Soliton hierarchies and matrix loop algebras Wen-Xiu Ma Department - PowerPoint PPT Presentation

Overview Generating scheme and symmetry algebra sl(2 , R )-soliton hierarchies so(3 , R )-soliton hierarchies Concluding remarks Soliton hierarchies and matrix loop algebras Wen-Xiu Ma Department of Mathematics and Statistics University of South Florida, USA (35th Workshop on Geometric Methods in Physics, Bialowieza, Poland; 26 June - 2 July, 2016)

Overview Generating scheme and symmetry algebra sl(2 , R )-soliton hierarchies so(3 , R )-soliton hierarchies Concluding remarks Outline Overview 1 Generating scheme and symmetry algebra 2 sl(2 , R )-soliton hierarchies 3 so(3 , R )-soliton hierarchies 4 Concluding remarks 5

Overview Generating scheme and symmetry algebra sl(2 , R )-soliton hierarchies so(3 , R )-soliton hierarchies Concluding remarks Outline Overview 1 Generating scheme and symmetry algebra 2 sl(2 , R )-soliton hierarchies 3 so(3 , R )-soliton hierarchies 4 Concluding remarks 5

Overview Generating scheme and symmetry algebra sl(2 , R )-soliton hierarchies so(3 , R )-soliton hierarchies Concluding remarks Outline Overview 1 Generating scheme and symmetry algebra 2 sl(2 , R )-soliton hierarchies 3 so(3 , R )-soliton hierarchies 4 Concluding remarks 5

Overview Generating scheme and symmetry algebra sl(2 , R )-soliton hierarchies so(3 , R )-soliton hierarchies Concluding remarks Outline Overview 1 Generating scheme and symmetry algebra 2 sl(2 , R )-soliton hierarchies 3 so(3 , R )-soliton hierarchies 4 Concluding remarks 5

Overview Generating scheme and symmetry algebra sl(2 , R )-soliton hierarchies so(3 , R )-soliton hierarchies Concluding remarks Outline Overview 1 Generating scheme and symmetry algebra 2 sl(2 , R )-soliton hierarchies 3 so(3 , R )-soliton hierarchies 4 Concluding remarks 5

Overview Generating scheme and symmetry algebra sl(2 , R )-soliton hierarchies so(3 , R )-soliton hierarchies Concluding remarks Overview 1 Generating scheme and symmetry algebra 2 sl(2 , R )-soliton hierarchies 3 so(3 , R )-soliton hierarchies 4 Concluding remarks 5

Overview Generating scheme and symmetry algebra sl(2 , R )-soliton hierarchies so(3 , R )-soliton hierarchies Concluding remarks Soliton equations u t = Φ n K 0 [ u ] ⇔ φ x = U ( u , λ ) φ or E φ = U ( u , λ ) φ ✓ ✏ u t = K 0 [ u ] ⇔ U t − V x + [ U , V ] = 0 � � or U t + UV − ( EV ) U = 0 ✒ ✑ spectral matrix U ⇔ recursion operator Φ

Overview Generating scheme and symmetry algebra sl(2 , R )-soliton hierarchies so(3 , R )-soliton hierarchies Concluding remarks KdV equation The KdV equation: u t − 3 2 b 1 uu x − 1 4 b 1 u xxx = 0 . Lax Pair: � � 0 1 U = , λ − u 0 − 1 λ + 1   4 u x 2 u V = b 1  .   − ( λ + 1 2 u )( u − λ ) − 1 1  4 u xx 4 u x

Overview Generating scheme and symmetry algebra sl(2 , R )-soliton hierarchies so(3 , R )-soliton hierarchies Concluding remarks NLS equations The nonlinear Schr¨ odinger equations: � p t = − 1 2 p xx + p 2 q , q t = 1 2 q xx − pq 2 . Lax Pair: � � − λ p U = , q λ − λ 2 + 1 λ p − 1 � � 2 pq 2 p x V = . λ 2 − 1 λ q + 1 2 q x 2 pq

Overview Generating scheme and symmetry algebra sl(2 , R )-soliton hierarchies so(3 , R )-soliton hierarchies Concluding remarks Symmetry and conservation law Symmetry: S is called a symmetry of u t = K ( u ), if [ K , S ] = K ′ [ S ] − S ′ [ K ] = 0 , P ′ [ S ] = ∂ � ∂ε P ( u + ε S ) ε =0 . � � This defines a commuting flow with u t = K ( u ). Conservation law: A conservation law is ∂ t T + ∂ x X = 0 when u t = K ( u ) . This gives a conserved density: � d T dx = 0 when u t = K ( u ) . dt

Overview Generating scheme and symmetry algebra sl(2 , R )-soliton hierarchies so(3 , R )-soliton hierarchies Concluding remarks The fundamental question Question: How to generate soliton equations with infinitely many symmetries and/or conservation laws? Starting point: Spectral problems on matrix loop algebras

Overview Generating scheme and symmetry algebra sl(2 , R )-soliton hierarchies so(3 , R )-soliton hierarchies Concluding remarks Overview 1 Generating scheme and symmetry algebra 2 sl(2 , R )-soliton hierarchies 3 so(3 , R )-soliton hierarchies 4 Concluding remarks 5

Overview Generating scheme and symmetry algebra sl(2 , R )-soliton hierarchies so(3 , R )-soliton hierarchies Concluding remarks Spectral problems Let g be a semisimple Lie algebra and its loop algebra g = g ⊗ C [ λ, λ − 1 ] . ˜ Choose a peudoregular element e 0 ( λ ): ( a) Ker ( ad e 0 ) ⊕ Im ( ad e 0 ) = ˜ g , ( b ) Ker ( ad e 0 ) is commutative . Spectral problem with linearly independent e i ∈ ˜ g , 0 ≤ i ≤ q : φ x = U φ, U = U ( λ, u ) = e 0 ( λ ) + u 1 e 1 ( λ ) + · · · + u q e q ( λ ) .

Overview Generating scheme and symmetry algebra sl(2 , R )-soliton hierarchies so(3 , R )-soliton hierarchies Concluding remarks Zero curvature equations Solve the stationary zero curvature equation � V i λ − i . V x = [ U , V ] , V = i ≥ 0 Select ∆ n so that V ( n ) = ( λ n V ) + + ∆ n where + means to take the polynomial part, satisfies V ( n ) − [ U , V ( n ) ] ∈ span ( e 1 , · · · , e q ) . x

Overview Generating scheme and symmetry algebra sl(2 , R )-soliton hierarchies so(3 , R )-soliton hierarchies Concluding remarks Zero curvature equations - P.D. Lax, Comm. Pure Appl. Math. , 21 (1968), 467-490. Lax pairs: U , V ( n ) or φ x = U φ, φ t n = V ( n ) φ. Zero curvature equations U t n − V ( n ) + [ U , V ( n ) ] = 0 x present a soliton hierarchy u t n = K n ( u ) , n ≥ 0 .

Overview Generating scheme and symmetry algebra sl(2 , R )-soliton hierarchies so(3 , R )-soliton hierarchies Concluding remarks Algebraic structure of Lax operators - W.X. Ma, J. Phys. A , 25 (1992), 5329-5343; 26 (1993), 2573-2582. An evolution equation u t = K ( u ) � U ′ [ K ] + f ( λ ) U λ − V x + [ U , V ] = 0 � φ x = U ( u , λ ) φ, φ t = V ( u , λ ) φ under λ t = f ( λ ).

Overview Generating scheme and symmetry algebra sl(2 , R )-soliton hierarchies so(3 , R )-soliton hierarchies Concluding remarks Commutators Introduce [ K , S ] = K ′ [ S ] − S ′ [ K ] , ] = V ′ [ S ] − W ′ [ K ] + [ V , W ] + gV λ − fW λ , [ [ V , W ] ]( λ ) = f ′ ( λ ) g ( λ ) − f ( λ ) g ′ ( λ ) , [ [ f , g ] where P ′ [ S ] = ∂ � ∂ǫ P ( u + ǫ S ) ǫ =0 . � �

Overview Generating scheme and symmetry algebra sl(2 , R )-soliton hierarchies so(3 , R )-soliton hierarchies Concluding remarks Algebraic structure - W.X. Ma, J. Phys. A , 26 (1993), 2573-2582. If U ′ [ K ] + f ( λ ) U λ − V x + [ U , V ] = 0 , U ′ [ S ] + g ( λ ) U λ − W x + [ U , W ] = 0 , then U ′ [ [ K , S ] ] + [ [ f , g ] ]( λ ) U λ − [ [ V , W ] ] x + [ U , [ [ V , W ] ]] = 0 .

Overview Generating scheme and symmetry algebra sl(2 , R )-soliton hierarchies so(3 , R )-soliton hierarchies Concluding remarks Lie algebraic structure - W.X. Ma, British J. Appl. Sci. Tech. , 3 (2013), 1336-1344. All ( K , V , f ) form a Lie algebra under the binary operation: [ [( K , V , f ) , ( S , W , g )] ] = ([ K , S ] , [ [ V , W ] ] , [ [ f , g ] ]) . That is, the above operation satisfies Bilinearity: [ [ α ( K 1 , V 1 , f 1 ) + β ( K 2 , V 2 , f 2 ) , ( K 3 , V 3 , f 3 )] ] = α [ [( K 1 , V 1 , f 1 ) , ( K 3 , V 3 , f 3 )] ] + β [ [( K 2 , V 2 , f 2 ) , ( K 3 , V 3 , f 3 )] ] . Anticommutativity: [ [( K 1 , V 1 , f 1 ) , ( K 2 , V 2 , f 2 )] ] = − [ [( K 2 , V 2 , f 2 ) , ( K 1 , V 1 , f 1 )] ] . The Jacobi Identity: [ [( K 1 , V 1 , f 1 ) , [ [( K 2 , V 2 , f 2 ) , ( K 3 , V 3 , f 3 )] ]] ] + cycle (1 , 2 , 3) = 0 .

Overview Generating scheme and symmetry algebra sl(2 , R )-soliton hierarchies so(3 , R )-soliton hierarchies Concluding remarks Symmetry algebras Symmetry algebras in (1+1)-dimensions: [ K m , K n ] = 0 , [ K n , τ s , m ] = ( m + γ + 1) K m + n , [ τ s , n , τ s , m ] = ( m − n ) τ s , m + n , where τ s , m = σ m +1 + t [ K s , σ m +1 ] . Graded symmetry algebras in higher-dimensions: � g = g i i ∈ Z for KP, MKP, etc.

Overview Generating scheme and symmetry algebra sl(2 , R )-soliton hierarchies so(3 , R )-soliton hierarchies Concluding remarks Trace and variational identities Semisimple Lie algebras: δ � tr ( V ∂ U ∂λ ) dx = λ − γ ∂ ∂λλ γ tr ( V ∂ U ∂ u ) , γ = − λ d d λ ln | tr ( V 2 ) | . δ u 2 Non-semisimple Lie algebras: δ � � V , ∂ U ∂λ � dx = λ − γ ∂ ∂λλ γ � V , ∂ U ∂ u � , γ = − λ d d λ ln � V , V � , 2 δ u where �· , ·� is an ad-invariant, symmetric and non-degenerate bilinear form.

Overview Generating scheme and symmetry algebra sl(2 , R )-soliton hierarchies so(3 , R )-soliton hierarchies Concluding remarks Overview 1 Generating scheme and symmetry algebra 2 sl(2 , R )-soliton hierarchies 3 so(3 , R )-soliton hierarchies 4 Concluding remarks 5

Overview Generating scheme and symmetry algebra sl(2 , R )-soliton hierarchies so(3 , R )-soliton hierarchies Concluding remarks Lie algebra sl(2 , R ) sl(2 , R ): [ e 1 , e 2 ] = 2 e 2 , [ e 1 , e 3 ] = − 2 e 3 , [ e 2 , e 3 ] = e 1 , where � � � � � � 1 0 0 1 0 0 e 1 = , e 2 = , e 3 = . 0 − 1 0 0 1 0