The Evans function and the stability of travelling waves

Transcription

1 The Evans function and the stability of travelling waves Jitse Niesen (University of Leeds) Collaborators: Veerle Ledoux (Ghent) Simon Malham (Heriot Watt) Vera Thümmler (Bielefeld) PANDA meeting, University of Nottingham, 16 September 2010

2 Contents 1. Travelling wave solutions 2. Stability of travelling waves Linear stability analysis Nonlinear stability 3. The Evans function Miss-distance function for Sturm Liouville problems Infinite interval Increasing the dimension Topological construction 4. Computational aspects Numerical algorithm Representation of subspaces 5. Other settings Integral equations Two-dimensional fronts 6. Conclusion

3 Travelling wave solutions Consider a PDE like the reaction diffusion equation where y(t, x) R d with x R. y t = Ky xx + F (y) Travelling waves are solutions to the equation that move with constant speed c while maintaining their shape. Example: The Fisher equation is y t = y xx + y(1 y). It models the propagation of genes. It admits the travelling wave solution ( ( )) x ct 2 y(t, x) = 1 + exp 6 t = 0 t = 5 with c = 5/ 6.

4 The KdV equation Formalism described here is not restricted to reaction diffusion, but also applicable to other equations like Korteweg De Vries (KdV): u t + u xxx + 6uu x = 0. The KdV equation admits soliton solutions: u(t, x) = 1 2 c sech2( 1 2 c(x ct) ). Recreation of soliton wave in Edinburgh

5 Contents 1. Travelling wave solutions 2. Stability of travelling waves Linear stability analysis Nonlinear stability 3. The Evans function Miss-distance function for Sturm Liouville problems Infinite interval Increasing the dimension Topological construction 4. Computational aspects Numerical algorithm Representation of subspaces 5. Other settings Integral equations Two-dimensional fronts 6. Conclusion

6 Stability of travelling waves Recall: The RD equation is y t = Ky xx + F (y). In the moving frame ξ = x ct, the PDE is y t = Ky ξξ + cy ξ + F (y). Travelling wave solutions are of the form y(t, ξ) = ŷ(ξ). Hence, they satisfy Kŷ ξξ + cŷ ξ + F (ŷ) = 0. Given a travelling wave ŷ, what is the fate of solutions whose initial conditions are small perturbations of ŷ? If any such solution stay close to ŷ, we say that the travelling wave is stable. In context of RD, use space of bounded uniformly continuous functions with sup norm. Orbital stability: If ŷ(ξ) is a travelling wave solution, then so is its translate ŷ(ξ ξ 0 ). So, we require for stability that small perturbations of ŷ(ξ) stay close to the family {ŷ(ξ ξ 0 ) : ξ 0 R}.

7 Linear stability analysis A natural approach to stability analysis is to linearize the equation about the travelling wave. Suppose that y(t, ξ) = ŷ(ξ) + ỹ(t, ξ) is a perturbation of the travelling wave at t = 0. y solves the full PDE: Ky ξξ + cy ξ + f (y) = y t Travelling wave ŷ solves: Kŷ ξξ + cŷ ξ + f (ŷ) = 0 Subtracting: Kỹ ξξ + cỹ ξ + f (ŷ + ỹ) f (ỹ) = ỹ t Linearized stability analysis: Set ỹ(ξ, t) = e λt ȳ(ξ) to get: Kỹ ξξ + cỹ ξ + Df (ŷ) ỹ = ỹ t Kȳ ξξ + cȳ ξ + Df (ŷ) ȳ = λȳ So, we need to study the spectrum of the differential operator L = K ξξ + c ξ + DF (ŷ). If L has an eigenvalue λ with Re λ 0, then the wave is unstable.

8 The spectrum of L The spectrum of L can be decomposed in two parts. The point spectrum σ pt consists of eigenvalues λ, for which Lȳ = λȳ has a solution. We have 0 σ pt. The essential spectrum σ ess typically contains open sets, and is asymptotically (as λ ) inside the parabolic region {λ : (Im λ) 2 C Re λ}. Im λ σ ess σ pt Re λ

9 Nonlinear stability vs linear stability If the spectrum of L is contained in the left half of the complex plane and bounded away from the imaginary axis, except for a simple eigenvalue at the origin, then the travelling wave is (nonlinearly) stable. (Henri 81) If the essential spectrum touches the imaginary axis at the origin, then one typically has to introduce weighted norms to prove stability. For many equations, the essential spectrum is easily computed. Hence, we now assume that σ ess is in the left half-plane, and we concentrate on the point spectrum. (Sandstede 02)

10 Solving the eigenvalue problem The usual approach for solving the eigenvalue problem Kȳ ξξ + cȳ ξ + Df (ŷ) ȳ = λȳ, ȳ 0 as ξ ± is to discretize the differential operator L by finite differences, finite elements, or some spectral method, and solve the resulting (huge) matrix eigenvalue problem. The discretization may create spurious eigenvalues, but sometimes one can prove that the spectrum of the matrix converges to the spectrum of L as the grid size ξ 0. Secondly, the infinite domain needs to be truncated to ξ [ L, L]. The exact asymptotic boundary conditions are nonlinear; replacing them by linear BCs leads again to spurious eigenvalues. (Bridges, Derks & Gottwald 02)

11 Contents 1. Travelling wave solutions 2. Stability of travelling waves Linear stability analysis Nonlinear stability 3. The Evans function Miss-distance function for Sturm Liouville problems Infinite interval Increasing the dimension Topological construction 4. Computational aspects Numerical algorithm Representation of subspaces 5. Other settings Integral equations Two-dimensional fronts 6. Conclusion

12 The miss-distance function Consider the Sturm Liouville problem u + q(x) u = λu for x [a, b] with boundary conditions u(a) = u(b) = 0. We can rewrite this in first-order form: [ ] y 0 1 = A(x, λ)y where A(x, λ) = q(x) λ 0 with boundary conditions y 1 (a) = y 1 (b) = 0. ( ) Denote by y (x) the solution of ( ) with y (a) = [ ] 0 1. The miss-distance function is D(λ) = y1 (b). Eigenvalues correspond to zeros of the miss-distance function.

13 The matching point ξ We are looking at the Sturm Liouville problem [ ] y 0 1 = A(x, λ)y where A(x, λ) = q(x) λ 0 with boundary conditions y 1 (a) = y 1 (b) = 0. ( ) Denote by y (x) the solution of ( ) with y (a) = [ ] 0 1. Denote by y + (x) the solution of ( ) with y + (b) = [ ] 0 1. The SLP ( ) has a solution if y + is a multiple of y. The miss-distance function, evaluated at ξ [a, b], is [ y D(λ) = det 1 (ξ) y 1 + (ξ) ] y2 (ξ) y 2 + (ξ). Eigenvalues correspond to zeros of the miss-distance function. For ξ = b, we get D(λ) = y1 (b), as before.

14 Problems on an infinite interval Consider the following Sturm Liouville problem with x R: [ ] y 0 1 = A(x, λ)y where A(x, λ) = q(x) λ 0 with boundary conditions y(x) 0 as x ±. ( ) Assume that q(x) 0, then eigenvalues of A(±, λ) are ± λ. Assume that λ C \ [0, ), then v u = [ 1 λ ] is the eigenvector corresponding to the unstable eigenvalue µ u = λ at x =. Let y be the sol-n of ( ) with e µ u x y (x) vu Let y + be the sol-n of ( ) with e µ+ s x y + (x) v s + as x. as x +. Define D(λ) = det [ y (ξ) y + (ξ) ], with ξ R arbitrary. Eigenvalues of ( ) correspond to zeros of D.

15 Increasing the number of dimensions Consider y = A(x, λ) y with y(x) C n for x R. We assume that there is a region Ω C such that for all λ Ω the matrices A ± (λ) = lim A(x, λ) exist and are hyperbolic; x ± A (λ) has k unstable eigenvalues µ 1,..., µ k ; A + (λ) has n k stable eigenvalues µ + 1,..., µ+ n k. Let y i be a solution with e µ i x y i (x) v i Let y + i be a solution with e µ+ i x y + i (x) v + i The Evans function is defined by as x. as x +. D(λ) = det [ y 1 (ξ) y k (ξ) y + 1 (ξ) y + n k (ξ) ]. It is analytic in Ω and its zeros corresponds to eigenvalues. (Evans 75; Alexander, Gardner & Jones 90; Sandstede 02)

16 A topological construction Evans function is analytic, so by argument principle winding number W (D(λ)) around curve Ω equals number of eigenvalues in Ω C. (Evans & Feroe 77) Im λ Re λ As above, EVP is y = A(x, λ) y and we assume that A ± (λ) is hyperbolic, A (λ) has k unstable eigenvalues and A + (λ) has n k stable eigenvalues for λ Ω. Let S = span{y 1,..., y k } Cn be unstable subspace. S evolves according to lifted equation S = l(a(x, λ)) S. Compactify x R to x [ 1, 1] with tanh transformation, and consider capped cylinder C = { 1, 1} int(ω) [ 1, 1] Ω.

17 A topological construction II Recall: S( x, λ) C n is k-dimensional unstable subspace; cylinder C = { 1, 1} int(ω) [ 1, 1] Ω with Ω C. Use S : C C k to build a k-dimensional bundle over C. This is the augmented unstable bundle E. Winding number W (D(λ)) equals first Chern number of E. Before: Winding number W (D(λ)) counts eigenvalues inside Ω. This construction is well-suited for singularly perturbed RD equations like FitzHugh Nagumo u t = u xx + u(u a)(1 u) v, v t = δv xx + ɛ(u γv) For small δ, the system splits in a fast and slow system. This induces a splitting of E in Whitney sum E s E f. Now use c 1 (E s E f ) = c 1 (E s ) + c 1 (E f ) to count eigenvalues and thus determine stability of travelling waves. (Alexander, Gardner & Jones 90)

18 Contents 1. Travelling wave solutions 2. Stability of travelling waves Linear stability analysis Nonlinear stability 3. The Evans function Miss-distance function for Sturm Liouville problems Infinite interval Increasing the dimension Topological construction 4. Computational aspects Numerical algorithm Representation of subspaces 5. Other settings Integral equations Two-dimensional fronts 6. Conclusion

19 Evaluating the Evans function numerically The definition can be used as a basis for a numerical algorithm: Compute the unstable eigenvectors v 1,..., v k of A. For i = 1,..., k, solve y = A(ξ, λ) y with initial condition y( L) = v i (where L is large) to get y i (ξ). Compute y + i (ξ) similarly, and calculate the determinant. The Evans function is analytic, so we can use the argument principle to count the number of eigenvalues in a given region. Alternatively, we can use Newton s method to solve D(λ) = 0 and locate the eigenvalues. (Pego, Smereka & Weinstein 93) If D (0) and D(λ) for large real λ have opposite signs, then there is at least one eigenvalue on positive real axis.

20 Problems Unstable space = solutions grow exponentially; for example, y i grows with rate µ i. Solution: Find y i by solving y = ( A(ξ, λ) µ i I ) y. Evans function is only analytic if the eigenvectors v ± i are analytic functions in λ. This won t happen automatically, and is impossible if eigenvalues µ ± i coalesce. Solution: See (Brin 02) and (Bridges et al. 02) for the first part, and below for the second part. If Re µ 1 > Re µ 2, then y 1 grows faster than y 2, so any errors in the y1 direction will dominate the y 2 solution. Solution: Do not look at the y i individually, but look at the subspace S = span{y1,..., y k } and lift eq n y = A(ξ, λ) y to S = l(a(ξ, λ)) S (as in topological construction).

21 How to represent subspaces? We need to evolve the k-dimensional subspace S(ξ, λ) = span{y 1,..., y k } Cn using S = l(a(ξ, λ)) S. Can represent S by exterior product y 1... y k Λk (C n ). Theoretically nice, determinant is exterior product, but dim Λ k (C n ) = ( n k) is large. (Brin 00, Allen & Bridges 02) Can represent S by k-frame Y = (y 1,..., y k ) Cnk (Stiefel manifold) and apply continuous orthogonalization for stability. This destroys analyticity, but can compensate for this. (Humpherys & Zumbrun 06) Use local coordinates for Grassmannian: another basis for S brings Y in the form [ I Z ]. Dimension is n(n k), but issues when S leaves coordinate patch = Schubert cell. (Ledoux, Malham & Thümmler 10)

22 Contents 1. Travelling wave solutions 2. Stability of travelling waves Linear stability analysis Nonlinear stability 3. The Evans function Miss-distance function for Sturm Liouville problems Infinite interval Increasing the dimension Topological construction 4. Computational aspects Numerical algorithm Representation of subspaces 5. Other settings Integral equations Two-dimensional fronts 6. Conclusion

23 Integral equations The Evans function approach can be applied more generally: Travelling waves in integral(-differential) equations, periodic travelling waves, multipulses, rotating and spiral waves. Coombes & Owen ( 04) analysed the neural field equation u(x, t) = w(y) With η(t) = αe αt, we can write this as 1 α tu(x, t) + u(x, t) = 0 η(s)f (u(x y, t s y )) ds dy. w(y)f (u(x y, t)) dy. If f (u) = Heaviside(u h), then we can look for travelling pulses with u > h on left and u < h on right. An Evans function can be defined to assess its stability. Model can be extended to allow for recovery.

24 Planar fronts Planar fronts in two dimensions were treated by Terman ( 90). The 1d front of the cubic autocatalysis equation u t = δu xx uv 2, v t = v xx + uv 2. extends naturally to a 2d front (here δ = 2):

25 Genuinely two-dimensional settings Recent work includes generalization of Evans function to genuine 2d, e.g., wrinkled front that develops out of planar front when it becomes unstable to transverse perturbations at higher δ: (Malevanets, Careta & Kapral 95)

26 Fourier decomposition As before, we want to solve K ȳ + cȳ ξ + Df (ŷ) ȳ = λȳ, where now is 2d Laplacian. Use that travelling wave is ŷ periodic in transverse direction: Expand ȳ and DF (Ũ) in a Fourier series over transverse variable. Substitute this in the eigenvalue problem and truncate high wave numbers ( k > K). We are left with a (big) system of ODEs, and the usual Evans function techniques apply. (Ledoux, Malham, N. & Thümmler 09) This converges as the cut-off K. (Gesztesy, Latushkin & Zumbrun 08) A different approach (perhaps more 2d but up to now theoretical) is taken by Deng & Nii ( 06) and Deng & Jones ( 10).

27 Conclusion The Evans function is a construction which can be used to find eigenvalues and thus assess stability of travelling waves and other coherent structures. Topological aspects mean that it is often possible to get enough information analytically to decide on stability (Chern numbers, parity argument). Numerical computations provide an alternative to the usual approach of discretization and solving a matrix eigenvalue problem (Evans function is clean but slow). Recently, Evans function has been making inroads into two-dimensional problems.