Bi-Hamiltonian flows and their geometric realizations Gloria Mar´ ı Beffa February, 2010 Gloria Mar´ ı Beffa Bi-Hamiltonian flows and their geometric realizations
Assume u : J ⊂ R 2 → R P 1 is a solution of the Schwarzian KdV evolution u 2 u t = u x S ( u ) = u xxx − 3 xx 2 u x � 2 � where S ( u ) = u xxx u x − 3 u xx is the Schwarzian derivative of u . 2 u x Gloria Mar´ ı Beffa Bi-Hamiltonian flows and their geometric realizations
Assume u : J ⊂ R 2 → R P 1 is a solution of the Schwarzian KdV evolution u 2 u t = u x S ( u ) = u xxx − 3 xx 2 u x � 2 � where S ( u ) = u xxx u x − 3 u xx is the Schwarzian derivative of u . 2 u x Then S ( u ) t = S ( u ) xxx + 3 S ( u ) x S ( u ) is a solution of the KdV equation. Gloria Mar´ ı Beffa Bi-Hamiltonian flows and their geometric realizations
Assume u : J ⊂ R 2 → R P 1 is a solution of the Schwarzian KdV evolution u 2 u t = u x S ( u ) = u xxx − 3 xx 2 u x � 2 � where S ( u ) = u xxx u x − 3 u xx is the Schwarzian derivative of u . 2 u x Then S ( u ) t = S ( u ) xxx + 3 S ( u ) x S ( u ) is a solution of the KdV equation. We say the Schwarzian KdV flow is a RP 1 geometric realization of the KdV flow. Gloria Mar´ ı Beffa Bi-Hamiltonian flows and their geometric realizations
Assume u : J ⊂ R 2 → R 3 is a solution of the Vortex filament flow evolution u t = κ B where κ is the curvature of the flow u and B is the binormal. Gloria Mar´ ı Beffa Bi-Hamiltonian flows and their geometric realizations
Assume u : J ⊂ R 2 → R 3 is a solution of the Vortex filament flow evolution u t = κ B where κ is the curvature of the flow u and B is the binormal. Then, curvature and torsion of the flow satisfy an equation equivalent to the Nonlinear Schr¨ odinger equation. If φ t = i φ xx + i � φ = κ e i τ dx , | 2 φ 2 | | φ | (Hasimoto, 72) Gloria Mar´ ı Beffa Bi-Hamiltonian flows and their geometric realizations
Assume u : J ⊂ R 2 → R 3 is a solution of the Vortex filament flow evolution u t = κ B where κ is the curvature of the flow u and B is the binormal. Then, curvature and torsion of the flow satisfy an equation equivalent to the Nonlinear Schr¨ odinger equation. If φ t = i φ xx + i � φ = κ e i τ dx , | 2 φ 2 | | φ | (Hasimoto, 72) We say the Vortex Filament flow is a 3-dimensional Euclidean geometric realization of the NLS equation. Gloria Mar´ ı Beffa Bi-Hamiltonian flows and their geometric realizations
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Geometric realizations of soliton solutions have interesting properties and behavior Gloria Mar´ ı Beffa Bi-Hamiltonian flows and their geometric realizations
(Loading movie...) Gloria Mar´ ı Beffa Bi-Hamiltonian flows and their geometric realizations
(Loading movie...) Gloria Mar´ ı Beffa Bi-Hamiltonian flows and their geometric realizations
Some other geometric realizations of completely integrable systems are: ◮ The Sine-Gordon and modified KdV equations have both an Euclidean 3-dimensional realization. Gloria Mar´ ı Beffa Bi-Hamiltonian flows and their geometric realizations
Some other geometric realizations of completely integrable systems are: ◮ The Sine-Gordon and modified KdV equations have both an Euclidean 3-dimensional realization. ◮ The Boussinesq equation can be realized as a flow of projective curves in RP 2 . Gloria Mar´ ı Beffa Bi-Hamiltonian flows and their geometric realizations
Some other geometric realizations of completely integrable systems are: ◮ The Sine-Gordon and modified KdV equations have both an Euclidean 3-dimensional realization. ◮ The Boussinesq equation can be realized as a flow of projective curves in RP 2 . ◮ The Adler-Gelfand-Dikii flows, or generalized n -dimensional KdV systems, have geometric realizations in RP n , for all n . Gloria Mar´ ı Beffa Bi-Hamiltonian flows and their geometric realizations
Some other geometric realizations of completely integrable systems are: ◮ The Sine-Gordon and modified KdV equations have both an Euclidean 3-dimensional realization. ◮ The Boussinesq equation can be realized as a flow of projective curves in RP 2 . ◮ The Adler-Gelfand-Dikii flows, or generalized n -dimensional KdV systems, have geometric realizations in RP n , for all n . ◮ The Sawada Kotera equation can be realized as a flow in equi-affine plane and also as a projective flow in RP 1 . Gloria Mar´ ı Beffa Bi-Hamiltonian flows and their geometric realizations
Some other geometric realizations of completely integrable systems are: ◮ The Sine-Gordon and modified KdV equations have both an Euclidean 3-dimensional realization. ◮ The Boussinesq equation can be realized as a flow of projective curves in RP 2 . ◮ The Adler-Gelfand-Dikii flows, or generalized n -dimensional KdV systems, have geometric realizations in RP n , for all n . ◮ The Sawada Kotera equation can be realized as a flow in equi-affine plane and also as a projective flow in RP 1 . ◮ The complexly coupled system of KdV equations has realizations as a conformal flow on the sphere S 2 , as a flow of star-shaped curves in the light cone of Lorentzian R 4 and as a flow in the 2-Grassmannian in R 4 . Gloria Mar´ ı Beffa Bi-Hamiltonian flows and their geometric realizations
Some other geometric realizations of completely integrable systems are: ◮ The Sine-Gordon and modified KdV equations have both an Euclidean 3-dimensional realization. ◮ The Boussinesq equation can be realized as a flow of projective curves in RP 2 . ◮ The Adler-Gelfand-Dikii flows, or generalized n -dimensional KdV systems, have geometric realizations in RP n , for all n . ◮ The Sawada Kotera equation can be realized as a flow in equi-affine plane and also as a projective flow in RP 1 . ◮ The complexly coupled system of KdV equations has realizations as a conformal flow on the sphere S 2 , as a flow of star-shaped curves in the light cone of Lorentzian R 4 and as a flow in the 2-Grassmannian in R 4 . ◮ Both decoupled systems of KdV equations and matrix mKdV equations have realizations as flows of Lagrangian planes, as flows of 2-Grassmannians in R 4 and as flows of even dimensional spinors. Gloria Mar´ ı Beffa Bi-Hamiltonian flows and their geometric realizations
Anco, Calini & Ivey, Doliwa & Santini, Langer & Perline, Mar´ ı-Beffa, Olver, Pinkall, Qu, Sanders & Wang, Sym, Terng & Thorbergsson, Yasui & Sasaki. Gloria Mar´ ı Beffa Bi-Hamiltonian flows and their geometric realizations
Anco, Calini & Ivey, Doliwa & Santini, Langer & Perline, Mar´ ı-Beffa, Olver, Pinkall, Qu, Sanders & Wang, Sym, Terng & Thorbergsson, Yasui & Sasaki. What do these completely integrable systems have in common? Gloria Mar´ ı Beffa Bi-Hamiltonian flows and their geometric realizations
Anco, Calini & Ivey, Doliwa & Santini, Langer & Perline, Mar´ ı-Beffa, Olver, Pinkall, Qu, Sanders & Wang, Sym, Terng & Thorbergsson, Yasui & Sasaki. What do these completely integrable systems have in common? First of all, all the geometric realizations live in homogeneous spaces , that is geometric manifolds of the form G / H with G a Lie group and H ⊂ G a closed subgroup. Gloria Mar´ ı Beffa Bi-Hamiltonian flows and their geometric realizations
Anco, Calini & Ivey, Doliwa & Santini, Langer & Perline, Mar´ ı-Beffa, Olver, Pinkall, Qu, Sanders & Wang, Sym, Terng & Thorbergsson, Yasui & Sasaki. What do these completely integrable systems have in common? First of all, all the geometric realizations live in homogeneous spaces , that is geometric manifolds of the form G / H with G a Lie group and H ⊂ G a closed subgroup. They are also all Bi-Hamiltonian systems. That is, they can be written as k t = P i δ H i , i = 1 , 2 for two choices of Hamiltonian structures P i (described by differential operators), and two choices of Hamiltonian operators H i . Gloria Mar´ ı Beffa Bi-Hamiltonian flows and their geometric realizations
Anco, Calini & Ivey, Doliwa & Santini, Langer & Perline, Mar´ ı-Beffa, Olver, Pinkall, Qu, Sanders & Wang, Sym, Terng & Thorbergsson, Yasui & Sasaki. What do these completely integrable systems have in common? First of all, all the geometric realizations live in homogeneous spaces , that is geometric manifolds of the form G / H with G a Lie group and H ⊂ G a closed subgroup. They are also all Bi-Hamiltonian systems. That is, they can be written as k t = P i δ H i , i = 1 , 2 for two choices of Hamiltonian structures P i (described by differential operators), and two choices of Hamiltonian operators H i .Furthermore, the Hamiltonian structures are compatible, that is, P 1 + P 2 is also a Hamiltonian structure. Gloria Mar´ ı Beffa Bi-Hamiltonian flows and their geometric realizations
Are the Hamiltonian structures linked to the background geometry of the realization? Gloria Mar´ ı Beffa Bi-Hamiltonian flows and their geometric realizations
Are the Hamiltonian structures linked to the background geometry of the realization? Not only they are linked, the background geometry generates them, determine many of their properties and guarantees geometric realizations. Gloria Mar´ ı Beffa Bi-Hamiltonian flows and their geometric realizations
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