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Integrable twisted hierarchies Derchyi Wu Motivation The Adler-Kostant- Symes Theorem Integrable twisted hierarchies Twisted with D 2 symmetries hierarchies of a splitting type Twisted flows of a splitting type Examples Derchyi Wu

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Integrable twisted hierarchies Derchyi Wu Motivation

The Adler-Kostant- Symes Theorem

Twisted hierarchies of a splitting type

Twisted flows of a splitting type Examples

Twisted hierarchies of a non-splitting type

Twisted flows of a non-splitting type The GMV equation

Inverse scattering theory

Twisted flows of a splitting type Twisted flows of a non-splitting type

Open problems References

MATH, Academia Sinica, Taiwan

Integrable twisted hierarchies with D2 symmetries

Derchyi Wu

Institute of Mathematics Academia Sinica, Taiwan

24 July 2012

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Integrable twisted hierarchies Derchyi Wu Motivation

The Adler-Kostant- Symes Theorem

Twisted hierarchies of a splitting type

Twisted flows of a splitting type Examples

Twisted hierarchies of a non-splitting type

Twisted flows of a non-splitting type The GMV equation

Inverse scattering theory

Twisted flows of a splitting type Twisted flows of a non-splitting type

Open problems References

MATH, Academia Sinica, Taiwan

Abstract

Two new integrable twisted hierarchies with D2 symmetries are constructed via the loop algebra factorization method. The splitting type factorization yields the generalized sinh-Gordon equation, this result justifies some far-reaching generalizations of the well-known connection between the sine-Gordon equation, the Backlund transformation, and surfaces with curvature −1. The non-splitting type factorization yields the Gerdjikov-Mikhailov-Valchev equation which is an anisotropic multicomponent generalization of the classical Heisenberg ferromagnetic equation and is one of the simplest twisted integrable systems. Special analytical features in the associated inverse scattering theory are discussed to solve the Cauchy problem of these twisted flows and to illustrate difficulties in solving the scattering theory of twisted hierarchies with Dk symmetries, k > 2.

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Integrable twisted hierarchies Derchyi Wu Motivation

The Adler-Kostant- Symes Theorem

Twisted hierarchies of a splitting type

Twisted flows of a splitting type Examples

Twisted hierarchies of a non-splitting type

Twisted flows of a non-splitting type The GMV equation

Inverse scattering theory

Twisted flows of a splitting type Twisted flows of a non-splitting type

Open problems References

MATH, Academia Sinica, Taiwan

Outline

1

Motivation The Adler-Kostant-Symes Theorem

2

Twisted hierarchies of a splitting type Twisted flows of a splitting type Examples

3

Twisted hierarchies of a non-splitting type Twisted flows of a non-splitting type The GMV equation

4

Inverse scattering theory Twisted flows of a splitting type Twisted flows of a non-splitting type

5

Open problems

6

References

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Integrable twisted hierarchies Derchyi Wu Motivation

The Adler-Kostant- Symes Theorem

Twisted hierarchies of a splitting type

Twisted flows of a splitting type Examples

Twisted hierarchies of a non-splitting type

Twisted flows of a non-splitting type The GMV equation

Inverse scattering theory

Twisted flows of a splitting type Twisted flows of a non-splitting type

Open problems References

MATH, Academia Sinica, Taiwan

The Adler-Kostant-Symes Theorem

1.1.1 The Adler-Kostant-Symes Theorem Many soliton equations can be obtained by finding some Lie algebra g, equipped w. an ad-invariant, non-degenerate bilinear form, a decomposition g = k + n, − → a symplectic manifold (co-adjoint orbit) ⊂ k⊥ ∼ = N∗, s.t. the equation is the Hamiltonian equation ut = J(∇F). It is a scheme developed over the years for constructing Hamiltonian systems having Lax representations [Adler,’78], [Drinfeld-Sokolov,’81], [Terng,’97].

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Integrable twisted hierarchies Derchyi Wu Motivation

The Adler-Kostant- Symes Theorem

Twisted hierarchies of a splitting type

Twisted flows of a splitting type Examples

Twisted hierarchies of a non-splitting type

Twisted flows of a non-splitting type The GMV equation

Inverse scattering theory

Twisted flows of a splitting type Twisted flows of a non-splitting type

Open problems References

MATH, Academia Sinica, Taiwan

Twisted flows of a splitting type

2.1.1 Loop groups and algebras [Terng,’07] Let U : real form of a complex s. s. Lie algebra G, σ1, σ2 : involutions of U, Ki : fixed point set of σi, U = Ki + Pi : the Cartan decomp. for U Ki . Assume K1 ∩ K2 = S1 × S2, K1 = S1 × K ′

1,

K2 = K ′

2 × S2

as direct product of subgroups.

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Integrable twisted hierarchies Derchyi Wu Motivation

The Adler-Kostant- Symes Theorem

Twisted hierarchies of a splitting type

Twisted flows of a splitting type Examples

Twisted hierarchies of a non-splitting type

Twisted flows of a non-splitting type The GMV equation

Inverse scattering theory

Twisted flows of a splitting type Twisted flows of a non-splitting type

Open problems References

MATH, Academia Sinica, Taiwan

Twisted flows of a splitting type

Define the loop groups by Lσ1 = {f : Aǫ, 1

ǫ

holo.

→ U | σ1(f(−λ)) = f(λ)}, Lσ1

+

= {f ∈ Lσ1|σ2

  • f( 1

λ)

  • = f(λ), f(1) ∈ K ′

2},

Lσ1

−

= {f ∈ Lσ1, f : C/Dǫ

holo.

→ U | f(∞) ∈ K ′

1}.

Thus we have a direct sum decomposition of the loop algebra Lσ1 = Lσ1

+ + Lσ1 − .

Let ˆ π± : Lσ1 → Lσ1

±

be the corres. projections.

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Integrable twisted hierarchies Derchyi Wu Motivation

The Adler-Kostant- Symes Theorem

Twisted hierarchies of a splitting type

Twisted flows of a splitting type Examples

Twisted hierarchies of a non-splitting type

Twisted flows of a non-splitting type The GMV equation

Inverse scattering theory

Twisted flows of a splitting type Twisted flows of a non-splitting type

Open problems References

MATH, Academia Sinica, Taiwan

Twisted flows of a splitting type

Finally, let A be a CSA ⊂ P1, σ2(A) ⊂ A, {a1, . . . , an} a basis of A, Ji,2j+1 = aiλ2j+1 + σ2(ai)λ−(2j+1) ∈ L+ ∩ A. 2.1.2 Definition : the twisted U

K1 -hierarchy (splitting type)

Given a, ˜ a ∈ A, the 2j + 1-th twisted U

K1 -flow is the compatibility condition

  • ∂x + ˆ

π+

  • MJa,1M−1

, ∂t + ˆ π+

  • MJ˜

a,2j+1M−1

= 0, for some M = M(x, t, λ) ∈ Lσ1

− .

→ a nonlinear partial differential system in b, v, where ˆ π+

  • MJa,1M−1

= bab−1λ + v + σ2(bab−1) 1 λ.

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Integrable twisted hierarchies Derchyi Wu Motivation

The Adler-Kostant- Symes Theorem

Twisted hierarchies of a splitting type

Twisted flows of a splitting type Examples

Twisted hierarchies of a non-splitting type

Twisted flows of a non-splitting type The GMV equation

Inverse scattering theory

Twisted flows of a splitting type Twisted flows of a non-splitting type

Open problems References

MATH, Academia Sinica, Taiwan

Examples

2.2.1 Splittings of loop groups Let the twisted U

K1 -hierarchy (splitting type) be defined by

U = O(J, J) =

  • x ∈ GL2n(R)| xt ˜

Jx = ˜ J

  • ,

U = o(J, J) = {ξ ∈ gl2n(R)| ξt ˜ J + ˜ Jξ = 0}, w. J = Iq,n−q = diag(

q times

  • −1, · · · , −1,

n−q times

  • 1, . . . , 1),

˜ J = J −J

  • ,

σi(x) = In+i,n−ixI−1

n+i,n−i,

i = 0, 1 and A = {ξ = D D

  • , D diagonal } ⊂ P0

be a maximal abelian subalgebra.

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Integrable twisted hierarchies Derchyi Wu Motivation

The Adler-Kostant- Symes Theorem

Twisted hierarchies of a splitting type

Twisted flows of a splitting type Examples

Twisted hierarchies of a non-splitting type

Twisted flows of a non-splitting type The GMV equation

Inverse scattering theory

Twisted flows of a splitting type Twisted flows of a non-splitting type

Open problems References

MATH, Academia Sinica, Taiwan

Examples

2.2.2 For (n, q) = (2, 0), a = ˜ a, the first twisted

O(J,J) O(J)×O(J) -flow is the trivial

linear system ∂tu = ∂xu, ∂tω = ∂xω, where b =     1 1 cos u(x, t) sin u(x, t) − sin u(x, t) cos u(x, t)     ∈ K ′

0,

v =     −ω(x, t) ω(x, t)     ∈ S0,

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Integrable twisted hierarchies Derchyi Wu Motivation

The Adler-Kostant- Symes Theorem

Twisted hierarchies of a splitting type

Twisted flows of a splitting type Examples

Twisted hierarchies of a non-splitting type

Twisted flows of a non-splitting type The GMV equation

Inverse scattering theory

Twisted flows of a splitting type Twisted flows of a non-splitting type

Open problems References

MATH, Academia Sinica, Taiwan

Examples

2.2.3 For (n, q) = (2, 0), a = ˜ a, the third twisted

O(J,J) O(J)×O(J) -flow is the 4th

  • rder partial differential system

∂tu = 1 18{ 10∂3

x u + ∂xu

  • 5(∂xu)2 − 12ω∂xu + 180 (cos u)2 − 90 + 15ω2

−8∂2

x ω − 4ω3 + ω

  • 24 − 48 (cos u)2

}, ∂tω = − 4 9 ∂4

x u + ∂2 x u[− 2

3 (∂xu)2 + 5 3 ω∂xu − 40 3 (cos u)2 + 20 3 − 2 3 ω2] −32 sin u (cos u)3 + 16 cos u sin u +∂xu[ 40 3 (∂xu) cos u sin u + 5 6 (∂xu) (∂xω) − 4 3 ω∂xω] + 5 9 ∂3

x ω + ∂xω[5

6 ω2 − 5 + 10 (cos u)2] − 8 3 ω2 cos u sin u.

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Integrable twisted hierarchies Derchyi Wu Motivation

The Adler-Kostant- Symes Theorem

Twisted hierarchies of a splitting type

Twisted flows of a splitting type Examples

Twisted hierarchies of a non-splitting type

Twisted flows of a non-splitting type The GMV equation

Inverse scattering theory

Twisted flows of a splitting type Twisted flows of a non-splitting type

Open problems References

MATH, Academia Sinica, Taiwan

Examples

2.2.4 Definition The 1-dimensional twisted

O(J,J) O(J)×O(J) -system is the compatibility condition

  • ∂xi + ˆ

π+

  • MJai ,1M−1

, ∂xj + ˆ π+

  • MJaj ,1M−1

= 0,

  • r equivalently
  • ∂xi + baib−1λ + vi + σ1(baib−1)

λ , ∂xj + bajb−1λ + vj + σ1(bajb−1) λ

  • = 0,

for 1 ≤ i, j ≤ n and some M = M(x1, · · · , xn, λ) ∈ Lσ0

− .

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SLIDE 12

Integrable twisted hierarchies Derchyi Wu Motivation

The Adler-Kostant- Symes Theorem

Twisted hierarchies of a splitting type

Twisted flows of a splitting type Examples

Twisted hierarchies of a non-splitting type

Twisted flows of a non-splitting type The GMV equation

Inverse scattering theory

Twisted flows of a splitting type Twisted flows of a non-splitting type

Open problems References

MATH, Academia Sinica, Taiwan

Examples

2.2.5 Submanifold geometry Let Mn

s ֒

→ R2n−1

s+q−1. Choose a local adapted ψ-orthnormal frame

{e1, · · · , en, en+1, · · · , e2n−1}, s.t. < ei, ej >= ǫiδij, 1 ≤ i, j ≤ 2n − 1 ǫi =

  • −1,

1 ≤ i ≤ s, n + 1 ≤ i ≤ n + q − 1, 1,

  • therwise

and dx =

n

  • i=1

ωiei, I =

n

  • i=1

ǫiωi ⊗ ωi, II =

2n−1

  • α=n+1

n

  • i=1

ǫαωi ⊗ ωα

i eα.

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Integrable twisted hierarchies Derchyi Wu Motivation

The Adler-Kostant- Symes Theorem

Twisted hierarchies of a splitting type

Twisted flows of a splitting type Examples

Twisted hierarchies of a non-splitting type

Twisted flows of a non-splitting type The GMV equation

Inverse scattering theory

Twisted flows of a splitting type Twisted flows of a non-splitting type

Open problems References

MATH, Academia Sinica, Taiwan

Examples

2.2.6 Theorem [Huang, ’86; C-Z-C, ’04; Ma-Wu, ’11] Let Mn

q ֒

→ R2n−1

2q−1 be timelike and of constant sectional curvature 1. Suppose

I =

n

  • i=1

ǫi(a1

i )2dxi ⊗ dxi,

II =

n

  • λ=2

n

  • i=1

ǫiǫn+λ−1a1

i aλ i dxi ⊗ dxien+λ−1.

Then A(x) =

  • ai

j

  • ∈ O(q, n − q) and the G-C and structure eqns are

A−1dA = δF t − JFδJ, ω = δF − JF tδJ, ∂xj ak

i = ak j fij, fii = 0, i = j,

ǫj∂xj fij + ǫi∂xi fji +

  • k=i, j

ǫkfikfjk = −a1

i a1 j , i = j,

∂xk fij = fikfkj, i = j = k which are called the generalized sinh-Gordon equations.

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Integrable twisted hierarchies Derchyi Wu Motivation

The Adler-Kostant- Symes Theorem

Twisted hierarchies of a splitting type

Twisted flows of a splitting type Examples

Twisted hierarchies of a non-splitting type

Twisted flows of a non-splitting type The GMV equation

Inverse scattering theory

Twisted flows of a splitting type Twisted flows of a non-splitting type

Open problems References

MATH, Academia Sinica, Taiwan

Examples

2.2.7 Theorem [Ma-Wu, ’11] Let Mn

q ֒

→ R2n−1

2q−1 be timelike and of constant sectional curvature 1. Suppose

I =

n

  • i=1

ǫi(a1

i )2dxi ⊗ dxi,

II =

n

  • λ=2

n

  • i=1

ǫiǫn+λ−1a1

i aλ i dxi ⊗ dxien+λ−1.

Then the G-C eqns (GShGE) are the 1-dimensional twisted

O(J,J) O(J)×O(J) -system

  • ∂xi + baib−1λ + vi + σ1(baib−1)

λ , ∂xj + bajb−1λ + vj + σ1(bajb−1) λ

  • = 0,

with b = In A−1

  • , ai = 1

2

  • ei

ei

  • ,

vi = ui

  • ,

u =

  • uidxi = δF − JF tδJ,

fij =   

1 a1

j

∂a1

i

∂xj ,

i = j, 0, i = j. Conversely, ....

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Integrable twisted hierarchies Derchyi Wu Motivation

The Adler-Kostant- Symes Theorem

Twisted hierarchies of a splitting type

Twisted flows of a splitting type Examples

Twisted hierarchies of a non-splitting type

Twisted flows of a non-splitting type The GMV equation

Inverse scattering theory

Twisted flows of a splitting type Twisted flows of a non-splitting type

Open problems References

MATH, Academia Sinica, Taiwan

Examples

2.2.8 Historical background

1

M2 ֒ → R3 constant curvature −1, line congruence − → the geometric Backlund transformation, sin-Gordon equation

2

Chern, Tenenblat-Terng, 80: Mn ֒ → R2n−1 constant sectional curvature −1, GSGE

geometric Backlund transformation; analytic Backlund transformation dX − Xω = DλAδ − XδAt DλX, Dλ = 1 2 (λI − 1 λ I1,n−1)

3

Ablowitz-Beals-Tenenblat, 86: linearization X = PQ−1, P, Q ∈ gl(n, C)

  • P

Q

  • satisfies a 2n × 2n linear system w. GSGE as the c. c.

4

Terng, 07: algebraic symmetries, twisted hierarchies θλ = baib−1λ + vi + σ1(baib−1) λ

  • dxi,

b ∈ O(n, n) O(n) × O(n) , v ∈ o(n) ⊕ 0n.

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Integrable twisted hierarchies Derchyi Wu Motivation

The Adler-Kostant- Symes Theorem

Twisted hierarchies of a splitting type

Twisted flows of a splitting type Examples

Twisted hierarchies of a non-splitting type

Twisted flows of a non-splitting type The GMV equation

Inverse scattering theory

Twisted flows of a splitting type Twisted flows of a non-splitting type

Open problems References

MATH, Academia Sinica, Taiwan

Twisted flows of a non-splitting type

3.1.1 Loop groups and algebras Let U = U(3), U = u(3), σi(x) = JixJ−1

i

, i = 1, 2 w. J1 = diag(1, −1, −1), J2 = diag(1, −1, 1) and u(3) = Ki + Pi. Then K1 = S +S1 K′

1,

S = K1 ∩ K2 = {i diag (α1, α2, α3) | αi ∈ R }, K′

1 = 0 ⊕ su(2),

S1 = S ∩ K′

1 = { i diag (0, α, −α) | α ∈ R}.

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Integrable twisted hierarchies Derchyi Wu Motivation

The Adler-Kostant- Symes Theorem

Twisted hierarchies of a splitting type

Twisted flows of a splitting type Examples

Twisted hierarchies of a non-splitting type

Twisted flows of a non-splitting type The GMV equation

Inverse scattering theory

Twisted flows of a splitting type Twisted flows of a non-splitting type

Open problems References

MATH, Academia Sinica, Taiwan

Twisted flows of a non-splitting type

Let L = {ξ(λ) =

  • j≤n0

ξjλj| ξj ∈ K1 if j is even, ξj ∈ P1 if j is odd}, L+ = {ξ(λ) =

  • |j|≤n0

ξjλj ∈ L| ξ−j = σ2(ξj), ξ0 ∈ S}, L− = {ξ(λ) =

  • j≤0

ξjλj ∈ L| ξ0 ∈ K′

1}.

Thus we have a non-direct sum decomposition L = L+ +S1 L−. Let ˆ π± be the projections ˆ π+(ξ) = πS(ξ0) +

  • 0<j≤n0
  • ξjλj + σ2(ξj) 1

λj

  • ,

ˆ π−(ξ) = πK′

1(ξ0) +

  • 0<j≤n0

( ξ−j − σ2(ξj) )λ−j, ξ = ˆ π+(ξ) +S1 ˆ π−(ξ), ξ0 = πS(ξ0) +S1 πK′

1(ξ0).

Let ξ(λ) ∼S1 ˜ ξ(λ) ⇐ ⇒ ξ(λ) − ˜ ξ(λ) is a constant loop in S1.

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Integrable twisted hierarchies Derchyi Wu Motivation

The Adler-Kostant- Symes Theorem

Twisted hierarchies of a splitting type

Twisted flows of a splitting type Examples

Twisted hierarchies of a non-splitting type

Twisted flows of a non-splitting type The GMV equation

Inverse scattering theory

Twisted flows of a splitting type Twisted flows of a non-splitting type

Open problems References

MATH, Academia Sinica, Taiwan

Twisted flows of a non-splitting type

Finally, let A = {i   d1 r r d1 d3   ∈ u(3) : r, d1, d3 ∈ R} be a CSA, and a = i   1 1   ∈ P1 ∩ A, ˆ J1,0 = aλ + σ2(a) 1 λ ∈ P1 ∩ A ∩ L+, ˆ Jk = ik−1akλk − i   d1 d1 d3   + ik−1σ2(ak) 1 λk ∈ A ∩ L+, for k ∈ {1, 2, . . . }. Note [ˆ J1,0, ˆ Jk] = 0.

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Integrable twisted hierarchies Derchyi Wu Motivation

The Adler-Kostant- Symes Theorem

Twisted hierarchies of a splitting type

Twisted flows of a splitting type Examples

Twisted hierarchies of a non-splitting type

Twisted flows of a non-splitting type The GMV equation

Inverse scattering theory

Twisted flows of a splitting type Twisted flows of a non-splitting type

Open problems References

MATH, Academia Sinica, Taiwan

Twisted flows of a non-splitting type

3.1.2 Definition : the twisted

U(3) U(1)×U(2) -hierarchy (non-splitting type)

The k-th twisted

U(3) U(1)×U(2) -flow is the compatibility condition

[L, M] = 0, where L = ∂x − ∂Ψ ∂x Ψ−1 = ∂x − λbab−1 − 1 λσ2(bab−1), M = ∂t − ∂Ψ ∂t Ψ−1, Ψ(x, t, λ) = m(x, t, λ)exˆ

J1,0+tˆ Jk ,

for some m = m(x, t, ·) ∈ L− and b(x, t) = m(x, t, ∞) ∈ K ′

1 and

lim

|x|→∞ b(x, t) = 1 or

  1 −1 1   .

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Integrable twisted hierarchies Derchyi Wu Motivation

The Adler-Kostant- Symes Theorem

Twisted hierarchies of a splitting type

Twisted flows of a splitting type Examples

Twisted hierarchies of a non-splitting type

Twisted flows of a non-splitting type The GMV equation

Inverse scattering theory

Twisted flows of a splitting type Twisted flows of a non-splitting type

Open problems References

MATH, Academia Sinica, Taiwan

Twisted flows of a non-splitting type

3.1.3 Remarks Set ∂Ψ ∂t Ψ−1 =

k

  • 1
  • Pjλj + σ2(Pj) 1

λj

  • + P0,

we have

1

The k-th twisted

U(3) U(1)×U(2) -flow is

[L, M] = 0, where L ∼S1 ∂x − ˆ π+(mˆ J1,0m−1), M ∼S1 ∂t − ˆ π+(mˆ Jkm−1)

2

The coefficients Pj, j = 0, and the first diagonal element of P0 of M are fixed functions of components of ∂s

x b, 0 ≤ s ≤ k − j.

→ The k-th twisted

U(3) U(1)×U(2) -flow CANNOT be written explicitly as a nonlinear

partial differential system in b.

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Integrable twisted hierarchies Derchyi Wu Motivation

The Adler-Kostant- Symes Theorem

Twisted hierarchies of a splitting type

Twisted flows of a splitting type Examples

Twisted hierarchies of a non-splitting type

Twisted flows of a non-splitting type The GMV equation

Inverse scattering theory

Twisted flows of a splitting type Twisted flows of a non-splitting type

Open problems References

MATH, Academia Sinica, Taiwan

The GMV equation

3.2.1 The Gerdjikov-Mikhailov-Valchev equation [GMV,’10] The GMV equation is an anisotropic multicomponent generalization of the classical Heisenberg ferromagnetic equation: i ut =( ux − u( u∗ · ux))x + 4 u( u∗ · J u) + A u, where

  • u∗

u = 1,

  • u(x, t) ∈ C2,

J = −1 1

  • ,

A = α β

  • , α, β ∈ R.
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Integrable twisted hierarchies Derchyi Wu Motivation

The Adler-Kostant- Symes Theorem

Twisted hierarchies of a splitting type

Twisted flows of a splitting type Examples

Twisted hierarchies of a non-splitting type

Twisted flows of a non-splitting type The GMV equation

Inverse scattering theory

Twisted flows of a splitting type Twisted flows of a non-splitting type

Open problems References

MATH, Academia Sinica, Taiwan

The GMV equation

3.2.2 The GMV equation has a Lax representation [L, M] = 0, with L = ∂x − bab−1λ − σ2(bab−1) 1 λ, M = ∂t − iba2b−1λ2−p1λ − p0 − σ2(p1) 1 λ−iσ2(ba2b−1) 1 λ2 , a = i   1 1   ∈ A, b(x, t) =   1 u −¯ v v ¯ u   ∈ K ′

1,

p1 = − a∗

  • a
  • ∈ P1,
  • a = (1 −

u u∗) ux,

  • u =

u v

  • ,

p0 = −i −2 u∗J u

  • J

u u∗ + u u∗J

  • − i diag (0, α, β) ∈ S.

Note L − ∂x ∈ L+, M − ∂t ∈ L+.

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SLIDE 23

Integrable twisted hierarchies Derchyi Wu Motivation

The Adler-Kostant- Symes Theorem

Twisted hierarchies of a splitting type

Twisted flows of a splitting type Examples

Twisted hierarchies of a non-splitting type

Twisted flows of a non-splitting type The GMV equation

Inverse scattering theory

Twisted flows of a splitting type Twisted flows of a non-splitting type

Open problems References

MATH, Academia Sinica, Taiwan

The GMV equation

3.2.3 Lemma If [L, M] = 0, L = ∂x − bab−1λ − σ2(bab−1) 1 λ, M − ∂t = −iba2b−1λ2 + · · · ∈ L+. Then M = ∂t − iba2b−1λ2−p′

1λ − p′ 0 − σ2(p′ 1) 1

λ−iσ2(ba2b−1) 1 λ2 , w. p′

1 − p1 = γbab−1 ∈ P1,

p′

0 − p0 = −i diag (α1, α2 − α, α3 − β),

and γ(x, t), α1(t), α2(t), α3(t) ∈ R.

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Integrable twisted hierarchies Derchyi Wu Motivation

The Adler-Kostant- Symes Theorem

Twisted hierarchies of a splitting type

Twisted flows of a splitting type Examples

Twisted hierarchies of a non-splitting type

Twisted flows of a non-splitting type The GMV equation

Inverse scattering theory

Twisted flows of a splitting type Twisted flows of a non-splitting type

Open problems References

MATH, Academia Sinica, Taiwan

The GMV equation

3.2.4 Theorem Write the second twisted

U(3) U(1)×U(2) -flow b(x, t) =

  1 u −¯ v v ¯ u  , then

  • u =

u v

  • satisfies the GMV equation

i ut = ( ux − u( u∗ · ux))x + 4 u( u∗ · J u) + A′ u, A′ = 4 2 + d3 − d1

  • r

−2 + d3 − d1 −4

  • .
slide-25
SLIDE 25

Integrable twisted hierarchies Derchyi Wu Motivation

The Adler-Kostant- Symes Theorem

Twisted hierarchies of a splitting type

Twisted flows of a splitting type Examples

Twisted hierarchies of a non-splitting type

Twisted flows of a non-splitting type The GMV equation

Inverse scattering theory

Twisted flows of a splitting type Twisted flows of a non-splitting type

Open problems References

MATH, Academia Sinica, Taiwan

The GMV equation

3.2.5 Remarks

1

The obstruction to constructing the GMV equation parametrized by an arbitrary pair (α, β) by the second twisted

U(3) U(1)×U(2) -flow is the commutativity condition

[ˆ J1,0, ˆ Jk] = 0.

2

Different (d1, d3)’s give different solutions to the second twisted

U(3) U(1)×U(2) -flow.

However, they correspond to the same GMV solution once d3 − d1’s are equal. →the GMV equation is part of the constraints

  • f the second twisted

U(3) U(1) × U(2) -flows.

slide-26
SLIDE 26

Integrable twisted hierarchies Derchyi Wu Motivation

The Adler-Kostant- Symes Theorem

Twisted hierarchies of a splitting type

Twisted flows of a splitting type Examples

Twisted hierarchies of a non-splitting type

Twisted flows of a non-splitting type The GMV equation

Inverse scattering theory

Twisted flows of a splitting type Twisted flows of a non-splitting type

Open problems References

MATH, Academia Sinica, Taiwan

The twisted

O(J,J) O(J)×O(J)-flows 4.1.1 Linearization scheme The initial valued problem of the twisted flows can be solved by an inverse scattering problem plus a linear (or a relatively trivial) evolution equation on the scattering data:

direct problem

  • L(x, 0, λ)

− → Ψ(x, 0, λ) − → V(0, λ)

  • tw. ↓ flow
  • spec. ↓ evol.

L(x, t, λ) ← − Ψ(x, t, λ) ← − V(t, λ)

  • inverse problem

eigenfunction LΨ = (∂x + bab−1λ + v + σ1(bab−1) 1 λ)Ψ = 0, scattering data V(t, λ) = essential part of ∂ ∂¯ λΨ(x, t, λ), spec evol. ∂V ∂t =

  • V, J˜

a,2j+1

  • .
slide-27
SLIDE 27

Integrable twisted hierarchies Derchyi Wu Motivation

The Adler-Kostant- Symes Theorem

Twisted hierarchies of a splitting type

Twisted flows of a splitting type Examples

Twisted hierarchies of a non-splitting type

Twisted flows of a non-splitting type The GMV equation

Inverse scattering theory

Twisted flows of a splitting type Twisted flows of a non-splitting type

Open problems References

MATH, Academia Sinica, Taiwan

The twisted

O(J,J) O(J)×O(J)-flows 4.1.2 Complex analysis plays a key role in the IVP of integrable systems. The main advantage is to provide a canonical approach, through the Cauchy integral theorem, to extract scattering data and to reconstruct potentials. eigenfunction LΨ = (∂x + bab−1λ + v + σ1(bab−1) 1 λ)Ψ = 0, Ψ(x, 0, λ) = m(x, 0, λ)e−x(λa+ 1

λ σ1(a))

scattering data V(0, λ) = essential part of ∂ ∂¯ λΨ(x, 0, λ) V(0, λ) =

  • e. p. { ∂

∂¯ λm(x, 0, λ), m(x, 0, ∞)},

slide-28
SLIDE 28

Integrable twisted hierarchies Derchyi Wu Motivation

The Adler-Kostant- Symes Theorem

Twisted hierarchies of a splitting type

Twisted flows of a splitting type Examples

Twisted hierarchies of a non-splitting type

Twisted flows of a non-splitting type The GMV equation

Inverse scattering theory

Twisted flows of a splitting type Twisted flows of a non-splitting type

Open problems References

MATH, Academia Sinica, Taiwan

The twisted

O(J,J) O(J)×O(J)-flows 4.1.2 Complex analysis plays a key role in the inverse scattering theory of nonlinear integrable systems. The main advantage is to provide a canonical approach, through the Cauchy integral theorem, to extract scattering data and derive a reconstruction formula of the potentials. eigenfunction LΨ = (∂x + bab−1λ + v + σ1(bab−1) 1 λ)Ψ = 0, Ψ(x, 0, λ) = m(x, 0, λ)e−x(λa+ 1

λ σ1(a))

= b(x, 0) ˘ m(x, 0, λ)e−x(λa+ 1

λ σ1(a))

scattering data V(0, λ) = essential part of ∂ ∂¯ λΨ(x, 0, λ) V(0, λ) =

  • e. p. { ∂

∂¯ λm(x, 0, λ), m(x, 0, ∞)}, =

  • e. p. { ∂

∂¯ λ ˘ m(x, 0, λ), ˘ m(x, 0, ∞) = 1}, =

  • e. p. {
  • ˘

m+ ˘ m−1

−

  • (x, 0, λ), ˘

m(x, 0, ∞) = 1}.

slide-29
SLIDE 29

Integrable twisted hierarchies Derchyi Wu Motivation

The Adler-Kostant- Symes Theorem

Twisted hierarchies of a splitting type

Twisted flows of a splitting type Examples

Twisted hierarchies of a non-splitting type

Twisted flows of a non-splitting type The GMV equation

Inverse scattering theory

Twisted flows of a splitting type Twisted flows of a non-splitting type

Open problems References

MATH, Academia Sinica, Taiwan

The twisted

O(J,J) O(J)×O(J)-flows Moreover, in view of the evol. operator MΨ(x, t, λ) =

  • ∂t + ˆ

π+

  • MJ˜

a,2j+1M−1

Ψ(x, t, λ) = 0, eigenfunction LΨ = (∂x + bab−1λ + v + σ1(bab−1) 1 λ)Ψ = 0, Ψ(x, t, λ) = m(x, t, λ)e

−x(λa+ 1

λ σ1(a))−t(λ2j+1˜

a+

1 λ2j+1 σ1(˜

a))

= b(x, t)

  • ˘

me−tJ˜

a,2j+1

  • e−x(λa+ 1

λ σ1(a))

scattering data V(t, λ) = essential part of ∂ ∂¯ λΨ(x, t, λ) V(t, λ) =

  • e. p. {
  • ˘

m+ ˘ m−1

−

  • (x, t, λ), ˘

m(x, t, ∞) = 1} = e−tJ˜

a,2j+1V(0, λ)etJ˜ a,2j+1.

So ∂V ∂t =

  • V, J˜

a,2j+1

  • .
slide-30
SLIDE 30

Integrable twisted hierarchies Derchyi Wu Motivation

The Adler-Kostant- Symes Theorem

Twisted hierarchies of a splitting type

Twisted flows of a splitting type Examples

Twisted hierarchies of a non-splitting type

Twisted flows of a non-splitting type The GMV equation

Inverse scattering theory

Twisted flows of a splitting type Twisted flows of a non-splitting type

Open problems References

MATH, Academia Sinica, Taiwan

The twisted

O(J,J) O(J)×O(J)-flows 4.1.3 Revised linearization scheme

direct problem

  • L(x, 0, λ)

− → Ψ(x, 0, λ) − → ˘ m(x, 0, λ) − → V(0, λ)

  • tw. ↓ flow

spec.↓evol. L(x, t, λ) ← − Ψ(x, t, λ)

×b

← − ˘ m(x, t, λ) ← − V(t, λ)

  • inverse problem

w. Direct problem solving Volterra integral equations, Inverse problem solving Riemann-Hilbert problems.

slide-31
SLIDE 31

Integrable twisted hierarchies Derchyi Wu Motivation

The Adler-Kostant- Symes Theorem

Twisted hierarchies of a splitting type

Twisted flows of a splitting type Examples

Twisted hierarchies of a non-splitting type

Twisted flows of a non-splitting type The GMV equation

Inverse scattering theory

Twisted flows of a splitting type Twisted flows of a non-splitting type

Open problems References

MATH, Academia Sinica, Taiwan

The twisted

O(J,J) O(J)×O(J)-flows 4.1.4 IVP: Find the gauge b(x)

1

Liouville’s theorem ∂ ˘ m ∂x = [ ˘ m(x, λ), λa + 1 λ σ1(a)] + 1 λ (σ1(a) − B) ˘ m − C′ ˘ m. Need to find {b, v} ∈ K ′

0 × S0 s.t. C′ = b−1vb + b−1 ∂b ∂x at first. 2

Introduce x = (x1, · · · , xn) = x(w1, · · · , wn), M( x, λ) = ˘ m(x, a, λ) = ˘ m(x, λ). Then ∂M ∂xj = [M, λaj + 1 λ σ1(aj)] + 1 λ

  • σ1(aj) − Bj
  • M − CjM,

Bj ∈ P0, Cj ∈ K0, and the compatibility condition ∂xj Ci − ∂xi Cj −

  • Ci, Cj
  • =
  • Bi, aj
  • −
  • Bj, ai
  • .
  • alg. sym. & diag. prop. =

⇒ Cj =

  • ˜

vj (∂j ˜ b)˜ b−1

slide-32
SLIDE 32

Integrable twisted hierarchies Derchyi Wu Motivation

The Adler-Kostant- Symes Theorem

Twisted hierarchies of a splitting type

Twisted flows of a splitting type Examples

Twisted hierarchies of a non-splitting type

Twisted flows of a non-splitting type The GMV equation

Inverse scattering theory

Twisted flows of a splitting type Twisted flows of a non-splitting type

Open problems References

MATH, Academia Sinica, Taiwan

The twisted

O(J,J) O(J)×O(J)-flows 4.1.5 IVP: Exterior differential systems

1

Let b =

  • 1

˜ b

  • ∈ K ′

0,

v =

  • wjvj =
  • wj
  • ˜

vj

  • ∈ S0,

A = −

n

  • j=1

wjbajb−1 dxj, B = −

n

  • j=1

wjbBjb−1 dxj, C =

n

  • j=1

wj(−bCjb−1 + (∂jb)b−1) dxj Ψ(x, λ) = b( x)M( x, λ)e−(λX+ 1

λ σ1(X)).

Then dΨ = λAΨ + 1 λ BΨ + CΨ,

d2Ψ=0

→ d(A − σ1(B)) + (A − σ1(B)) ∧ C + C ∧ (A − σ1(B)) = 0, bdry cond of B σ1(C) = C → A = σ1(B).

slide-33
SLIDE 33

Integrable twisted hierarchies Derchyi Wu Motivation

The Adler-Kostant- Symes Theorem

Twisted hierarchies of a splitting type

Twisted flows of a splitting type Examples

Twisted hierarchies of a non-splitting type

Twisted flows of a non-splitting type The GMV equation

Inverse scattering theory

Twisted flows of a splitting type Twisted flows of a non-splitting type

Open problems References

MATH, Academia Sinica, Taiwan

The twisted

U(3) U(1)×U(2)-flows 4.2.1 Remarks

1

The gauge is found via an exterior differential system derived from a 1-dimensional system as. w. a Cartan subalgebra with higher rank in [MW,’11]. → the extended twisted U(4) U(2) × U(2) -spectral problem.

2

Twisted

U(3) U(1)×U(2) - and extended twisted U(4) U(2)×U(2) -hierarchies share the same

reduction group and Ψ − → V

  Ψ11 Ψ12 Ψ13 1 Ψ21 Ψ22 Ψ23 Ψ31 Ψ32 Ψ33    ← −    V11 V12 V13 1 V21 V22 V23 V31 V32 V33   

3

Despite u(4) = ˜ K1 ⊕ ˜ P1 vs

  • (2, 2) = K0 ⊕ P0,

˜ P1 antisymmetric P0 symmetric, we have IVP of twisted U(4) U(2) × U(2) -hier. ≈ IVP of twisted O(2, 2) O(2) × O(2) -hier.

slide-34
SLIDE 34

Integrable twisted hierarchies Derchyi Wu Motivation

The Adler-Kostant- Symes Theorem

Twisted hierarchies of a splitting type

Twisted flows of a splitting type Examples

Twisted hierarchies of a non-splitting type

Twisted flows of a non-splitting type The GMV equation

Inverse scattering theory

Twisted flows of a splitting type Twisted flows of a non-splitting type

Open problems References

MATH, Academia Sinica, Taiwan

Open problems

Difficulties for general diherdral groups

1

  • Bi, aj
  • −
  • Bj, ai
  • =
  • ˜

M0σ(ai) ˜ M−1 , aj

  • −
  • ˜

M0σ(aj) ˜ M−1 , ai

  • =

˜ M0σ(ai) ˜ M−1 aj − aj ˜ M0σ(ai) ˜ M−1 − ˜ M0σ(aj) ˜ M−1 ai + ai ˜ M0σ(aj) ˜ M−1 . Note σ( ˜ M0σ(ai) ˜ M−1 aj) = −           g−1

1

ei · · · g−1

2

ei · · · . . . . . . ... . . . ... g−1

n−1ei

g−1

N ei

· · ·                    · · · g2ej ... . . . ... · · · gN−1e · · · = −diag

  • g−1

1

eig2ej, g−1

2

eig3ej, · · · , g−1

N eig1ej

  • ??? =

−diag

  • eig2ejg−1

1

, eig3ejg−1

2

, · · · , eig1ejg−1

N

  • =

ai ˜ M0σ(aj) ˜ M−1 .

slide-35
SLIDE 35

Integrable twisted hierarchies Derchyi Wu Motivation

The Adler-Kostant- Symes Theorem

Twisted hierarchies of a splitting type

Twisted flows of a splitting type Examples

Twisted hierarchies of a non-splitting type

Twisted flows of a non-splitting type The GMV equation

Inverse scattering theory

Twisted flows of a splitting type Twisted flows of a non-splitting type

Open problems References

MATH, Academia Sinica, Taiwan

Open problems

Problems for solitons

1

We have succeeded in deriving a local B¨ ackland transformation theory for the GShGE in [MW,’11] via a loop group factorization approach. We wish to study an analytical interpretation of the theory. Namely, justifying that taking a B¨ ackland transform is an action of adding poles.

2

A B¨ ackland transformation theory for the twisted

U(4) U(2)×U(2) -flows should be obtained

by adapting the theory for the GSGE. However, the extended scattering data is not preserved under these transformations. So the approach yields no GMV or twisted

U(3) U(1)×U(2) -solitons.

However, it should be possible to find explicit twisted

U(3) U(1)×U(2) -solitons. It could

result in explicit PDEs for twisted

U(3) U(1)×U(2) -flows.

slide-36
SLIDE 36

Integrable twisted hierarchies Derchyi Wu Motivation

The Adler-Kostant- Symes Theorem

Twisted hierarchies of a splitting type

Twisted flows of a splitting type Examples

Twisted hierarchies of a non-splitting type

Twisted flows of a non-splitting type The GMV equation

Inverse scattering theory

Twisted flows of a splitting type Twisted flows of a non-splitting type

Open problems References

MATH, Academia Sinica, Taiwan

References

  • M. Adler: On a trace functional for formal pseudo differential operators and

the symplectic structure of the Korteweg-de Vries type equations. Invent.

  • Math. 50 (1978/79), no. 3, 219ï¿ 1

2 V24

  • M. Ablowitz, R. Beals, K. Tenenblat: On the solution of the generalized wave

and generalized sine Gordon equations. Stud. Appl. Math. 74 (1986), no. 3, 177–203.

  • Q. Chen, D. Zuo, Y. Cheng: Isometric immersions of pseudo-Riemannian

space forms. J. Geometry and Physics 52 (2004), 241-262.

  • V. G. Drinfeld, V. V. Sokolov: Equations of Korteweg-de Vries type, and simple

Lie algebras. (Russian) Dokl. Akad. Nauk SSSR 258 (1981), no. 1, 11ï¿ 1

2 V16

  • H. Ma, D. Wu: Twisted hierarchies associated with the generalized

sine-Gordon equation. Journal of Mathematical Physics, 52 (2011), no. 9, 093704, 33 pp.

  • C. L. Terng: Soliton equations and differential geometry. J. Differential Geom.

45 (1997), no. 2, 407ï¿ 1

2V445.

  • C. L. Terng: Soliton hierarchies constructed from involutions. Fourth

International Congress of Chinese Mathematicians, 367ï¿ 1

2 V381, AMS/IP

  • Stud. Adv. Math., 48, Amer. Math. Soc., Providence, RI, 2010
  • D. Wu: A twisted integrable hierarchy with D2 symmetry arXiv:1204.6548