YANG-BAXTER AND REFLECTION MAPS FROMOPTICAL SOLITONS INTERACTIONS

Vincent Caudrelier

Department of Mathematics

City University London

LAPTh, 18 September 2014

References: based on joint work with Cheng Zhang

Discovery and first solutions of the set-theoretical reflection equation:

• Vector Nonlinear Schrodinger Equation on the half-line, J. Phys. A45,(2012), 105201.

• Yang-Baxter and reflection maps from vector solitons with a boundary,Nonlinearity 27 (2014) 1081.

Study and applications in the context of fully discrete integrable systems anddiscrete di↵erential geometry with Nicolas Crampe (not covered today):

• Set-theoretical reflection equation: Classification of reflection maps, J.Phys. A46, (2013), 095203.

• Integrable boundary for quad-graph systems: Three-dimensional boundaryconsistency, SIGMA 10 (2014).

Plan

1. Vector nonlinear Schrodinger (NLS) equation on the half-line

•Model and soliton solutions via nonlinear mirror image method

• Polarization e↵ect at the boundary

2. Further study of vector soliton solutions

• Interactions of vector solitons: factorization and set-theoretical Yang-Baxter equation

• Case with a boundary: factorization and set-theoretical reflection equation•Moral of the story: Yang-Baxter and reflection equation not just for quan-tum matrices

3. Conclusions and outlook

1.1 Vector NLS on the half-line: model and soliton solutions

•Model of wave propagation in nonlinear Kerr optical medium:

(i@

t

+ @

2

x

)R + 2R

†RR = 0 ,

for a vector field R = (R

1

, . . . , R

n

)

T of 1 + 1 variables (x, t).

• Case n = 2 is the historical Manakov model representing propagatingpolarized light.

• On the full line x 2 R, the model is integrable and amenable to theso-called Inverse Scattering Method (ISM) = nonlinear Fouriertransform [Manakov ’74]

ISM:R(x, 0)

F�! {S(k, 0), kj

, C

j

(0)}#

R(x, t)

F�1 � {S(k, t), kj

, C

j

(t)}

• F and F�1 obtained by spectral analysis of an auxiliary problem involvinga so-called Lax pair U, V :

(@

x

(x, t, k) = U(x, t, k) (x, t, k) ,

@

t

(x, t, k) = V (x, t, k) (x, t, k) .

• Compatibility of this linear auxiliary problem yields the zero-curvature con-dition which is equivalent to the nonlinear PDE of interest

xt

=

xt

, 8k , U

t

� V

x

+ [U, V ] = 0, (i@

t

+ @

2

x

)R = �2R†RR .

where

U = (�ik⌃3

+Q) , V = (�2ik2⌃3

+G) ,

with

⌃

3

=

✓1In

0

0 �1

◆, Q =

✓0 R

�R†0

◆and G = 2kQ� iQ

x

⌃

3

� iQ

2

⌃

3

,

• Beauty of the method is that, as R evolves in time nonlinearly, S(k) andC

j

evolve linearly:

@

t

S(k, t) = 2ik

2

[�

3

, S(k, t)] ,

d

dt

C

j

(t) = �4ik2j

C

j

(t) .

• Inverse part allows to obtainedR(x, t) from evolved scattering data S(k, t)and C

j

(t). Technical!

Pure soliton case

• S(k, t) plays no role and R(x, t) completely controlled by k

j

2 C withcorresponding norming constants C

j

2 Cn, j = 1, . . . , N . N is thenumber of solitons.

• Explicitely,

R(x, t) = �2i�1 1 . . . 1

�µ

�1(x, t)

0

B@C

1

e

�2ik⇤1

x�4ik⇤21

t

...

C

N

e

�2ik⇤N

x�4ik⇤2N

t

1

CA ,

where

µ

m`

(x, t) = �

m`

1In

+

NX

j=1

e

2i(k

j

�k⇤m

)x+4i(k

2

j

�k⇤2m

)t

(k

⇤`

� k

j

)(k

j

� k

⇤m

)

C

m

C

†j

.

Why is it called an N -soliton solution?

• for N = 1, the solution is characterized by k = v + iA and polarizationp = C/||C||

R(x, t) = pA

e

if (x,t)

coshA(x� vt� x

0

)

• For N � 2, the large time asymptotic behaviour of the solution is

R(x, t) =

NX

j=1

p±j

A

j

e

if

j

(x,t)

coshA

j

(x� v

j

t� x

±j

)

, t! ±1 .

An N -soliton solution splits into a train of N one-soliton solutions.

Case with a boundary at x = 0

• Restrict to half-line x > 0, impose integrable boundary conditions atx = 0,

• Construct solutions using nonlinear version of mirror image technique.

• Both finding integrable BC and implementing nonlinear mirror image tech-nique achieved by using Backlund transformations technique combinedwith folding.

• Backlund transformations are transformations between solutions of PDEs.

Example: The Cauchy-Riemann equations

u

x

= v

y

, u

y

= �vx

are an example of (auto) BT for the Laplace equation u

xx

+ u

yy

= 0.

Procedure for scalar NLS (Habibullin ’89; Bibikov, Tarasov ’89; Biondini,Hwang ’09):

• Take two copies of the Lax pair presentation and connect them by a 2⇥2

matrix L(x, t, k)

(IS1)

(@

x

= U ,

@

t

= V ,

, (IS2)

(@

x

e =

eU

e ,

@

t

e =

eV

e ,

with e = L .

• The Backlund matrix L must satisfy

L

x

=

eUL� LU , L

t

=

eV L� LV . (1)

• Put (IS1) on (0,1), (IS2) on (�1, 0).

• Impose (1) at x = 0 and eQ(x, t) = Q(�x, t) (folding). Then, (1) atx = 0 gives allowed integrable boundary conditions (BC) on Q.

• Implement mirror image technique: apply ISM to appropriate extensionof Q(x, t) for all x 2 R and extract solution on x > 0.

Application to vector NLS [Caudrelier, Zhang ’12-’13]

• The simplest time-independent integrable boundary conditions for vectorNLS are

(@

x

� ↵)R(x, t)

x=0

= 0 , ↵ 2 R , (Robin BC) ,(R

i

(0, t) = 0 , i = 1 . . . , k ,

@

x

R

i

(0, t) = 0 , i = k + 1, . . . , n

, (mixed Neumann-Dirichlet BC) .

• The reconstruction formulas of the full line for R(x, t) are still valid whenrestricted to x > 0 provided symmetry constraints hold on scatteringdata.

For pure solitons case, the mirror image constraints are

• N = 2J and, without loss of generality, kj+J

= �k⇤j

, j = 1, . . . , J

• The corresponding norming constants Cj+J

= C

0j

and C

j

satisfy

C

0j

C

†j

= �B(�kj

)M(k

j

) , j = 1, . . . , J .

where M(k

j

) is a matrix depending on all the norming constants and

B(k) =

k � i↵

k + i↵

1In

or B(k) =

0

@�

1

. . .�

n

1

A, �

j

= ±1 .

• Pictorially

2J , . . . , J + 2 , J + 1

��� 1 , 2 , . . . J , t! �1 ,

J , . . . , 2 , 1

��� J + 1 , J + 2 , . . . 2J , t!1 ,

1.2 E↵ect of the boundary

Important observation:

• (i@t

+ @

2

x

)R + 2R

†RR = 0 invariant under global U(n) transformation

R! U R.

• But mixed BC are not!(R

i

(0, t) = 0 , i = 1 . . . , k ,

@

x

R

i

(0, t) = 0 , i = k + 1, . . . , n

) the boundary has a polarization e↵ect: transmission between modesas solitons reflect.

• Easily seen on one-soliton solution from our explicit results.

• Choose n = 2 (two-dimensional polarizations) and boundary matrix

B

0

=

✓1 0

0 �1

◆,

• First mode R

1

(left) satisfies Neumann BC @

x

R

1

(0, t) = 0 and secondmode R

2

(right) satisfies Dirichlet BC R

2

(0, t) = 0.

0

5

10

x-2

02

t

»R1Hx,tL»

0

510

x-20

2

t

»R2Hx,tL»

• Now choose transformation

U =

✓cos ✓ sin ✓

� sin ✓ cos ✓

◆,

with ✓ 2 R and consider the boundary matrix B = U B

0

U

�1. ✓ mea-sures the ”deviation” from the natural boundary basis corresponding tomixed Neumann-Dirichlet BC.

• Now, clear transmission of ”mass” from mode 1 to mode 2.

0

5

10

x-2

02

t

»R1Hx,tL»

0

5

10

x-2

02

t

»R2Hx,tL»

2.1 Further study of vector soliton solutions

• Back to scalar solitons on the line: behave like particles through collisionsalthough they are classical, extended, nonlinear objects! (Video)

• Through collision shape, velocity and amplitude and number of solitonsare retained. Only e↵ect is phase and position shifts

R(x, t) =

NX

j=1

A

j

e

if

j

(x,t)

coshA

j

(x� v

j

t� x

±j

)

, t! ±1 .

• Formulas for shifts in f

j

and for x+j

�x

�j

show phenomenon of factoriza-tion of collisions into two-body collisions.

• Consistency in scalar case ensured by additivity (hence commutativity) ofthe two-body shifts.

The general problem of factorization of interactions

•When the objects are quantum particles, this gives rise to the Yang-Baxterequation for the quantum scattering matrix S

S

12

(k

1

, k

2

)S

13

(k

1

, k

3

)S

23

(k

2

, k

3

) = S

23

(k

2

, k

3

)S

13

(k

1

, k

3

)S

12

(k

1

, k

2

)

• But what is the analog of this when the objects are solitons? Is factoriza-tion still present for vector solitons?

• Answer obtained by looking again at large time asymptotics of solutions

R(x, t) =

NX

j=1

p±j

A

j

e

if

j

(x,t)

coshA

j

(x� v

j

t� x

±j

)

, t! ±1 ,

and keeping track of the relations between outgoing and incoming polar-izations.

• Results: Factorisation property still holds! [Tsuchida ’04; Ablowitz, Prinari, Trubatch

’04; Caudrelier, Zhang ’13]

• The map {p�j

}N1

7! {p+j

}N1

factorises into a succession of two-bodycollision maps of the form

R(k

1

, k

2

) : (p�1

,p�2

) 7! (p+1

,p+2

) ,

p+1

=

I

n

+

✓k

⇤1

� k

2

k

⇤1

� k

⇤2

� 1

◆p�2

(p�2

)

†

(p�2

)

†p�2

!p�1

,

p+2

=

I

n

+

✓k

2

� k

⇤1

k

2

� k

1

� 1

◆p�1

(p�1

)

†

(p�1

)

†p�1

!p�2

.

• Consistency ensured by the fact that R(k

1

, k

2

) satisfies the so-called set-theoretical Yang-Baxter equation on the set (CPn�1)3

R12

(k

1

, k

2

)�R13

(k

1

, k

3

)�R23

(k

2

, k

3

) = R23

(k

2

, k

3

)�R13

(k

1

, k

3

)�R12

(k

1

, k

2

)

What’s going on?

• Important to realize that behind factorization lies the permutation groupwith generators s

i

, i = 1, . . . , N � 1 and relations

s

i

s

i+1

s

i

= s

i+1

s

i

s

i+1

, s

2

i

= id .

• Now roughly speaking, quantum YBE deals with linear representations ofthe permutation groups i.e. (quantum) matrices.

• The quantum R matrix is a parameter-dependent solution of the equation

R

12

(u, v) �R13

(u, w) �R23

(v, w) = R

23

(v, w) �R13

(u, w) �R12

(u, v)

understood as an identity in End(V

⌦3) for some vector space V . Oper-

ation � simply matrix multiplication in that case.

• The indices show in which copy of V inside V ⌦ V ⇥ V the matrix R

acts nontrivially.

• In 1990, Drinfeld emphasized the importance the set-theoretical YBE :roughly, it amounts to the most general study of the representations of

R

12

(u, v) �R13

(u, w) �R23

(v, w) = R

23

(v, w) �R13

(u, w) �R12

(u, v)

• R12

any (nonlinear) map on X ⇥X to itself, X any set, � compositionof maps.

• As we have just seen, vector soliton collisions provide examples of suchnonlinear representation of the permutation group.

• Beauty in integrable systems: quantum and classical systems share thesame fundamental structures! One ”simply” has to allow for more generalrepresentations of these structures.

2.2 Case with a boundary [Caudrelier, Zhang ’13-’14]

Questions:

• Does factorization survive the presence of a boundary?

• If so, how is the consistency ensured in this case?

Answers:

• Yes, for the integrable boundary conditions that we found above viaBacklund transformation technique.

• Asymptotic analysis shows that a new object appears to describe soliton-boundary collisions. This new object has to satisfy the boundary analogof the set-theoretical YBE: the set-theoretical reflection equation.

• Pictorially, e↵ect of the boundary on polarizations extracted from R mapmodulo mirror image relations

p�j

Rj+J,j

(k

j+J

, k

j

)

ep+j+J

p�j+J

ep+j

!

p�j

Bj

(k

j

)

p+j

•We get the boundary map B : (p�, k) 7! (p+,�k⇤) where

p+ =

I

n

+

k � k

⇤

k + k

⇤p�p�†

p�†p�

!B(k)p� .

• Consistency of factorization with a boundary ensured by the reflectionequation

•We found that B satisfies the set-theoretical reflection equation

R12

(k

1

, k

2

) � B1

(k

1

) �R21

(�k⇤1

, k

2

) � B2

(k

2

)

= B2

(k

2

) �R12

(k

1

,�k⇤2

) � B1

(k

1

) �R21

(�k⇤1

,�k⇤2

)

Comments

• The observation on the relation between permutation group and the Yang-Baxter equation goes over to the case with reflection .

• The appropriate generalization of the permutation group is the finite Cox-eter group of typeBC

N

which is the permutation group with an additionalgenerator b satisfying b

2

= id and

bs

1

bs

1

= s

1

bs

1

b .

• The quantum reflection equation, introduced in 1998 by Sklyanin dealswith linear representations of this group.

•We discovered the set-theoretical reflection equation which is the non-linear, abstract generalization of this problem: given a map R on a setX ⇥X , find a map B on X such that

R12

� B1

�R21

� B2

= B2

�R12

� B1

�R21

2.3 Moral of the story

• Quantum and classical integrable systems share the most fundamentalstructures of integrability: one ”simply” has to go beyond the realm oflinear representations to see it.

• Both YBE and RE have set-theoretical (nonlinear) versions: they are notjust for quantum (linear) objects. Solutions arise from study of classicalintegrable PDEs.

• The Yang-Baxter equation: unifying principle behind quantum integrablesystems/quantum groups and classical integrable systems/Poisson-Lie groups.

• In fact, factorization of processes seems to be the deepest unifying prin-ciple in integrability.

•Warning: not to be confused with the classical YBE for the classical r-matrix!

3 Conclusions and perspectives

Some natural questions: a lot left to do (exciting?!)

•More examples from physical soliton systems? Applications for solitoncontrol?

• Classification problem: given a set X and a Yang-Baxter map, find allpossible reflection maps. First results available [Caudrelier, Crampe, Zhang ’13]

• There exists a deep mathematical connection between quantum groupsand Poisson-Lie groups:

– Usually seen through the classical r-matrix approach and the theory ofLie bialgebras (common ”limit” of the two theories).

– But set-theoretical YBE seems to give a direct link...

–What is the story with reflection? Only known on the quantum side sofar.

THANK YOU!