Fredholm determinants and computing the stability of travelling waves

Transcription

1 Fredholm determinants and computing the stability of travelling waves Simon J.A. Malham, Margaret Beck (HW), Issa Karambal (NAIS), Veerle Ledoux (Ghent), Robert Marangell (UNC), Jitse Niesen (Leeds), Vera Thümmler (D-fine) Heriot Watt University, Edinburgh, UK NWCS 2012, Seattle June 13th 16th

2 Spectral problems Parabolic nonlinear systems on R T: t U = B U + c x U + F(U). Travelling wave U c. Small perturbations U satisfy: B U + c x U + DF(U c )U =λu. Main solution approaches: Projection and iteration. Shooting and matching. Operator determinants.

3 Setup OnR: B xx U + c x U + DF(U c )U =λu x U = P, x P = B 1( λ DF(U c ) ) U cb 1 P. Y = A(x;λ) Y. Forλ C: matching condition x E(λ) := e 0 TrA(ξ;λ) dξ det ( Y 1 Y k Y+ k+1 ) Y+ n x = e 0 TrA(ξ;λ) dξ det ( Y Y +)

4 Synonyms Carrying the same information are: In one dimension d = 1: Evans matrix; Matrix of transmission coefficients; Fredholm operators; Titchmarsh Weyl matrix-function; Grassmannian Riccati flow. In multi-dimensions d > 1: Fredholm operators; Dirichlet-to-Neumann map; Fredholm Grassmannian flow.

5 Numerical issues Computational domain. Different exponential growth rates. Polynomial complexity. Flow singularities?! Where to match? Retaining analyticity? How to project transversely. How to approximate the Fredholm Grassmannian flow.

6 Fredholm determinants Eval problem: (L 0 + v(x) λ id)u = 0 Inverse K 0 (λ) (L 0 λ id) 1 is compact λ eval det F ( id + K0 (λ) v ) = 0 Operator x ( A 0 (λ) + R(x) ) is Fredholm Inverse K 0 (λ) ( x A 0 (λ)) 1 is compact λ eval det F ( id + K0 (λ) R ) = 0

7 Trace class and Hilbert Schmidt operators For Hilbert space H: tr K ϕ i, Kϕ i H = tr G(x; x) dx R i 1 Trace class operators: K J1 (H) tr K Hilbert Schmidt operators: K J2 (H) tr K K Birman Schwinger operator is trace class: K(λ) R 1/2 K 0 (λ) R R 1/2

8 Evans function and operator determinants Definition Set d(λ) det F ( id + K(λ) ) and E(λ) = detc n( Y Y +). Theorem (Issa Karambal (2012a)) 1 If D(λ) is matrix of transmission coefficients: ) 2 If c(λ) det C n( Y 0 Y+ 0 then d(λ) det C nd(λ) d(λ) = E(λ) c(λ) i.e. det F (L 0 + v(x) λid) = det F (L 0 λ id) det 1 ( id K(λ) v )

9 Fredholm expansion where tr K m = = i 1 < <i m det(id + K) = tr K m i 1 < <i m det m 0 ϕi1... ϕ im, Kϕ i1... Kϕ im [ ϕip, Kϕ iq H ] p,q {1,...,m} = 1 det [ G(x i, x j ) ] dx 1...dx m m! R m H m Bornemann 2010, Karambal 2012b

10 Stiefel and Grassmann manifolds Stiefel manifold: V(n, k) ={k-frames centred at the origin}. Grassmann manifold: Gr(n, k) ={k-dimensional subspaces ofc n }. Fibre bundle: π:v(n, k) Gr(n, k) V(n, k)/gl(k) π: k-frame spanning k-plane

11 Representation Example: Gr(3, 2) ( ) 1 0 ; 0 1 ( ) ; 0 1 e ( ) (0,0,1) (1,0,0) e e 2 (0,1,0) 3

12 Representation II π: Y = y i u y i Example coordinate patch: y i =. ŷ k+1,1 ŷ k+1,2 ŷ k+1,k ŷ k+2,1 ŷ k+2,2 ŷ k+2,k ŷ n,1 ŷ n,2 ŷ n,k Local chartu i C (n k)k given by y i ŷ.

13 Grassmannian flows Y = A(x, Y) Y Substitute decomposition Y = y i u: y i u + y i u = (A i + A i ŷ) u Project ontoi th andith rows: ŷ = c + d ŷ ŷ(a + b ŷ) and u = (a + b ŷ) u where a = A i i, b = A i i, c = A i i and d = A i i.

14 Boussinesq system PDE: u tt = (1 c 2 ) u xx + 2c u xt u xxxx (u 2 ) xx. Solitary waves with sech 2 profile. D(λ) 1.5 x λ Y ξ Figure: Evans function for c = 1/4 with GGEM-RK and x = 8 (left panel). Entries of y i forλ= (right panel).

15 Boussinesq: error vs matching point error in the eigenvalue CO RK Riccati RK Mobius M GGEM LG GGEM RK Riccati QOGE matching point Figure: Error in the eigenvalue for different choices of the matching point: N = 512.

16 Autocatalytic fronts t u =δ u + c x u uv m, t v = v + c x v + uv m. error in the eigenvalue CO RK Riccati RK Mobius M GGEM LG matching point Figure: Error in the eigenvalue whenδ=0.1 and m = 9: N = 256.

17 Transverse Fourier basis OnR T we have: B U + c x U + DF(U c )U =λu. On the Fourier modes k = K, K + 1,...,K: x Û k = ˆP k, K x ˆPk =λb 1 Û k + (k/ L) 2 Û k B 1 ˆDk ν Û ν c B 1 ˆPk. ν= K

18 Computing travelling waves: freezing method Substitute U(x, y, t) = V(x γ(t), y, t) into original PDE: t V = B V +γ (t) x V + F(V), ( ) T ) 0 = x ˆV(x, y, t) (ˆV(x, y, t) V(x, y, t) dx dy. R T (Developed by Beyn and Thümmler.)

19 Wrinkled front: Evans function forδ= x D(λ) D(λ) λ x λ x 10 4

20 Wrinkled front: Evans function forδ=3 2 x D(λ) λ x 10 3 D(λ) 6 x x λ x 10 3 λ x 10 4 D(λ) D(λ) 4 x λ 1.6 x 10 3

21 Wrinkled front: Eigenvalues for δ = 3 K Eigenvalues (Evans function) Eigenvalues (ARPACK)

22 Wrinkled front: contour integration argument (multiples of π) i 3i i 1.5 λ Figure: Left panel: contour. Right panel: arg ( D(λ) ) whenλtransverses the top half.δ = 3.

23 Singularities and Schubert cycles Ledoux & M. 2009, Aljasser 2012 and Beck & M. 2012

24 Review: Grassmannian Sufficient to compute the subspace spanned by Y [Y 1 Y k ] With respect toc n =C k C n k graph of y Lin(C k ;C n k ), i.e. { (idc k y)(z): z C k} realizes such a k-plane. GL(C n ) acts on Gr(n, k) via the Mobius map ( ) ( ) ( ) α β id id = γ δ y (γ +δy)(α +βy) 1 whereα GL(C k ),δ GL(C n k ),β C k (n k) andγ C (n k) k

25 Multi-dimensional scenario SupposeH=H 1 H 2 eg.h 1 = H s+ 1 2( Ω) andh 2 = H s 1 2( Ω) Gr(H) is a Hilbert manifold modelled onj 2 (H 1 ;H 2 ) For y J 2 (H 1 ;H 2 ): { (id H1 y)(z): z H 1 } carves a subspace ofh GL res (H) acts on Gr(H) via the Mobius map ( ) ( ) ( ) α β id id = γ δ y (γ +δy)(α +βy) 1 whereα Fred(H 1 ),δ Fred(H 2 ),β J 2 (H 2 ;H 1 ) and γ J 2 (H 1 ;H 2 ).

26 References 1 J.C., Alexander, R. Gardner, and C.K.R.T. Jones, A topological invariant arising in the stability analysis of traveling waves, J. Reine Angew. Math. 410 (1990), pp A.Y. Aljasser, Computing the Evans function for the stability of combustion waves, PhD Thesis (Heriot Watt), J. Deng and C.K.R.T. Jones, Multi-dimensional Morse index theorems and a symplectic view of elliptic boundary value problems, Transactions of the AMS, J. Humpherys and K. Zumbrun, An efficient shooting algorithm for Evans function calculations in large systems, Physica D 220 (2006), pp

27 References II 1 I. Karambal, The Evans function and operator determinants, submitted to Physica D, 2012a. 2 I. Karambal, manuscript in preparation, 2012b. 3 V. Ledoux, S.J.A. Malham and V. Thümmler, Grassmannian spectral shooting, Mathematics of Computation 79 (2010), pp V. Ledoux, S.J.A. Malham, J. Niesen and V. Thümmler, Computing stability of multi-dimensional travelling waves, SIADS 8(1) (2009), pp