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 Twisted hierarchies of a non-splitting type Institute of Mathematics Twisted flows of a Academia Sinica, Taiwan non-splitting type The GMV equation Inverse 24 July 2012 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 Derchyi Wu Motivation The Adler-Kostant- Abstract Symes Theorem Two new integrable twisted hierarchies with D 2 symmetries are constructed via the Twisted hierarchies of a loop algebra factorization method. The splitting type factorization yields the splitting type generalized sinh-Gordon equation, this result justifies some far-reaching Twisted flows of a generalizations of the well-known connection between the sine-Gordon equation, the splitting type Backlund transformation, and surfaces with curvature − 1. The non-splitting type Examples factorization yields the Gerdjikov-Mikhailov-Valchev equation which is an anisotropic Twisted multicomponent generalization of the classical Heisenberg ferromagnetic equation hierarchies of a non-splitting and is one of the simplest twisted integrable systems. type Twisted flows of a Special analytical features in the associated inverse scattering theory are discussed non-splitting type to solve the Cauchy problem of these twisted flows and to illustrate difficulties in The GMV equation solving the scattering theory of twisted hierarchies with D k symmetries, k > 2. Inverse scattering theory Twisted flows of a splitting type Twisted flows of a non-splitting type Open problems References MATH, Academia Sinica, Taiwan
Outline Integrable twisted hierarchies Motivation 1 Derchyi Wu The Adler-Kostant-Symes Theorem Motivation The Adler-Kostant- Symes Twisted hierarchies of a splitting type 2 Theorem Twisted flows of a splitting type Twisted Examples hierarchies of a splitting type Twisted flows of a splitting type Twisted hierarchies of a non-splitting type 3 Examples Twisted flows of a non-splitting type Twisted The GMV equation hierarchies of a non-splitting type Twisted flows of a Inverse scattering theory 4 non-splitting type Twisted flows of a splitting type The GMV equation Twisted flows of a non-splitting type Inverse scattering theory Open problems Twisted flows of a 5 splitting type Twisted flows of a non-splitting type References 6 Open problems References MATH, Academia Sinica, Taiwan
The Adler-Kostant-Symes Theorem Integrable twisted hierarchies Derchyi Wu 1.1.1 The Adler-Kostant-Symes Theorem Motivation Many soliton equations can be obtained by finding The Adler-Kostant- Symes Theorem some Lie algebra g , Twisted equipped w. an ad-invariant, non-degenerate bilinear form , hierarchies of a splitting type a decomposition g = k + n , Twisted flows of a splitting type Examples a symplectic manifold (co-adjoint orbit) ⊂ k ⊥ ∼ = N ∗ , − → Twisted hierarchies of a non-splitting s.t. the equation is the Hamiltonian equation type Twisted flows of a non-splitting type u t = J ( ∇ F ) . The GMV equation Inverse scattering theory It is a scheme developed over the years for constructing Hamiltonian systems Twisted flows of a splitting type having Lax representations [Adler,’78], [Drinfeld-Sokolov,’81], [Terng,’97]. Twisted flows of a non-splitting type Open problems References MATH, Academia Sinica, Taiwan
Twisted flows of a splitting type Integrable twisted hierarchies Derchyi Wu 2.1.1 Loop groups and algebras [Terng,’07] Motivation Let The Adler-Kostant- Symes Theorem U : real form of a complex s. s. Lie algebra G , Twisted hierarchies of a σ 1 , σ 2 : involutions of U , splitting type Twisted flows of a K i : fixed point set of σ i , splitting type Examples U = K i + P i : the Cartan decomp. for U . Twisted K i hierarchies of a non-splitting type Assume Twisted flows of a non-splitting type K 1 ∩ K 2 = S 1 × S 2 , The GMV equation Inverse K 1 = S 1 × K ′ K 2 = K ′ 1 , 2 × S 2 scattering theory Twisted flows of a as direct product of subgroups. splitting type Twisted flows of a non-splitting type Open problems References MATH, Academia Sinica, Taiwan
Twisted flows of a splitting type Integrable twisted hierarchies Derchyi Wu Define the loop groups by Motivation The Adler-Kostant- Symes holo . L σ 1 = { f : A ǫ, 1 → U | σ 1 ( f ( − λ )) = f ( λ ) } , Theorem ǫ Twisted � � f ( 1 hierarchies of a L σ 1 { f ∈ L σ 1 | σ 2 = f ( λ ) , f ( 1 ) ∈ K ′ = λ ) 2 } , splitting type + Twisted flows of a splitting type holo . L σ 1 { f ∈ L σ 1 , f : C / D ǫ → U | f ( ∞ ) ∈ K ′ = 1 } . Examples − Twisted hierarchies of a Thus we have a direct sum decomposition of the loop algebra non-splitting type L σ 1 = L σ 1 + + L σ 1 − . Twisted flows of a non-splitting type The GMV equation Let Inverse π ± : L σ 1 → L σ 1 ˆ scattering ± theory be the corres. projections. 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 Integrable twisted hierarchies Finally, let Derchyi Wu A be a CSA ⊂ P 1 , σ 2 ( A ) ⊂ A , Motivation The Adler-Kostant- { a 1 , . . . , a n } a basis of A , Symes Theorem J i , 2 j + 1 = a i λ 2 j + 1 + σ 2 ( a i ) λ − ( 2 j + 1 ) ∈ L + ∩ A . Twisted hierarchies of a splitting type Twisted flows of a 2.1.2 Definition : the twisted U splitting type K 1 -hierarchy (splitting type) Examples a ∈ A , the 2 j + 1-th twisted U Given a , ˜ K 1 -flow is the compatibility condition Twisted hierarchies of a non-splitting � � MJ a , 1 M − 1 � � a , 2 j + 1 M − 1 �� type ∂ x + ˆ π + , ∂ t + ˆ π + MJ ˜ = 0 , Twisted flows of a non-splitting type The GMV equation for some M = M ( x , t , λ ) ∈ L σ 1 − . Inverse → a nonlinear partial differential system in b , v , where scattering theory � MJ a , 1 M − 1 � = bab − 1 λ + v + σ 2 ( bab − 1 ) 1 Twisted flows of a splitting type π + ˆ λ. Twisted flows of a non-splitting type Open problems References MATH, Academia Sinica, Taiwan
Examples Integrable twisted hierarchies Derchyi Wu 2.2.1 Splittings of loop groups Let the twisted U K 1 -hierarchy (splitting type) be defined by Motivation The Adler-Kostant- Symes � � Theorem x ∈ GL 2 n ( R ) | x t ˜ Jx = ˜ U = O ( J , J ) = J , Twisted hierarchies of a { ξ ∈ gl 2 n ( R ) | ξ t ˜ J + ˜ U = o ( J , J ) = J ξ = 0 } , splitting type Twisted flows of a splitting type w. Examples q times n − q times � J Twisted � � �� � � �� � 0 hierarchies of a ˜ J = I q , n − q = diag ( − 1 , · · · , − 1 , 1 , . . . , 1 ) , J = , non-splitting 0 − J type Twisted flows of a σ i ( x ) = I n + i , n − i xI − 1 n + i , n − i , i = 0 , 1 non-splitting type The GMV equation and � 0 Inverse � scattering D A = { ξ = , D diagonal } ⊂ P 0 theory D 0 Twisted flows of a splitting type be a maximal abelian subalgebra. Twisted flows of a non-splitting type Open problems References MATH, Academia Sinica, Taiwan
Examples Integrable twisted hierarchies Derchyi Wu O ( J , J ) Motivation 2.2.2 For ( n , q ) = ( 2 , 0 ) , a = ˜ a , the first twisted O ( J ) × O ( J ) -flow is the trivial The Adler-Kostant- Symes linear system Theorem ∂ t u = ∂ x u , ∂ t ω = ∂ x ω, Twisted hierarchies of a where splitting type Twisted flows of a splitting type 1 0 0 0 Examples 0 1 0 0 ∈ K ′ b = 0 , Twisted 0 0 cos u ( x , t ) sin u ( x , t ) hierarchies of a non-splitting 0 0 − sin u ( x , t ) cos u ( x , t ) type Twisted flows of a 0 − ω ( x , t ) 0 0 non-splitting type ω ( x , t ) 0 0 0 The GMV equation v = ∈ S 0 , 0 0 0 0 Inverse scattering 0 0 0 0 theory Twisted flows of a splitting type Twisted flows of a non-splitting type Open problems References MATH, Academia Sinica, Taiwan
Examples Integrable twisted hierarchies Derchyi Wu O ( J , J ) 2.2.3 For ( n , q ) = ( 2 , 0 ) , a = ˜ a , the third twisted O ( J ) × O ( J ) -flow is the 4th Motivation The Adler-Kostant- order partial differential system Symes Theorem 1 � 5 ( ∂ x u ) 2 − 12 ω∂ x u + 180 ( cos u ) 2 − 90 + 15 ω 2 � Twisted 18 { 10 ∂ 3 ∂ t u = x u + ∂ x u hierarchies of a splitting type � 24 − 48 ( cos u ) 2 � Twisted flows of a x ω − 4 ω 3 + ω − 8 ∂ 2 } , splitting type Examples − 4 x u [ − 2 3 ( ∂ x u ) 2 + 5 3 ω∂ x u − 40 3 ( cos u ) 2 + 20 3 − 2 9 ∂ 4 x u + ∂ 2 3 ω 2 ] Twisted ∂ t ω = hierarchies of a non-splitting − 32 sin u ( cos u ) 3 + 16 cos u sin u type Twisted flows of a non-splitting type + ∂ x u [ 40 3 ( ∂ x u ) cos u sin u + 5 6 ( ∂ x u ) ( ∂ x ω ) − 4 3 ω∂ x ω ] The GMV equation Inverse + 5 x ω + ∂ x ω [ 5 6 ω 2 − 5 + 10 ( cos u ) 2 ] − 8 scattering 3 ω 2 cos u sin u . 9 ∂ 3 theory Twisted flows of a splitting type Twisted flows of a non-splitting type Open problems References MATH, Academia Sinica, Taiwan
Recommend
More recommend
