,*π*_{n}) is a permutation *π*∈*S*_{n} that has no fixed points. An excedance of *π* is a pair (*i*,*π*_{i}) such that *π*_{i} > *i*. We call *i* the excedance position and *π*_{i} the excedance value. Similarly, a nonexcedance is a pair (*i*,*π*_{i}) such that *π*_{i} < *i*.

A plabic graph is a planar undirected graph *G* drawn inside a disk with *n* boundary vertices 1,…,*n* placed in counterclockwise order around the boundary of the disk, such that each boundary vertex *i* is incident to a single edge.* Each internal vertex is colored black or white.

Let *Y*_{λ} denote the Young diagram of the partition *λ*. A Le diagram *L* = (*λ*,*D*)_{k,n} of type (*k*,*n*) is a Young diagram *Y*_{λ} contained in a *k* × (*n* - *k*) rectangle together with a filling *D*: *Y*_{λ} → {0,+} that has the Le property: There is no 0 that has a + above it in the same column and a + to its left in the same row. A Le diagram is irreducible if each row and each column contains at least one +.

See Fig. 1 for an example of an irreducible Le diagram.

(11, Theorem 17.2) Let be a positroid cell in (*Gr*_{kn})_{≥0}. For 1 ≤ *r* ≤ *n*, let *I*_{r} be the element of , which is lexicographically minimal with respect to the order *r* < *r* + 1 < ⋯ < *n* < 1 < 2 < …*r* - 1. Then is a Grassmann necklace of type (*k*,*n*).

(11, Lemma 16.2) Given an irreducible Grassmann necklace , define a derangement by requiring that if *I*_{i+1} = (*I*_{i}∖{*i*})∪{*j*} for *j* ≠ *i*, then *π*(*j*) = *i*.^{†} Indices are taken modulo *n*. Then is a bijection from irreducible Grassmann necklaces of type (*k*,*n*) to derangements with *k* excedances. The excedances of are in positions *I*_{1}.

If the positroid cell is indexed by the Grassmann necklace , the derangement *π*, and the Le diagram *L*, then we also refer to this cell as , , and . The bijections above preserve the indexing of cells, that is, .

## Soliton Solutions to the KP Equation

Here we explain how to obtain a soliton solution to the KP equation from a point of (*Gr*_{kn})_{≥0}.

### From the Grassmannian to the *τ*-Function.

We start by fixing real parameters *κ*_{j} such that *κ*_{1} < *κ*_{2}⋯ < *κ*_{n}, which are generic, in the sense that the sums *κ*_{jm} are all distinct for 2 ≤ *d* ≤ *k*.

Let {*E*_{j}; *j* = 1,…,*n*} be a set of exponential functions in defined by If denotes , then the Wronskian determinant with respect to *x* of *E*_{1},…,*E*_{n} is defined by

Let *A* be a full-rank *k* × *n* matrix. We define a set of functions {*f*_{1},…,*f*_{k}} by where (…)^{T} denotes the transpose of the vector (…). The *τ*-function of *A* is defined by [1]It is easy to verify that *τ*_{A} depends only on which point of (*Gr*_{kn})_{≥0} the matrix *A* represents.

Applying the Binet–Cauchy identity to the fact that for *i* = 1,…,*k*, we get [2]where *E*_{I}(*x*,*y*,*t*) with *I* = {*j*_{1},…,*j*_{k}} is defined by Therefore if *A*∈(*Gr*_{kn})_{≥0}, then *τ*_{A} > 0 for all .

Thinking of *τ*_{A} as a function of *A*, we note from Eq. **2** that the *τ*-function encodes the information of the Plücker embedding. More specifically, if we identify each function *E*_{I} with *I* = {*j*_{1},…,*j*_{k}} with the wedge product *E*_{j1}∧⋯∧*E*_{jk}, then the map , *A*↦*τ*_{A} has the Plücker coordinates as coefficients.

### From the *τ*-Function to Solutions of the KP Equation.

The KP equation was proposed by Kadomtsev and Petviashvili in 1970 (1), in order to study the stability of the one-soliton solution of the Korteweg–de Vries (KdV) equation under the influence of weak transverse perturbations. The KP equation also gives an excellent model to describe shallow water waves (13).

It is well-known (see, e.g., ref. 14) that the *τ*-function defined in Eq. **1** provides a soliton solution of the KP equation, [3]Note that if *A*∈(*Gr*_{kn})_{≥0}, then *u*_{A}(*x*,*y*,*t*) is regular.

## From Soliton Solutions to Soliton Graphs

One can visualize such a solution *u*_{A}(*x*,*y*,*t*) in the *xy* plane by drawing level sets of the solution for each time *t*. For each , we denote the corresponding level set by Fig. 2 depicts both a three-dimensional image of a solution *u*_{A}(*x*,*y*,*t*), as well as multiple level sets *C*_{r}(0). Note that these level sets are lines parallel to the line of the wave peak.

To study the behavior of *u*_{A}(*x*,*y*,*t*) for , we set where . From Eq. **2**, we see that, generically, *τ*_{A} can be approximated by .

Let *f*_{A}(*x*,*y*,*t*) be the closely related function [4]Clearly a given term dominates *f*_{A}(*x*,*y*,*t*) if and only if its exponentiated version dominates .

Given a solution *u*_{A}(*x*,*y*,*t*) of the KP equation as in Eq. **3**, we define its contour plot for each *t* = *t*_{0} to be the locus in where *f*_{A}(*x*,*y*,*t* = *t*_{0}) is not linear.

provides an approximation of the location of the wave crests.

It follows from Definition 7 that is a one-dimensional piecewise linear subset of the *xy* plane.

If each *κ*_{i} is an integer, then is a tropical curve in .

Note that each region of the complement of in is a domain of linearity for *f*_{A}(*x*,*y*,*t*_{0}), and hence each region is naturally associated to a *dominant exponential* Δ_{J}(*A*)*E*_{J}(*x*,*y*,*t*_{0}) from the *τ*-function Eq. **2**. We call the line segments comprising *line solitons*.^{‡} Some of these line solitons have finite length, whereas others are unbounded and extend in the *y* direction to ± ∞. We call these *unbounded* line solitons. Note that each line soliton represents a balance between two dominant exponentials in the *τ*-function.

(6, Proposition 5) The dominant exponentials of the *τ*-function in adjacent regions of the contour plot in the *xy* plane are of the form *E*(*i*,*m*_{2},…,*m*_{k}) and *E*(*j*,*m*_{2},…,*m*_{k}).

According to Lemma 2, those two exponential terms have *k* - 1 common phases, so we call the soliton separating them a *line soliton of type* [*i*,*j*]. Locally we have with , so the equation for this line soliton is [5]Note that the ratio of the Plücker coordinates labeling the regions separated by the line soliton determines the location of the line soliton.

Consider a line soliton given by Eq. **5**. Compute the angle Ψ_{[i,j]} between the line soliton and the positive *y* axis, measured in the counterclockwise direction, so that the negative *x* axis has an angle of and the positive *x* axis has an angle of . Then tan Ψ_{[i,j]} = *κ*_{i} + *κ*_{j}. Therefore we refer to *κ*_{i} + *κ*_{j} as the slope of the [*i*,*j*] line soliton (see Fig. 2).

We will be interested in the combinatorial structure of a contour plot, that is, the pattern of how line solitons interact with each other. To this end, in Definition 9 we will associate a *soliton graph* to each contour plot.

Generically we expect a point of a contour plot at which several line solitons meet to have degree 3; we regard such a point as a trivalent vertex. Three line solitons meeting at a trivalent vertex exhibit a *resonant interaction* (this corresponds to the *balancing condition* for a tropical curve). One may also have two line solitons that cross over each other, forming an *X* shape: We call this an *X* crossing, but do not regard it as a vertex. In general, there exists a phase shift at each *X* crossing. However, we ignore them in this paper as explained in footnote ‡. Vertices of degree greater than 4 are also possible.

A contour plot is called generic if all interactions of line solitons are at trivalent vertices or are *X* crossings.

The following definition of *soliton graph* forgets the metric data of the contour plot, but preserves the data of how line solitons interact and which exponentials are dominant.

Let be a generic contour plot with *n* unbounded line solitons. Color a trivalent vertex black (respectively, white) if it has a unique edge extending downward (respectively, upward) from it. Label each region with the dominant exponential *E*_{I} and each edge (line soliton) by the type [*i*,*j*] of that line soliton. Preserve the topology of the metric graph, but forget the metric structure. Embed the resulting graph with bicolored vertices and *X* crossings into a disk with *n* boundary vertices,replacing each unbounded line soliton with an edge that ends at a boundary vertex. We call this labeled graph a soliton graph.

See Fig. 3 for an example of a soliton graph. Although we have not labeled all regions or all edges, the remaining labels can be determined using Lemma 2.

## Permutations and Soliton Asymptotics

Given a contour plot , where *A* belongs to an irreducible positroid cell and *t*_{0} is arbitrary, we show that the labels of the unbounded solitons allow us to determine which positroid cell *A* belongs to. Conversely, given *A* in the irreducible positroid cell , we can predict the asymptotic behavior of the unbounded solitons in .

Suppose *A* is an element of an irreducible positroid cell in (*Gr*_{kn})_{≥0}. Consider the contour plot for any time *t*_{0}. Then there are *k* unbounded line solitons at *y*≫0, which are labeled by pairs [*e*_{r},*j*_{r}] with *e*_{r} < *j*_{r}, and there are *n* - *k* unbounded line solitons at *y* ≪ 0, which are labeled by pairs [*i*_{r},*g*_{r}] with *i*_{r} < *g*_{r}. We obtain a derangement in *S*_{n} with *k* excedances by setting *π*(*e*_{r}) = *j*_{r} and *π*(*g*_{r}) = *i*_{r}. Moreover, *A* must be an element of the cell .

The first part of this theorem follows from work of Chakravarty and Kodama (ref. 5, Prop. 2.6 and 2.9 and ref. 6, Theorem 5). Our contribution is that the derangement *π* is precisely the derangement labeling the cell that *A* belongs to. This fact is the first step toward establishing that various other combinatorial objects in bijection with positroid cells (Grassmann necklaces and plabic graphs) carry useful information about the corresponding soliton solutions.

We now give a concrete algorithm for writing down the asymptotics of the soliton solutions of the KP equation.

Fix generic parameters *κ*_{1} < … < *κ*_{n}. Let *A* be an element from an irreducible positroid cell in (*Gr*_{kn})_{≥0}. (So *π* must have *k* excedances.) For any *t*_{0}, the asymptotic behavior of the contour plot —i.e., its unbounded line solitons, and the dominant exponentials in its unbounded regions—can be read off from *π* as follows.

For

*y*≫0, there is an unbounded line soliton of type [*i*,*π*(*i*)] for each excedance*π*(*i*) >*i*. From left to right, list these solitons in decreasing order of the quantity*κ*_{i}+*κ*_{π(i)}.For

*y*≪ 0, there is an unbounded line soliton of type [*π*(*j*),*j*] for each nonexcedance*π*(*j*) <*j*. From left to right, list these solitons in increasing order of*κ*_{j}+*κ*_{π(j)}.Label the unbounded region for

*x*≪ 0 with the exponential*E*_{i1,…,ik}, where*i*_{1},…,*i*_{k}are the excedance positions of*π*.Use Lemma 2 to label the remaining unbounded regions of the contour plot.

Consider the positroid cell corresponding to *π* = (6,7,1,2,8,3,9,4,5)∈*S*_{9}. The algorithm of Theorem 5 gives rise to the picture in Fig. 4. If one reads the dominant exponentials in counterclockwise order, starting from the region at the left, then one recovers the Grassmann necklace from Example 1. Also note that . See Theorem 7.

## Grassmann Necklaces and Soliton Asymptotics

One particularly nice class of positroid cells is the *TP* or *totally positive Schubert cells*. These are the positroid cells indexed by Le diagrams, which are filled with all +’s, or equivalently, the positroid cells indexed by derangements *π* such that *π*^{-1} has at most one descent. When is a TP Schubert cell, we can make a link between the corresponding soliton solutions of the KP equation and Grassmann necklaces.

Let *A* be an element of a TP Schubert cell , and consider the contour plot for an arbitrary time *t*_{0}. Let the index sets of the dominant exponentials of the unbounded regions of be denoted *R*_{1},…,*R*_{n}, where *R*_{1} labels the region at *x* ≪ 0, and *R*_{2},…,*R*_{n} label the regions in the counterclockwise direction from *R*_{1}. Then (*R*_{1},…,*R*_{n}) is a Grassmann necklace and .

Theorem 7 is illustrated in Example 6.

Theorem 7 does not hold if we replace “TP Schubert cell” by “positroid cell.”

## From Soliton Graphs to Generalized Plabic Graphs

In this section we associate a *generalized plabic graph* *Pl*(*C*) to each soliton graph *C*. We then show that from *Pl*(*C*)—whose only labels are on the boundary vertices—we can recover the labels of the line solitons and dominant exponentials of *C*.

A generalized plabic graph is a connected graph embedded in a disk with *n* boundary vertices labeled 1,…,*n* placed in any order around the boundary of the disk, such that each boundary vertex *i* is incident to a single edge. Each internal vertex must have degree at least two and is colored black or white. Edges are allowed to form *X* crossings (this is not considered to be a vertex).

We now generalize the notion of trip from ref. 11, Section 13.

Given a generalized plabic graph *G*, the trip *T*_{i} is the directed path that starts at the boundary vertex *i* and follows the “rules of the road”: It turns right at a black vertex, left at a white vertex, and goes straight through an *X* crossing. Note that *T*_{i} will also end at a boundary vertex. The trip permutation *π*_{G} is the permutation such that *π*_{G}(*i*) = *j* whenever the trip starting at *i* ends at *j*.

We use these trips to associate a canonical labeling of edges and regions to each generalized plabic graph.

Given a generalized plabic graph *G* with *n* boundary vertices, start at each boundary vertex *i* and label every edge along trip *T*_{i} with *i*. Such a trip divides the disk containing *G* into two parts: the part to the left of *T*_{i} and the part to the right. Place an *i* in every region that is to the left of *T*_{i}. After repeating this procedure for each boundary vertex, each edge will be labeled by up to two numbers (between 1 and *n*), and each region will be labeled by a collection of numbers. Two regions separated by an edge labeled *ij* will have region labels *S* and (*S*∖{*i*})∪{*j*}. When an edge is assigned two numbers *i* < *j*, we write [*i*,*j*] on that edge, or {*i*,*j*} or {*j*,*i*} if we do not wish to specify the order of *i* and *j*.

Fix an irreducible cell of (*Gr*_{kn})_{≥0}. To each soliton graph *C* coming from a point of that cell we associate a generalized plabic graph *Pl*(*C*) by

labeling the boundary vertex incident to the edge {

*i*,*π*_{i}} by*π*_{i}=*π*(*i*),forgetting the labels of all edges and regions.

See Fig. 5 for the generalized plabic graph *Pl*(*C*) corresponding to the soliton graph *C* from Fig. 3.

Fix an irreducible cell of (*Gr*_{kn})_{≥0}, and consider a soliton graph *C* coming from a point of that cell. Then the trip permutation associated to the plabic graph *Pl*(*C*) is *π*, and by labeling edges and regions of *Pl*(*C*) according to Definition 12, we will recover the original labels in *C*.

We invite the reader to apply Definition 12 to Fig. 5, and then compare the result to Fig. 3.

By Theorem 8, we can identify each soliton graph *C* with its generalized plabic graph *Pl*(*C*).

## Soliton Graphs for Positroid Cells When *t* ≪ 0

In this section we give an algorithm for producing a generalized plabic graph *G*_{-}(*L*) from the Le diagram *L* of a positroid cell . It turns out that this generalized plabic graph gives rise to the soliton graph for a generic point of the cell , at time *t* ≪ 0 sufficiently small.

Given a Le diagram *L*, construct *G*_{-}(*L*) as follows:

Start with a Le diagram

*L*contained in a*k*× (*n*-*k*) rectangle. Label its southeast border by the numbers 1 to*n*, starting from the northeast corner. Replace 0’s and +’s by “crosses” and “elbows.” From each label*i*on the southeast border, follow the associated “pipe” northwest, and label its destination by*i*as well.Add an edge, and one white and one black vertex to each elbow, as shown in the upper right of Fig. 6. Forget the labels of the southeast border. If there is an endpoint of a pipe on the east or south border whose pipe starts by going straight, then erase the straight portion preceding the first elbow.

Forget any degree 2 vertices, and forget any edges of the graph that end at the southeast border of the diagram. Denote the resulting graph

*G*_{-}(*L*).After embedding the graph in a disk with

*n*boundary vertices, we obtain a generalized plabic graph, which we also denote*G*_{-}(*L*). If desired, stretch and rotate*G*_{-}(*L*) so that the boundary vertices at the west side of the diagram are at the north instead.

Fig. 6 illustrates the steps of Algorithm 9. Note that this produces the graph from Fig. 5.

Let *L* be a Le diagram and *π* = *π*(*L*). Then *G*_{-}(*L*) has trip permutation *π*. Label its edges and regions according to the rules of the road. When is a TP Schubert cell, then *G*_{-}(*L*) coincides with the soliton graph *G*_{t}(*u*_{A}), provided that and *t* ≪ 0 sufficiently small. When is an arbitrary positroid cell, we can realize *G*_{-}(*L*) as “most” of a soliton graph *G*_{t}(*u*_{A}) for and *t* ≪ 0. Moreover, we can construct *G*_{t}(*u*_{A}) from *G*_{-}(*L*) by extending the unbounded edges of *G*_{-}(*L*) and introducing *X* crossings as necessary so as to satisfy the conditions of Theorem 5.

## Reduced Plabic Graphs and Cluster Algebras

The most important plabic graphs are those that are *reduced* (11, Section 12). Although it is not easy to characterize reduced plabic graphs (they are defined to be plabic graphs whose *move-equivalence class* contains no graph to which one can apply a *reduction*), they are important because of their application to cluster algebras and parameterizations of cells.

Let *A* be a point of a TP Schubert cell, let *t*_{0} be an arbitrary time, and suppose that the contour plot is generic and has no *X* crossings. Then the soliton graph associated to is a reduced plabic graph.

Cluster algebras are a class of commutative rings with a remarkable combinatorial structure, which were defined by Fomin and Zelevinsky (15). Scott (16) proved that Grassmannians have a cluster algebra structure.

(16) The coordinate ring of the (affine cone over the) Grassmannian has the structure of a cluster algebra. Moreover, the set of labels of the regions of any reduced plabic graph for the TP Grassmannian comprises a cluster for this cluster algebra.

Scott’s strategy in ref. 16 was to show that certain labelings of alternating strand diagrams for the TP Grassmannian gave rise to clusters. However, alternating strand diagrams are in bijection with reduced plabic graphs (11), and under this bijection, Scott’s labelings of alternating strand diagrams correspond to the labelings of regions of plabic graphs induced by the various trips in the plabic graph.

The set of Plücker coordinates labeling regions of a generic soliton graph with no X vertices for the TP Grassmannian is a cluster for the cluster algebra associated to the Grassmannian.

Conjecturally, every positroid cell of the totally nonnegative Grassmannian also carries a cluster algebra structure, and the Plücker coordinates labeling the regions of any reduced plabic graph for should be a cluster for that cluster algebra. In particular, the TP Schubert cells should carry cluster algebra structures. Therefore we conjecture that Corollary 13 holds with “TP Schubert cell” replacing “TP Grassmannian.” Finally, there should be a suitable generalization of Corollary 13 for arbitrary positroid cells.

## The Inverse Problem

The *inverse problem* for soliton solutions of the KP equation is the following: Given a time *t* together with the contour plot of a soliton solution, can one reconstruct the point of (*Gr*_{k,n})_{≥0} that gave rise to the solution?

Fix *κ*_{1} < … < *κ*_{n} as usual. Consider a generic contour plot of a soliton solution coming from a point *A* of a positroid cell , for *t* ≪ 0. Then from the contour plot together with *t* we can uniquely reconstruct the point *A*.

The strategy of the proof is as follows: From the contour plot together with *t*, we can reconstruct the value of each of the dominant exponentials (Plücker coordinates) labeling regions of the graph. We have shown how to use the Le diagram to construct the soliton graph for a positroid cell when *t* ≪ 0 is sufficiently small, which allows us to identify what is the set of Plücker coordinates that label regions of the graph. We then show that this collection of Plücker coordinates contains a subset of Plücker coordinates, which Talaska (17) showed were sufficient for reconstructing the original point of .

Using Theorem 14, Corollary 13, and the cluster algebra structure for Grassmannians, we can prove the following:

Consider a generic contour plot of a soliton solution coming from a point *A* of the TP Grassmannian, at an arbitrary time *t*. If the contour plot has no X crossings, then from the contour plot together with *t* we can uniquely reconstruct the point *A*.

## Triangulations of a Polygon and Soliton Graphs

We now explain how to use triangulations of an *n*-gon to produce all soliton graphs for the TP Grassmannian (*Gr*_{2,n})_{>0}.

Let *T* be a triangulation of an *n*-gon *P*, whose *n* vertices are labeled by the numbers 1,2,…,*n*, in counterclockwise order. Therefore each edge of *P* and each diagonal of *T* is specified by a pair of distinct integers between 1 and *n*. The following procedure yields a labeled graph Ψ(*T*). See Fig. 7.

Put a black vertex in the interior of each triangle in

*T*.Put a white vertex at each of the

*n*vertices of*P*that is incident to a diagonal of*T*; put a black vertex at the remaining vertices of*P*.Connect each vertex which is inside a triangle of

*T*to the three vertices of that triangle.Erase the edges of

*T*, and contract every pair of adjacent vertices that have the same color. This produces a new graph*G*with*n*boundary vertices, in bijection with the vertices of the original*n*-gon*P*.Add one unbounded ray to each of the boundary vertices of

*G*, so as to produce a new (planar) graph Ψ(*T*). Note that Ψ(*T*) divides the plane into regions; the bounded regions correspond to the diagonals of*T*, and the unbounded regions correspond to the edges of*P*.

The graphs Ψ(*T*) constructed above are soliton graphs for (*Gr*_{2,n})_{>0}, and conversely, any generic soliton graph with no X crossings for (*Gr*_{2,n})_{>0} comes from this construction.

Flipping a diagonal in a triangulation corresponds to a mutation in the cluster algebra. In our setting, each mutation may be considered as an evolution along a flow of the KP hierarchy defined by the symmetries of the KP equation.

## Acknowledgments

The first author was partially supported by National Science Foundation (NSF) Grant DMS-0806219, and the second author was partially supported by NSF Grant DMS-0854432 and a Sloan fellowship.

## Footnotes

- ↵
^{1}To whom correspondence should be addressed. E-mail: williams{at}math.berkeley.edu.

Author contributions: Y.K. and L.K.W. designed research; Y.K. and L.K.W. performed research; and Y.K. and L.K.W. wrote the paper.

The authors declare no conflict of interest.

This article is a PNAS Direct Submission.

↵

^{*}The convention of ref. 11 was to place the boundary vertices in clockwise order.↵

^{†}Actually Postnikov’s convention was to set*π*(*i*) =*j*above, so the permutation we are associating is the inverse one to his.↵

^{‡}In general, there exist phase shifts that also appear as line segments (see ref. 6). However the phase shifts depend only on the*κ*parameters, and we ignore them in this paper.

## References

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- Kadomtsev BB,
- Petviashvili VI

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- Sato M

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- Gantmacher F,
- Krein M

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- Schoenberg I

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- Lehrer GI

- Lusztig G

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- Postnikov A

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- Hirota R

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