Integrable Flows for Starlike Curves in Centroaffine Space A. Calini - PowerPoint PPT Presentation

Integrable Flows for Starlike Curves in Centroaffine Space A. Calini 1 T. Ivey 1 ı-Beffa 2 G. Mar´ 1 College of Charleston 2 University of Wisconsin, Madison VDM60: Nonlinear Evolution Equations and Linear Algebra Cagliari, Sardinia, Italy. September 2-5, 2013

Outline Introduction Starlike curves in centroaffine R 3 Hamiltonian structure on starlike loops Bi-hamiltonian curve flows Connection with the Boussinesq hierarchy

Introduction For integrable PDE describing geometric evolutions of curves, the differential invariants ( curvatures ) play a fundamental role in revealing integrability. Example: Vortex Filament Flow γ ( x , t ) ∈ R 3 : position vector of an evolving space curve. x : arclength parameter. κ, τ : differential invariants (curvature and torsion). binormal evolution cubic focusing NLS Hasimoto Map i q t + q xx + 2 | q | 2 q = 0 γ t = γ x × γ xx = κ B ✲ � τ dx q = 1 2 κ e i This work: investigates the relation between a non-stretching curve evolution in centro-affine space and the completely integrable PDE system for the differential invariants, by seeking a natural Hamiltonian formulation of the curve flow.

Inspiration: Hamiltonian setting for the Vortex Filament Flow Pre-symplectic structure on the space of curves: � | X , γ ′ , Y | d x . ω γ ( X , Y ) = (Marsden & Weinstein, 1983) γ A geometric recursion operator X j + 1 = − T × X ′ j plus the non-stretching condition generate an infinite hierarchy (Langer & Perline, 1991): � Commuting flows : γ t = X j ρ j d x Conserved integrals : ρ 0 = � T � , X 1 = κ B ρ 1 = − τ, X 2 = 1 2 κ 2 T + κ ′ N + κτ B , 2 κ 2 , ρ 2 = 1 X 3 = κ 2 τ T + ( 2 κ ′ τ + κτ ′ ) N 2 κ 2 τ, ρ 3 = 1 + ( κτ 2 − κ ′′ − 1 2 (( κ ′ ) 2 + κ 2 τ 2 ) − 1 2 κ 3 ) B 8 κ 4 , ρ 4 = 1 . . . . . . � where X j + 1 is the Hamiltonian vector field for ρ j d x

Starlike curves in centroaffine R 3 – The “isometry group” is SL ( 3 , R ) without translations. – A smooth curve γ : I ⊂ R → R 3 is starlike if | γ, γ ′ , γ ′′ | � = 0. This condition is invariant under the linear action of SL ( 3 , R ) . � | γ, γ ′ , γ ′′ | 1 / 3 d x . A curve γ is – Define centroaffine arclength arclength parametrized if | γ, γ ′ , γ ′′ | = 1 . (1) From now on, let γ ( x ) be a starlike curve with arclength parameter x . Differentiating (1) with respect to x gives | γ, γ ′ , γ ′′′ | = 0, implying γ ′′′ = p 0 γ + p 1 γ ′ . p 0 , p 1 are the differential invariants ( Wilczynski invariants ) of γ , and compare with Euclidean torsion and curvature (resp.): – If p 0 = 0 then γ is planar. – If p 0 = 1 2 p ′ 1 then γ lies on a conic in RP 2 .

Hamiltonian structure on starlike loops M ⊂ Map ( S 1 , R 3 ) be the space of starlike loops γ : S 1 → R 3 Let � parametrized by centroaffine arclength. The vector field X = δ X γ = a γ + b γ ′ + c γ ′′ ( a , b , c smooth, periodic) is in T γ � M (i.e. is non-stretching ) if δ X | γ, γ ′ , γ ′′ | = 0: a + b ′ + 1 3 ( c ′′ + 2 p 1 c ) = 0 . For γ ∈ � M , the closed skew-symmetric 2-form � X , Y ∈ T γ � | X , γ ′ , Y | d x , ω γ ( X , Y ) = M γ gives pre-symplectic structure on � M with 2-dimensional kernel spanned by Z 1 = γ ′′ − 2 Z 0 = γ ′ , 3 p 1 γ.

Hamiltonian vector fields: examples Using the correspondence between Hamiltonians H ∈ C ∞ ( � M ) and Hamiltonian vector fields X H : ∀ X ∈ T γ � d H [ X ] = ω γ ( X , X H ) , M (2) find that: 1. Z 1 = γ ′′ − 2 3 p 1 γ is the Hamiltonian vector field for � | γ, γ ′ , γ ′′ | 1 / 3 d x (total arclength); γ 2. Z 2 = p 1 γ ′′ + ( p 0 − p ′ 1 ) γ ′ + ( 2 3 ( p ′′ 1 − p 2 1 ) − p ′ 0 ) γ is the � Hamiltonian vector field for ( − p 1 ) d x (minus the total γ curvature). Remark: Correspondence (2) is not an isomorphism. For those H ’s for which X H exists, X H is defined up to addition of elements in the kernel of ω γ .

General curvature evolutions 1 . (Then Z 2 = k 1 γ ′′ + k 2 γ ′ + . . . ). Switch to k 1 = p 1 , k 2 = p 0 − p ′ Let γ t = r 0 γ + r 1 γ ′ + r 2 γ ′′ be a general non-stretching flow (i.e. with 1 − 1 r 0 = − r ′ 3 ( r ′′ 2 + 2 k 1 r 2 ) .) Then, the differential invariants evolve by � k 1 � � r 1 � = P , k 2 r 2 t where P is a skew-adjoint 5th order matrix differential operator: � � − 2 D 3 + Dk 1 + k 1 D − D 4 + D 2 k 1 + 2 Dk 2 + k 2 D , D 4 − k 1 D 2 + 2 k 2 D + Dk 2 3 ( D 5 + k 1 Dk 1 − k 1 D 3 − D 3 k 1 ) + [ k 2 , D 2 ] 2 with D = D x . Remark: P plays a key role in the integrability of γ t = Z 1 .

Bi-hamiltonian formulation The curvature evolution induced by γ t = Z 1 = γ ′′ − 2 3 k 1 γ can be written in Hamiltonian form in two distinct ways: � k 1 � � k 1 � = P E ρ 1 , = Q E ρ 3 , ( ‡ ) k 2 k 2 t t where 1 ) 2 + k 2 k ′ 1 + ( k 2 ) 2 + 1 3 ( k ′ 9 k 3 ρ 3 = 1 ρ 1 = k 2 , 1 are conserved densities, E is the Euler operator � � � T � ( − D ) j ∂ f ( − D ) j ∂ f E f = , , ∂ k ( j ) ∂ k ( j ) j ≥ 0 j ≥ 0 1 2 � 0 � D and Q = . D 0 Since P , Q form a compatible pair of Hamiltonian operators (related to the Adler-Gel’fand-Dikii bracket for sl ( 3 ) and its companion), ( ‡ ) is a bi-Hamiltonian system.

A double hierarchy: Recursion Operators The curvature evolution induced by γ t = Z 2 = k 1 γ ′′ + κ 2 γ ′ + . . . is also bi-Hamiltonian for P and Q , with respect to the densities: ρ 4 = 1 3 ( k ′′ 1 ) 2 + k ′′ 1 ( k ′ 2 − k 2 1 ) − k 1 ( k ′ 1 ) 2 +( k ′ 2 ) 2 − k 2 1 k ′ 2 + 1 9 k 4 1 + 2 k 1 k 2 ρ 2 = k 1 k 2 , 2 . Define a sequence of evolution equations for k 1 , k 2 � k 1 � ∂ = F j [ k 1 , k 2 ] , ∂ t j k 2 via the recursion F j + 2 = PQ − 1 F j , with initial data given by � k ′ � � � k ′′ 1 + 2 k ′ 1 2 F 0 = , F 1 = , 2 k ′ 3 ( k 1 k ′ 1 − k ′′′ 1 ) − k ′′ 2 2 and a sequence of conserved densities given by E ρ j + 2 = Q − 1 P E ρ j , ρ 0 = k 1 , ρ 1 = k 2 . with

Connection with the Boussinesq hierarchy The curvature evolution induced by γ t = Z 1 : � k 1 � � � k ′′ 1 + 2 k ′ ∂ 2 = , 2 3 ( k 1 k ′ 1 − k ′′′ 1 ) − k ′′ k 2 ∂ t 2 is equivalent to the Boussinesq equation � q 0 � � − 1 � 1 − 2 ∂ 6 q ′′′ 3 q 1 q ′′ = 1 2 q ′ ∂ t q 1 0 under the change of variables k 1 = − q 1 , k 2 = 1 2 q ′ 1 − q 0 . (See also Chou & Qu, 2002.) We show: ◮ The curvature evolution induced by γ t = Z 2 is equivalent to the second nontrivial flow in the Boussinesq hierarchy; ◮ The recursion operator PQ − 1 is equivalent to the Boussinesq recursion operator.

Relation with centroaffine curve flows Theorem: Each flow of the Boussinesq hierarchy is the curvature evolution induced by a geometric flow for centroaffine curves in R 3 . Proof: Define X ρ j := Z j = ( E ρ j ) 1 γ ′ + ( E ρ j ) 2 γ ′′ + r 0 γ, ( r 0 given by the non-stretching condition), with ρ j the j -th Boussinesq conserved density. � k 1 � Then, γ t = Z j induces the curvature evolution = F j . k 2 t � γ ( − ρ j ) d x and E ρ j + 2 = Q − 1 P E ρ j (the next Theorem: Let H ( γ ) = density after ρ j in the Boussinesq hierarchy). Then ∀ X ∈ T γ � d H [ X ] = ω γ ( X , Z j + 2 ) M . That is, γ t = Z j , j ≥ 2 is a Hamiltonian evolution with Hamiltonian � γ ( − ρ j − 2 ) d x .

Summary Non-stretching vector fields Conserved densities Z 0 = γ ′ ρ 0 = k 1 Z 1 = γ ′′ − 2 3 k 1 γ ρ 1 = k 2 Z 2 = k 1 γ ′′ + k 2 γ ′ + . . . ρ 2 = k 1 k 2 � 1 � 1 + 2 k 2 ) γ ′′ + γ ′ + . . . 1 ) 2 + k ′ Z 3 = ( k ′ 3 k 2 1 − 2 3 k ′′ 1 − k ′ ρ 3 = 1 3 ( k ′ 1 k 2 + 1 9 k 3 1 + k 2 2 2 1 + 4 k 1 k 2 ) γ ′′ + ( 2 1 ) 2 + k ′′ 3 k ( 4 ) Z 4 = ( − k ′′′ 1 − 2 k ′′ 2 + 2 k 1 k ′ + k ′′′ ρ 4 = 1 3 ( k ′′ 1 ( k ′ 2 − k 2 1 ) − k 1 ( k ′ 1 ) 2 1 2 1 ) 2 − 2 k 1 k ′ 2 ) γ ′ + . . . 2 ) 2 − k 2 − 2 k 1 k ′′ 1 − ( k ′ 2 + 4 9 k 3 1 + 2 k 2 +( k ′ 1 k ′ 2 + 1 9 k 4 1 + 2 k 1 k 2 2 ◮ The γ ′ and γ ′′ coefficients of Z j match the components of E ρ j . ◮ Densities satisfy the recursion relation E ρ j + 2 = Q − 1 P E ρ j � k 1 � ◮ γ t = Z j induces curvature evolution = P E ρ j = Q E ρ j + 2 . k 2 t � ◮ For j ≥ 2, Z j is a Hamiltonian vector field for − ρ j − 2 d x .

An interesting sub-hierarchy Example: The Vortex Filament Flow hierarchy has integrable sub-hierarchies preserving geometric invariants, e.g.: ◮ Under the even flows X 2 j , planar curves remain planar. ◮ For each constant τ 0 , there is a sequence of linear combinations of the X j that preserves the constant torsion condition τ = τ 0 . (These flows induce the mKdV hierarchy for κ .) Fact: centroaffine curves with p 0 = 1 2 p ′ 1 (i.e. k 2 = − k ′ 1 ) lie on a quadric cone through the origin in R 3 . Theorem: If γ lies on a cone at time zero, and evolves under any of the following curve flows, then it stays on the same cone Z 0 , Z 3 , Z 4 , Z 7 , Z 8 , Z 11 , Z 12 , . . . ( ∗ ) Remark: γ t = Z 3 restricted to a conical curve induces the Kaup-Kuperschmidt (KK) equation for k 1 (Chou & Qu, 2002): ( k 1 ) t = k ′′′′′ − 5 k 1 k ′′′ 1 − 25 2 k ′ 1 k ′′ 1 + 5 k 2 1 k ′ 1 . 1 In fact, we show that the sequence ( ∗ ) realizes the KK hierarchy, when restricted to conical curves.