Summary
Introduction
Sato formulation of the KP hierarchy
Line soliton solutions of KP-II
Gr (N, M)
Grassmannians and multisoliton KP-II solutions:
an algebro-geometric classification
SISSA - 2013
Simonetta Abenda
Dipartimento di Matematica e CIRAM, Università di Bologna
Trieste, 6-8/2/2013
Simonetta Abenda
Grassmannians and multisoliton KP-II solutions: an algebro-geo
Summary
Introduction
Sato formulation of the KP hierarchy
Line soliton solutions of KP-II
Gr (N, M)
1
Introduction
2
Sato formulation of the KP hierarchy
3
Line soliton solutions of KP-II
The heat hierarchy
(M − N, N)-soliton solutions
4
Gr (N, M)
Description of points in Gr0TNN (1, M) via divisors
Description of points in Gr0TNN (N, M) via systems of divisors
Simonetta Abenda
Grassmannians and multisoliton KP-II solutions: an algebro-geo
Summary
Introduction
Sato formulation of the KP hierarchy
Line soliton solutions of KP-II
Gr (N, M)
Introduction
KP − II equation : (−4ut + 6uux + uxxx )x + 3uyy = 0,
Real bounded regular (M − N, N) - line soliton solutions
u(x, y , t) = 2∂x2 log(τ (x, y , t)).
X
τ (x, y , t) = Wr(f1 , . . . , fN ) =
ξ(I)EI ,
I
• f1 , . . . , fN solutions to Heat hierarchy s.t. fi =
M
X
aij Ej
j=1
• ξ(I) ≥ 0 , ∀I = {1 ≤ i1 < i2 < · · · < iN }.
Simonetta Abenda
Grassmannians and multisoliton KP-II solutions: an algebro-geo
Summary
Introduction
Sato formulation of the KP hierarchy
Line soliton solutions of KP-II
Gr (N, M)
ξ(I) are the Plücker coordinates of the corresponding point in the
totally non-negative part of the Grassmannian Gr TNN (N, M).
Gr (N, M) is the set of the N-dimensional subspaces in RM .
The characterization of the cell decomposition of Gr TNN (N, M) is
a classical subject (e.g. classical problem of totally positive
matrices) and many results have been obtained by Fomin,
Zelevinski and collaborators in the framework of cluster algebras.
Postnikov [2006]: combinatorial classification of N × M totally
non-negative TNN matrices Gr TNN (N, M) via colored planar
directed networks and Le diagrams.
Simonetta Abenda
Grassmannians and multisoliton KP-II solutions: an algebro-geo
Summary
Introduction
Sato formulation of the KP hierarchy
Line soliton solutions of KP-II
Gr (N, M)
Kodama [2004-2010], Biondini, Chakravarthy [2006],
Chakravarthy-Kodama [2007-9], Kodama-Williams [2011]:
correspondence between asymptotic soliton lines and the
combinatorial Grassmannian structure of Gr TNN (N, M).
N = number of
asymptotic line - solitons (y → +∞),
N
…
y
…
M-N
(M − N) = number of
asymptotic line - solitons (y → −∞).
x
Simonetta Abenda
Grassmannians and multisoliton KP-II solutions: an algebro-geo
Summary
Introduction
Sato formulation of the KP hierarchy
Line soliton solutions of KP-II
Gr (N, M)
Dubrovin-Natanzon [1988] proved that real finite-gap solutions to
KP-II are associated to θ-functions on M-curves (=genus g
algebraic curves with maximal number g + 1 of real ovals).
The multi-soliton solutions which we consider here are associated
to rational curves and may be obtained (theoretically) as limits of
finite-gap solutions.
A-Grinevich: an alternative classification of N × M irreducible
TNN points in Gr0TNN (N, M) via systems of divisors.
Our main motivation in introducing this system of coordinates in
Gr TNN (N, M) is directly linked to the intent of giving an effective
characterization of such multi-soliton solutions as limits of real
finite-gap solutions on M-curves.
Simonetta Abenda
Grassmannians and multisoliton KP-II solutions: an algebro-geo
Summary
Introduction
Sato formulation of the KP hierarchy
Line soliton solutions of KP-II
Gr (N, M)
Example: Gr0TNN (1, 4)
Figura : τ (x, y , t) =
4
X
aj Ej (x, y , t),
ai =
j=1
k1
d1
k2
Γ =CP 1
g1
k3
P
(kj − d1 )(kj − d2 )
Q0
(kj − g1 ) l (kj − kl )
d2
Γ =CP 1
k
4
D
D
vanishing cycles
Simonetta Abenda
Grassmannians and multisoliton KP-II solutions: an algebro-geo
Summary
Introduction
Sato formulation of the KP hierarchy
Line soliton solutions of KP-II
Gr (N, M)
KP hierarchy
Lax operator L = ∂x + u2 ∂x−1 + u3 ∂x−2 + · · · ,
ui = ui (t),
t = (x, y , t, . . . ),
KP hierarchy :
Bn = (Ln )+
Lψ(t, k) = kψ(t, k),
∂tn ψ(t, k) = Bn ψ(t, k),
Gauge transformation : L
7→
n ≥ 1.
∂x = W −1 LW
Dressing operator : W = 1 − w1 ∂x−1 − w2 ∂x−2 − · · ·
then:
uKP = u2 = ∂x w1
Simonetta Abenda
Grassmannians and multisoliton KP-II solutions: an algebro-geo
Summary
Introduction
Sato formulation of the KP hierarchy
Line soliton solutions of KP-II
Gr (N, M)
The KP hierarchy is generated by the inverse gauge (dressing)
transformation L = W ∂x W −1 . The KP linear system
Lψ = kψ,
∂tn ψ = Bn ψ, n ≥ 1.
is obtained by the dressing action ψ = W ψ0 where the vacuum
eigenfunction ψ0 satisfies
∂x ψ0 = kψ0 ,
∂tn ψ0 = ∂xn ψ0 = k n ψ0 , n ≥ 1.
In the following we use the normalization
ψ0 (t, k) = e
θ(t,k)
,
θ(t, k) =
∞
X
k n tn .
n=1
Simonetta Abenda
Grassmannians and multisoliton KP-II solutions: an algebro-geo
Summary
Introduction
Sato formulation of the KP hierarchy
Line soliton solutions of KP-II
Gr (N, M)
The heat hierarchy
(M − N, N)-soliton solutions
The heat hierarchy
Suppose that f1 , . . . , fN satisfy the heat hierarchy
(t1 = x, t2 = y , t3 = t)
∂tn fi = ∂xn fi ,
n ≥ 1,
let
τ (x, y , t) = Wrx (f1 , . . . , fN ),
then
u(x, y , t) = 2∂x2 log (τ (x, y , t)),
is a solution to KP-II.
Simonetta Abenda
Grassmannians and multisoliton KP-II solutions: an algebro-geo
Summary
Introduction
Sato formulation of the KP hierarchy
Line soliton solutions of KP-II
Gr (N, M)
The heat hierarchy
(M − N, N)-soliton solutions
Here: restricted class of solutions to heat hierarchy
fi (t) =
M
X
aij Ej (t),
aij ∈ R ,
i = 1, . . . , N,
j=1
A = (aij ) has maximal rank N, is irreducible with all N × N minors
are non-negative once in RRE form.
X
Ej (t) ≡ ψ0 (t, kj ) = exp(θ(t, kj )),
θ(t, k) =
k n tn ,
n≥1
and
k1 < k2 < · · · < kM .
Then τ (x, y , t) = Wrx (f1 , . . . , fN ) has no zeros in the (x, y )-plane
for all t, and the corresponding KP–II solution
u(x, y , t) = 2∂x2 log(τ ) is non-singular.
Simonetta Abenda
Grassmannians and multisoliton KP-II solutions: an algebro-geo
Summary
Introduction
Sato formulation of the KP hierarchy
Line soliton solutions of KP-II
Gr (N, M)
The heat hierarchy
(M − N, N)-soliton solutions
Plücker coordinates and soliton solutions
{f1 , . . . , fN } solutions of the linear heat hierarchy of the form
fi =
M
X
aij Ej ,
i = 1, . . . , N.
j=1
X
τ = Wrx (f1 , . . . , fN ) =
ξ(i1 , . . . , iN )E (i1 , . . . , iN ),
1≤i1 <···<iN ≤M
then minors ξ(i1 , . . . , iN ) are Plücker coordinates of corresponding
point in Gr (N, M) and E (i1 , . . . , iN ) = Wrx (Ei1 , . . . , EiN ).
In this case the associated dressing operator takes the form
W =1−
N
X
wj ∂x−j ,
W ∂xN fi = 0,
i = 1, . . . , N.
j=1
Simonetta Abenda
Grassmannians and multisoliton KP-II solutions: an algebro-geo
Summary
Introduction
Sato formulation of the KP hierarchy
Line soliton solutions of KP-II
Gr (N, M)
Description of points in Gr0TNN (1, M) via divisors
Description of points in Gr0TNN (N, M) via systems of divisors
Gr (N, M)
Gr (N, M) is the set of N-dimensional subspaces of RM .
Example 1: Gr (1, 2) is the set of all lines passing through the
origin. Then clearly
Gr (1, 2) = {(1 : a), : a ∈ R} ∪ {(0 : 1)} ∼
= R ∪ {∞}, which is the
projective line RP 1 and can be identified with the circle S 1 .
S
Example 2: Gr (1, M) = RP M−1 = M
j=1 Fj , (Schubert
decomposition) where cell
Fj = {(0 : · · · : 0 : 1 : a1 : · · · : aM−j ) ; ak ∈ R}
Simonetta Abenda
Grassmannians and multisoliton KP-II solutions: an algebro-geo
Summary
Introduction
Sato formulation of the KP hierarchy
Line soliton solutions of KP-II
Gr (N, M)
Description of points in Gr0TNN (1, M) via divisors
Description of points in Gr0TNN (N, M) via systems of divisors
N × M matrices in RREF and decomposition of Gr (N, M)
Let A = (aij ) be a N × M matrix of maximal rank N. It may be
put in reduced row echelon form (RREF) with a distinguished set
of pivot columns I = {1 ≤ i1 < i2 < · · · < iN ≤ M} such that
ARRE (I) = IdN×N .
For any given A its uniquely defined RREF corresponds to a point
of Gr (N, M).
The set of matrices A whose RREF has the same pivot set I forms
a cell WI which provides the Schubert decomposition of the
Grassmannian:
[
Gr (N, M) =
WI .
I
dim(Gr (N, M)) = N(M − N)
Simonetta Abenda
Grassmannians and multisoliton KP-II solutions: an algebro-geo
Summary
Introduction
Sato formulation of the KP hierarchy
Line soliton solutions of KP-II
Gr (N, M)
Description of points in Gr0TNN (1, M) via divisors
Description of points in Gr0TNN (N, M) via systems of divisors
The totally non-negative (TNN) Grassmannian Gr TNN (N, M) is
the subset of Gr (N, M) that can be represented by (N × M)
matrices A in RREF whose maximal minors (i.e. Plücker
coordinates) are non-negative. The decomposition of
Gr TNN (N, M) by Postnikov is a refinement of Schubert cell
decomposition.
Simonetta Abenda
Grassmannians and multisoliton KP-II solutions: an algebro-geo
Summary
Introduction
Sato formulation of the KP hierarchy
Line soliton solutions of KP-II
Gr (N, M)
Description of points in Gr0TNN (1, M) via divisors
Description of points in Gr0TNN (N, M) via systems of divisors
Postnikov classification of Gr TNN (N, M)
Postnikov [2006] Gr TNN (N, M) is decomposed into TNN positroid
TNN such that the Plücker coordinates
cells SM
ξ(I) > 0,
if I ∈ M,
ξ(I) = 0,
if I 6∈ M,
where M is a matroid of rank N on the set [M] = {1, . . . , M}, i.e.
a non–empty collection of N-element subsets that satisfies the
exchange axiom: if I , J ∈ M and i ∈ I then there exists j ∈ J such
that (I \{i}) ∪ {j} ∈ M. Natural connection with asymptotic line
directions of u(x, y , t) if associate minimal lexicographic order.
Example: The totally positive part of Gr (N, M) is its unique
positroid cell of top dimension N(M − N) (totally positive matrices
in classical sense).
Simonetta Abenda
Grassmannians and multisoliton KP-II solutions: an algebro-geo
Summary
Introduction
Sato formulation of the KP hierarchy
Line soliton solutions of KP-II
Gr (N, M)
Description of points in Gr0TNN (1, M) via divisors
Description of points in Gr0TNN (N, M) via systems of divisors
The decomposition in positroid cells is a refinement of Schubert
cell decomposition.
Postnikov has labeled the cells in TNN Gr TNN (N, M) using
combinatiorial objects
Grassmann necklaces of type (N, M);
decorated permutations π : on M elements with N excedances;
equivalence classes of reduced plabic graphs of type (N, M)
Le diagrams of type (N, M).
Postnikov associates positive weights to the edges of the planar
graph. These positive weights are local coordinates on the
TNN from which the associated N × M matrix A is
positroid cell SM
directly reconstructed in RREF.
TNN and so to
His construction allows to control glueing of cells SM
obtain the partial ordering in the refined decomposition.
Simonetta Abenda
Grassmannians and multisoliton KP-II solutions: an algebro-geo
Summary
Introduction
Sato formulation of the KP hierarchy
Line soliton solutions of KP-II
Gr (N, M)
Description of points in Gr0TNN (1, M) via divisors
Description of points in Gr0TNN (N, M) via systems of divisors
In the KP setting matrix A in RREF is irreducible:
in each column there is a non zero element: if j-column is
(0, . . . , 0)T then the kj phase is missing, so the KP-solution
corresponds to a point in Gr (N, M − 1).
in each row there is a non zero element other then the pivot:
if i-th row is (0, . . . , 0, 1, 0 . . . , 0) then via a Darboux
transformation, the KP-solution corresponds to a point in
Gr (N − 1, M − 1).
Simonetta Abenda
Grassmannians and multisoliton KP-II solutions: an algebro-geo
Summary
Introduction
Sato formulation of the KP hierarchy
Line soliton solutions of KP-II
Gr (N, M)
Description of points in Gr0TNN (1, M) via divisors
Description of points in Gr0TNN (N, M) via systems of divisors
Kodama-Williams [2011]:
TNN which are
consider the subset of positroid cells SM
irreducible in Postnikov classification;
define a contour plot at fixed time Ct0 (u) (tropical curve in R2
if kj integer) which provides an approximation of wave crest
locus;
under genericity assumptions associate to soliton graph a
unique reduced plabic graph of Postnikov decomposition, i.e.
the matrix A in the totally positive part of the top cell in
Gr TNN (N, M).
Simonetta Abenda
Grassmannians and multisoliton KP-II solutions: an algebro-geo
Summary
Introduction
Sato formulation of the KP hierarchy
Line soliton solutions of KP-II
Gr (N, M)
Description of points in Gr0TNN (1, M) via divisors
Description of points in Gr0TNN (N, M) via systems of divisors
Description of points in Gr0TNN (N, M) in terms of divisors
A-Grinevich: associate to each point in Gr0TNN (N, M) (TNN and
irreducible part of Gr (N, M)) a system of divisors.
Model: Gr0TNN (1, M) = F1TNN = {(1 : a1 : · · · : aM−1 ), aj ∈ R+ } .
f = E1 + a1 E2 + · · · + aM−1 EM ,
Ej = ψ0 (t, kj )
w1
ψ = 1−
ψ0 ,
w1 = fx /f .
k
P
Idea: 0 = W1 f ≡ (∂x − w1 )f
⇐⇒
0= M
j=1 Resk=kj Φ(k, t).
Simonetta Abenda
Grassmannians and multisoliton KP-II solutions: an algebro-geo
Summary
Introduction
Sato formulation of the KP hierarchy
Line soliton solutions of KP-II
Gr (N, M)
Description of points in Gr0TNN (1, M) via divisors
Description of points in Gr0TNN (N, M) via systems of divisors
We introduce
χ1 (t)
ψD (k, t) = 1 +
ψ0 (k, t)
k − γ1
with natural normalization χ1 (0) = 0 and satisfying
(k − γ1 )ψD = W1 ψ0 = kψ.
Then γ1 = w1 (0) ∈]k1 , kM [ and generically γ1 ∈
due to irreducibility and positivity. We take
ψU (k, t) = c0 (t) +
SM−1
j=1
]kj , kj+1 [
c1 (t)
cM−2 (t)
+ ··· +
,
k − δ1
k − δM−2
with the compatible normalization c0 (0) = 1, cl (0) = 0,
l = 1, . . . , M − 2.
Simonetta Abenda
Grassmannians and multisoliton KP-II solutions: an algebro-geo
Summary
Introduction
Sato formulation of the KP hierarchy
Line soliton solutions of KP-II
Gr (N, M)
Description of points in Gr0TNN (1, M) via divisors
Description of points in Gr0TNN (N, M) via systems of divisors
ψD (kj , t) = ψU (kj , t),
j = 1, . . . , M.
(1)
Theorem a) If D = (γ1 , δ1 , . . . , δM−2 ) s.t. D∩]kj , kj+1 [6= ∅,
(j = 1, . . . , M − 1) then system (1) possesses a unique solution
(χ1 , c0 , . . . , cM−2 ). Moreover w1 = γ1 − χ1 is associated a real
and regular (M − 1, 1)-soliton solution to KP-II and
fD (t) =
M
X
Q
βj E j ,
βj =
j=1
− δl )
.
Q0
(kj − γ1 ) s (kj − ks )
l (kj
b) Viceversa, let w1 = fx /f and γ1 = w1 (0) 6= kj . Then (1) gives a
unique divisor D (1) = (δ1 , . . . , δM−2 ) and uniquely defined
functions cl , l = 0, . . . , M − 2. Moreover D = γ1 ∪ D (1) satisfies
the compatibility condition D∩]kj , kj+1 [6= ∅, (j = 1, . . . , k − 1) and
Wr (f , fD ) = 0.
Simonetta Abenda
Grassmannians and multisoliton KP-II solutions: an algebro-geo
Summary
Introduction
Sato formulation of the KP hierarchy
Line soliton solutions of KP-II
Gr (N, M)
Description of points in Gr0TNN (1, M) via divisors
Description of points in Gr0TNN (N, M) via systems of divisors
Example: Gr0TNN (1, 4)
k1
d1
Γ =CP
1
k2
g1
P
k3
d2
k
4
Γ =CP 1
D
D
vanishing cycles
Simonetta Abenda
Grassmannians and multisoliton KP-II solutions: an algebro-geo
Summary
Introduction
Sato formulation of the KP hierarchy
Line soliton solutions of KP-II
Gr (N, M)
Description of points in Gr0TNN (1, M) via divisors
Description of points in Gr0TNN (N, M) via systems of divisors
Example: the 1 soliton solution
1
1
u(x, y , t) = (k2 − k1 )2 sech2 ( (θ(k2 , t) − θ(k1 , t) + θ0 )).
2
2
Postnikov: A = [1, x + y ]
x
k1
1
1
k2
A-Grinevich:
k 1 − γ1
A = 1, −
k 2 − γ1
g1 = γ1
y
g1
Simonetta Abenda
Grassmannians and multisoliton KP-II solutions: an algebro-geo
Summary
Introduction
Sato formulation of the KP hierarchy
Line soliton solutions of KP-II
Gr (N, M)
Description of points in Gr0TNN (1, M) via divisors
Description of points in Gr0TNN (N, M) via systems of divisors
The general case Gr0TNN (N, M)
fi =
M
X
aij Ej ,
i = 1, . . . , N,
Ej = ψ0 (t, kj )
j=1
A = (aij ) TNN and irreducible N × M matrix in RREF.
w1
wN − · · · − N ψ0 ,
w1 = τx /τ, τ = Wr(f1 , . . . , fN ).
ψ = 1−
k
k
Idea: for i = 1, . . . , N
0=
(∂xN
−
w1 ∂xN−1 . . . wN )fi
⇐⇒
0=
M
X
Resk=kj Φ(i) (k, t).
j=1
We need a system of divisors! There is not a unique choice!
Simonetta Abenda
Grassmannians and multisoliton KP-II solutions: an algebro-geo
Summary
Introduction
Sato formulation of the KP hierarchy
Line soliton solutions of KP-II
Gr (N, M)
Description of points in Gr0TNN (1, M) via divisors
Description of points in Gr0TNN (N, M) via systems of divisors
Description of points in Gr0TNN (N, M) in terms of divisors
D = (γ1 , . . . , γN ) :


 γ1 + · · · + γN = w1 (0),
..
.


γ1 · · · γN = (−1)N wN (0),
Generically γ1 < γ2 < · · · < γN and γl ∈]kjl , kjl +1 [, l = 1, . . . , N.
(i)
(i)
D (i) = (δ1 , . . . , δM−2 ),
i = 1, . . . , N
For fixed D = (γ1 , . . . , γN ), N ≥ 2, there is NOT a unique choice
of the system of divisors D (i) , i = 1, . . . , N, since we have plenty of
choice to fix a basis of solutions f1 , . . . , fN to the heat hierarchy
corresponding to the same point in Gr0TNN (N, M). Anyway it is
always possible to fix UNIQUELY D (i) , i = 1, . . . , N, corresponding
to the matrix A in RRE form.
Simonetta Abenda
Grassmannians and multisoliton KP-II solutions: an algebro-geo
Summary
Introduction
Sato formulation of the KP hierarchy
Line soliton solutions of KP-II
Gr (N, M)
Description of points in Gr0TNN (1, M) via divisors
Description of points in Gr0TNN (N, M) via systems of divisors
Example: Top cell (maximal dimension): A in RREF

M−N−1
Q
(i)


(kj − δl )



l=1


, j = i or j = N + 1, . . . , M
 N
0
Q
Q
aij =
(kj − γr ) (kj − ks )


r =1
s6=j






0,
j = 1, . . . , i − 1, i + 1 . . . , N,
(i)
(i)
D (i) = (δ1 , . . . , δM−N−1 , k1 , . . . , ki−1 , ki+1 , . . . , kN )
i
k1
d1
(i)
k2
g1
(i)
kj
g N k J+1
(i)
D0 = (δ1 , . . . , δM−N−1 , γ1 , . . . , γN ),
(i)
D0 ∩]kj , kj+1 [ 6= ∅,
∀i = 1, . . . , N,
Simonetta Abenda
i
d M-N-1 k M
i = 1, . . . , N
j = 1, . . . , M − 1.
Grassmannians and multisoliton KP-II solutions: an algebro-geo
Summary
Introduction
Sato formulation of the KP hierarchy
Line soliton solutions of KP-II
Gr (N, M)
Description of points in Gr0TNN (1, M) via divisors
Description of points in Gr0TNN (N, M) via systems of divisors
Example: top cell in Gr0TNN (3, 6)
P
Simonetta Abenda
Grassmannians and multisoliton KP-II solutions: an algebro-geo
Summary
Introduction
Sato formulation of the KP hierarchy
Line soliton solutions of KP-II
Gr (N, M)
Description of points in Gr0TNN (1, M) via divisors
Description of points in Gr0TNN (N, M) via systems of divisors
We are able to give coordinates of points in an open dense
subset of any cell of Gr TNN via a canonical compatible system
(i)
(i)
of divisors D = (γ1 , . . . , γN ) and D (i) = (δ1 , . . . , δM−2 ),
i = 1, . . . , N;
We have explicit form of the representative matrix A = (aij ) in
function of the divisor;
We solve the inverse problem, i.e. given the Plücker
coordinates of a point in Gr0TNN (N, M), we reconstruct the
canonical system of divisors associated to it;
Our cell decomposition is compatible with the one by
Postnikov restricted to the irreducible component.
We are presently constructing the explicit correspondence
between the combinatorial description by Postnikov and our
description in function of systems of canonical divisors.
Simonetta Abenda
Grassmannians and multisoliton KP-II solutions: an algebro-geo