tropical limit of solitons, yang-baxter maps and beyond · most frequently used notion of...

Geometry&Integrability Bidifferential calculus bDT Tropical limit KdV YB maps Boussinesq KP Conclusions

Tropical limit of solitons,

Yang-Baxter maps and beyond

Folkert Muller-HoissenMPI for Dynamics and Self-Organization, Gottingen, Germany

based on joint work with

Aristophanes DimakisUniversity of the Aegean, Greece

Local and Nonlocal Geometry of PDEs and Integrability, Trieste, 11 October 2018

on the occasion of Joseph Krasil’shchik’s 70th anniversary


Prelude: Geometry and Integrability

This section explains in particular how we construct exact solutionswhich are then explored in a “tropical limit”.It also bridges to a main topic of this meeting.


A geometric notion of integrability• Most frequently used notion of integrability of a PDDE:

The PDDE arises as compatibility condition of a set of linearequations (Lax pair).

• For some PDEs in two dimensions: compatibility conditioncan be expressed as vanishing curvature of a connection.But this differential-geometric framework is too narrow. Forexample, we also have to treat discrete equations.

• There is a generalization to a framework of noncommutativegeometry, which still keeps the simple computational rules.On an algebraic level, this means dealing with

associative algebra→ diff. calculus→ connection→ curvature

• Integrability:

Equation (e.g., PDDE)⇐⇒ zero curvature

Here the connection depends on constituents of the equation(dependent and possibly independent variables of a PDDE).


Zero curvature condition on associative algebras

A unital associative algebra, over a field K(Ω, δ) a differential calculus over A

Here Ω =⊕

k≥0 Ωk is a graded algebra with Ω0 = A and

A-bimodules Ωk , and δ is a derivation of degree one.

A connection on a left A-module Γ is a linear map∇ : Γ→ Ω1 ⊗A Γ, such that ∇(f γ) = δf ⊗A γ + f∇γ, for allf ∈ A and γ ∈ Γ. It extends to Ω⊗A Γ, hence we can define thecurvature as ∇2. The essence of integrability of some equationmay then be expressed as

∇2 = 0 ⇐⇒ equation


Simplifying assumption: Γ has a basis bµ, µ = 1, . . . ,m.Using the summation convention, we have γ = γµ b

µ and

∇γ = (δγµ − γν Aνµ)⊗A bν ∇2γ = γµ Fδ[A]µν ⊗A bν

Fδ[A]µν = δAµν − AµκAκν

where A is the m ×m matrix of elements of Ω1 defined by∇bµ = A

µν ⊗A bν .

However, in most cases the above characterization of integrabilityis not strong enough. Rather, one needs a connection that dependson a (“spectral”) parameter and the stronger condition that thecurvature vanishes for all of its values.

Bidifferential calculus (Dimakis & M-H 2000) is the special casewhere ∇ is linear in such a parameter. In particular, one meets thissituation in case of the (anti-) self-dual Yang-Mills equation.


Bidifferential calculus

δ = d + ν d , A = A + ν B .

The zero curvature condition (∀ν) is then equivalent to

Fd[A] = 0 = Fd[B] dA + dB + AB + B A = 0

Setting B = 0 (typically one of the two “gauge potentials” A andB can be transformed to zero by a gauge transformation), this is

dA = 0 Fd[A] = 0

The first equation can be solved by setting A = dφ, withφ ∈ Mat(m,m,A), the algebra of m ×m matrices over A. Then

d dφ+ dφ dφ = 0

Alternatively, we can solve Fd[A] = 0 by setting A = (dg) g−1 withan invertible g ∈ Mat(m,m,A). Then dA = 0 results in

d [(dg) g−1] = 0

Many integrable equations are realizations of one of these eqs.


Remark: Frolicher-Nijenhuis theoryLet Ω be the algebra of differential forms on a manifold M and dthe exterior derivative. A Nijenhuis tensor is a tensor field N oftype (1,1) (or a fiber preserving endomorphism of TM) on M,with vanishing Nijenhuis torsion

T (N)(X ,Y ) := [NX ,NY ]− N([NX ,Y ] + [X ,NY ]− N[X ,Y ]) ,

where X ,Y are any vector fields on M. Then

d = iN d

extends the above to a bidifferential calculus. Here we look at Nas a vector-valued 1-form, and iN acts via contraction of a vectorfield and a 1-form. According to Frolicher-Nijenhuis theory, anyderivation d of degree one, which anti-commutes with d andsatisfies d2 = 0, is of the form dN .There are applications to integrable systems (bi-Hamiltonianstructures). Magri: construction of “Lenard chain” of conservedquantities. Works more generally in bidifferential calculus.


IntegrabilityLet ∆ ∈ Mat(n, n,A) and α ∈ Mat(n, n,Ω1) satisfy

d∆ + [α,∆] = (d∆) ∆ dα + α2 = (dα) ∆

The linear system

dθ = A θ + (dθ) ∆ + θ α

for θ ∈ Mat(m, n,A), then has the integrability condition

Fd[A] θ − (dA) θ∆ = 0

If ∆ is such that this implies separate vanishing of bothsummands, dA = 0 = Fd[A] is integrable in the sense that itarises as the integrability condition of a linear system. Also:

dη = −η A + Γ dη + β η =⇒ dA = 0 = Fd[A]

for η ∈ Mat(n,m,A), if Γ ∈ Mat(n, n,A) and β ∈ Mat(n, n,Ω1)satisfy

dΓ− [β, Γ] = ΓdΓ dβ − β2 = Γdβ


A binary Darboux transformationLet ∆, λ, Γ, κ satisfy the respective equations and φ0 be a solutionof

d dφ+ dφ dφ = 0

Furthermore, let θ, η be solutions of the linear systems

dθ = (dφ) θ + (dθ) ∆ + θ α dη = −η dφ+ Γ dη + β η

Let Ω be a solution of the (compatible) linear equations

Γ Ω− Ω ∆ = η θ

dΩ = (dΩ) ∆− (dΓ) Ω + β Ω + Ωα + (dη) θ

Thenφ = φ0 − θΩ−1 η

is a new solution of d dφ+ dφ dφ = 0.

There is an analogous result for d [(dg) g−1] = 0.


Generating solutions for matrix KdV, KP, ...

Let u = 2φx (m × n) be the dependent variable.

seed solution φ0

linear system adjoint linear system

for θ (m × N) for χ (N × n)

compatible linear systemfor “Darboux potential” Ω

↓new solution

φ = φ0 − θΩ−1 χ

For soliton solutions of matrix KdV: φ0 = 0.y θ = θ0 e

ϑ(P), χ = eϑ(P) χ0, with ϑ(P) = x P + t P3.P is a constant N × N matrix.Similar, but more involved, for Boussinesq, KP, etc.


Tropical limit of solitons

and “particle” interactions


Main thoughts

• “Tropical limit” of a soliton solution in two space-timedimensions: wave crest limit.

• It associates with the wave solution a classical point particlepicture.

• In case of matrix KdV, e.g., matrix data are related by aYang-Baxter map along the tropical limit graph.

• In 2d integrable QFT models, YB equation expressesfactorization of the multi-particle scattering matrix: the latterdecomposes into 2-particle interactions. Similarly, we think ofa multi-soliton solution also as being composed of 2-solitoninteractions. However, because of the wave nature of solitons,there are no definite events at which the interaction takesplace. But the tropical limit takes waves to “point particles”and then indeed determines events at which an interactionoccurs. Does this mean that YB holds?


“Tropical limit” of KP soliton solutions

Writing u = 2 (log τ)xx ,

the function τ of a soliton solution of the scalar KP hierarchy, orsome of its reductions, has the form

τ =∑I

eΘI with ΘI linear in x , y , t

Then (Maslov dequantization)

limε→0

ε log∑I

eΘI /ε = maxΘI

defines the tropical limit of log τ .In the region, where ΘI > ΘJ for all J 6= I , we have u = 0 (sinceΘI is linear in x). Thus, the tropical limit of solitons has supporton the boundaries of dominating phase regions.


Remark

“Tropicalization”:

+ 7→ ⊕ := max

· 7→ := +

Characteristic of tropical geometry: piecewise linear.


KdV: 2-soliton solutionKorteweg-deVries (KdV) equa-tion:

4 ut − uxxx − 3 (u2)x = 0

The black line segments in thefigure constitute the

tropical limit graphin the xt-plane, of a 2-soliton solu-tion. It is the support of the vari-able u in the tropical limit.

t↑→ x

This looks like a space-time scattering diagram for two pointparticles (exchanging a “virtual” particle).The tropical limit gives a precise meaning to what contour plotshave shown us long ago (“resonance”).


Matrix KdV

4 ut − uxxx − 3 (u K u)x = 0

where K is a constant n ×m matrix and u an m × n matrix.

The tropical limit of a soliton solution, at fixed time, now consistsof a graph with matrix data attached to its edges.

What are the rules according to which such data are distributedover the graph ?


Tropical limit of matrix KdV solitons

We have

φ =F

τwhere τ = det Ω

and F is a matrix. Now we define the tropical limit graph usingthe function τ , which is not in general a “tau function” of thescalar KdV equation.Observation: support of matrix solution generically coincides withsupport of τ .


Yang-Baxter maps

Let X be a set. A map

R : X × X −→ X × X

is called a Yang-Baxter map if it satisfies the (quantum)Yang-Baxter equation (Yang 1967, Baxter 1972)

R12 R13 R23 = R23 R13 R12

Both sides act on X × X × X . The indices of Rij specify on whichfactors the map R acts. In general, R will depend on parametersthat change with these indices.

Solutions of the Yang-Baxter equation play a crucial role in2-dimensional quantum integrable models and exactly solvablemodels of statistical mechanics. Here mostly:R : V ⊗ V −→ V ⊗ V , or R : V ⊕ V −→ V ⊕ V , V vector space.


A simple example: particles in one dimension

Classical Mechanics. Scattering of two particles on the real line,with masses m1,m2, incoming velocities v1, v2, outgoing velocitiesv ′1, v

′2. They are subject to

• momentum conservation: m1 v1 + m2 v2 = m1 v′1 + m2 v

′2

• energy conservation: 12m1 v

21 + 1

2m2 v22 = 1

2m1 v′1

2 + 12m2 v

′2

2

Solving for the outgoing velocities:

(v ′1, v′2) = (v1, v2)R(m1,m2)

with

R(m1,m2) =

(m1−m2m1+m2

2m1m1+m2

2m2m1+m2

m2−m1m1+m2

)

(see, e.g., T.E. Kouloukas 2017)x

t

This matrix is an R-matrix, corresponding to a linear map solutionof the Yang-Baxter equation. We’ll meet it again later on.


Yang-Baxter relation in 3-particle scattering, schematically:

t↑→ x

=


Matrix KdV Yang-Baxter mapNormalized (tr(Ku) = 1) values of u along boundary segments:u1,in, u2,in and u1,out, u2,out for N = 2. Writing

uj ,in =ξj ,in ⊗ ηj ,inηj ,inKξj ,in

, uj ,out =ξj ,out ⊗ ηj ,outηj ,outKξj ,out

determines ξ1,in/out and η1,in/out up to scalings.

ξ1,out = α−1/2in

(1m −

2p2

p2 − p1

ξ2,in ⊗ η2,in

η2,inKξ2,inK)ξ1,in

ξ2,out = α−1/2in

(1m −

2p1

p1 − p2

ξ1,in ⊗ η1,in

η1,inKξ1,inK)ξ2,in

η1,out = α−1/2in η1,in

(1n −

2p2

p2 − p1Kξ2,in ⊗ η2,in

η2,inKξ2,in

)η2,out = α

−1/2in η2,in

(1n −

2p1

p1 − p2Kξ1,in ⊗ η1,in

η1,inKξ1,in

)with αin = 1 + 4p1p2

(p1−p2)2

η1,inKξ2,in η2,inKξ1,in

η1,inKξ1,in η2,inKξ2,in.

If K = I : A. Veselov and V.M. Goncharenko (2003, 2004).


The vector KdV R-matrix

In this case (n = 1) we have a linear map

(u1,out, u2,out) = (u1,in, u2,in)R(p1, p2)

R(pi , pj) =

( pi−pjpi+pj

2pipi+pj

2pjpi+pj

pj−pipi+pj

)

Setting pi = mi (masses), this is exactly the R-matrix governingthe scattering of classical point particles in one dimension !But the physical picture is different.


Why YB holds

t↑→ x

111

211

221

222

122

112

121

1 2 3

3

1 2

2 1

1 3

12 3

2 3

3 21

-15 -10 -5 0 5 10

0

5

10

15

20

25

111

211

221

222

122112

212

12 3

3 21

3 1

3

2 1

2 3

1 3

1 2

2

-10 -5 0 5 10 15

-25

-20

-15

-10

-5

0

Tropical limit graphs of a 3-soliton solution at a negative,respectively positive value of the next KdV hierarchy variable.The polarizations do not depend on the variables!Hence in both cases initial and final polarizations are the same.Therefore the Yang-Baxter equation holds.


Matrix Boussinesq equation

Potential version (u = 2φx):

φtt − 4βφxx +1

3φxxxx + 2 (φxKφx)x − 2 (φxKφt − φtKφx) = 0

This is also a reduction of potential KP, but after a transformationthat introduces the β term. We assume β > 0.

t↑→ x

11

21

12

22

1 2

2 1

12

11

21

12

22

1

1

2

2

1 2

11

21

12

221

2

21

1 2

These are tropical limit graphs of 2-soliton solutions. Again, thereis a Yang-Baxter map at work. But it is known that there is more.


Inelastic collision

t↑→ x

This is the most elementarybinary tree.Analog in case of KP: Milesresonance.There is also the reverseprocess.Not Yang-Baxter cases ...

Scalar and vector Boussinesq do not ad-mit regular solutions with more complicatedtrees.Only “degenerate” cases, like the one shownon the right.

32

13

12

3133

11


Solitons via binary Darboux ... and a trickIn the Boussinesq case, the linear system possesses solutions of theform

θ =∑a

θa eϑ(Pa) , ϑ(P) = P x + P2 t ,

whereP3a = 3β Pa + C

We only consider the case where C and Pa are diagonal.Parametrizing the entries of C as

ci = 2β3/2 1− 45λ2i + 135λ4

i − 27λ6i

(1 + 3λ2i )3

the roots can nicely be expressed as

pi ,1 = −√β(1 + 6λi − 3λ2

i

)1 + 3λ2

i

, pi ,2 = −√β(1− 6λi − 3λ2

i

)1 + 3λ2

i

,

pi ,3 =2√β(1− 3λ2

i

)1 + 3λ2

i

. Correspondingly for adjoint linear system.


”Pure” soliton solutions

... essentially exclude inelastic collisions and splittings. ThenYang-Baxter holds along a tropical limit graph.

In the vector case, the corresponding R matrix reads

R(λi , λj) =

λi−λjλi+λj

1−λi−λj−3λiλj1−λi+λj+3λiλj

2λiλi+λj

1+3λ2j

1−λi+λj+3λiλj

2λjλi+λj

1+3λ2i

1−λi+λj+3λiλj

λj−λiλi+λj

1+λi+λj−3λiλj1−λi+λj+3λiλj

The KdV and the Boussinesq YB maps are reductions of anonlinear YB map obtained for KP.


Why YB holds (for “pure” solitons)

t↑→ x

111

211

221

222

122112

121

1

2

3

1

2

3

1 2

1

213

32 3

111

211 221

222

122112

212

1

2

3

1

2

3

3

1

13

2 3

2

1 2

Tropical limit graphs of 3-soliton solutions of a matrix Boussinesqequation. The graphs correspond to large negative, respectivelypositive values of the next hierarchy variable(ϑ(P) = P x + P2 t + P4 s).


Matrix potential KP equation

Let u = 2φx . We consider the m× n matrix potential KP equation

4φxt − φxxxx − 3φyy − 6(φxKφx)x + 6(φxKφy − φyKφx) = 0


Soliton solutions (via bDT with φ0 = 0)

θ =A∑

a=1

θa eϑ(Pa) χ =

M∑j=1

e−ϑ(Qj ) χj

where Pa and Qj are constant N × N matrices, θa, χj are constantm × N, respectively N × n matrices, and

ϑ(P) = x P + y P2 + t P3 + s P4 + · · ·

If, for all a, j , the matrices Pa and Qj have no common eigenvalue,there are unique solutions Wja of the Sylvester equations

QjWja −WjaPa = χjKθa a = 1, . . . ,A, j = 1, . . . ,M .

Then the equations for Ω are solved by

Ω = Ω0 +A∑

a=1

M∑j=1

e−ϑ(Qj ) Wja eϑ(Pa)

and φ = −θΩ−1χ yields a soliton solution if Ω−1 exists.


Pure soliton solutionsThis is the subclass of soliton solutions with A = M = 1 andP1 = diag(p1,1, . . . , pN,1) = diag(p1, . . . , pN) andQ1 = diag(p1,2, . . . , pN,2) = diag(q1, . . . , qN). Then

φ =F

ττ =

∑I∈1,2N

µI eϑI , F =

∑I∈1,2N

MI eϑI

with constants µI (assumed > 0), constant m× n matrices MI , and

ϑI =N∑i=1

ϑ(pi ,ai ) if I = (a1, . . . , aN) ∈ 1, 2N

The tropical limit graph of a solution is defined as the tropicallimit of τ .

• The matrix KdV solitons correspond to solutions from thisclass (via qi = −pi ).

• The YB map generalizes that of KdV (and Boussinesq).


Why YB holds

y↑→ x

111 121

211

221

222

112

122

111

211

112

212

122

222

221

These are contour plots in the xy -plane of a 3-soliton solution atnegative, respectively positive time t.The polarizations do not depend on the independent variables.Hence in both cases initial and finial polarizations are the same.Therefore the Yang-Baxter equation holds.


Another subclass of KP solitons and the pentagon equation

Now we set N = 1, so that Pa = pa and Qj = qj . The bDT, withvanishing seed, then yields

φ =1

τ

A∑a=1

M∑i=1

φai τai

τ =A∑

a=1

M∑i=1

τai , τai = µai eϑai

µai =χiKθapa − qi

, φai =θa χi

µai, ϑai = ϑ(pa)− ϑ(qi )

If A = 1 or M = 1, the tropical limit graph is a rooted(generically) binary tree.


Binary trees

aj

bj

cj

dj

B

B12

aj

bj

cj

djB

B23

The figures show tropical limit graphs, in the xy -plane, of asolution with A = 4 and M = 1 at t 0 and t 0, respectively.

A binary operation B(λ) : V × V −→ V , (ξ, η) 7→ λ ξ + (1− λ) η,enters the stage, satisfying the local tetragon equation

B12

(pa − pbpa − pc

) B(pa − pcpa − pd

)= B23

(pb − pcpb − pd

) B(pa − pbpa − pd

)acting to the left.


A pentagon relationAs a consequence,

T(pa − pbpa − pc

,pa − pcpa − pd

)=(pa − pbpa − pd

,pb − pcpb − pd

)then satisfies the pentagon equation

T23 T13 T12 = T12 T23

This is the pentagon Tamari lattice. The two chains correspond toa negative, resp. positive value of the next KP hierarchy variable.


Vector KP, and beyond trees

The Yang-Baxter R-matrix and the binary map jointly rule thedistribution of polarizations over the limit graph. They satisfy aconsistency condition,

B12

(pa − pbpa − pc

)R

(pa − pcpa − qk

,qi − qjpa − qk

)= R23

(pb − pcpb − qj

,qi − qjpb − qj

)R12

(pa − pbpa − qj

,qi − qjpa − qj

)B23

(pa − pbpa − pc

)

Also this only applies to a subset of vector KP solutions!

To understand the full set of vector (and moreover matrix) KPsolutions in such a way is still an open problem.


Conclusions

• For matrix KdV, in a space-time picture there are incoming(t → −∞) and outgoing (t → +∞) solitons, with attachedpolarizations. The corresponding map was found to satisfy the2-simplex, i.e. Yang-Baxter equation (Veselov, Goncharenko).The tropical limit yields a deeper understanding !

• We addressed, more generally, the matrix KP equation(Dimakis, M-H: LMP (2018)), and explored its Boussinesqreduction (Dimakis, M-H, Chen: arXiv:1805.09711). Only asubclass of soliton solutions is ruled by YB !

• Beyond “pure” Boussinesq (and KP) solitons: more structure.For binary-tree-shaped solutions, a binary map solving a localtetragon equation, implying a pentagon equation, entersthe stage (Dimakis, M-H: TMP (2018)). These equationsbelong to the family of polygon equations.

• Vector KP: parameter-dependent R-matrix y new solutionsof the 3-simplex (tetrahedron, Zamolodchikov) equation.


References

1. A. Dimakis and F. Muller-Hoissen,“Matrix KP: tropical limit and Yang-Baxter maps”,Lett. Math. Phys. (online Sept. 2018)(arXiv:1708.05694 [nlin.SI]).

2. A. Dimakis and F. Muller-Hoissen,“Matrix Kadomtsev-Petviashvili equation: tropical limit,Yang-Baxter and pentagon maps”,Theor. Math. Phys. 196 (2018) 1164(arXiv:1709.09848 [nlin.SI]).

3. A. Dimakis, F. Muller-Hoissen and X.-M. Chen,“Matrix Boussinesq solitons and their tropical limit”,arXiv:1805.09711 [nlin.SI].


Thanks for your attention !

and all the best to Josephfor his post 70 period !

tropical limit of solitons, yang-baxter maps and beyond · most frequently used notion of...

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