1. Instability of local deformations of an elastic filament with S. Lafortune & S. Madrid-Jaramillo Department of Mathematics University of Arizona University of Arizona Tucson AZ S. Lafortune & J.L., Physica D 182, 103-124 (2003) S. Lafortune & J.L., submitted to SIMA S. Lafortune, J.L. & S. Madrid-Jaramillo, submitted to Chaos

2. Outline Motivations – Dynamics of an elastic filament – The coupled Klein-Gordon equations – Special solutions and numerical simulations of pulse solutions Evans function methods [S. Lafortune & J.L., Physica D 182, 103-124 (2003)] – Linearization about pulse solutions – Behavior of the Evans function at the origin – Behavior of the Evans function at infinity – Instability criterion – Numerical evaluation of the Evans function [S. Lafortune, J.L. & S. Madrid-Jaramillo, submitted to Chaos] Hamiltonian formalism [S. Lafortune & J.L, submitted to SIMA] – Hamiltonian form of the coupled Klein-Gordon equations – Spectral stability criterion Conclusions

3. Dynamic of an elastic filament Consider an elastic filament kept under tension and subject to constant twist There is a critical value of the applied twist above which the filament undergoes a writhing bifurcation

4. Description of near-threshold dynamics The Kirchhoff equations The coupled Klein-Gordon equations Elastic properties and parametrization of elastic filament Near-threshold dynamics “Reconstruction” of elastic filament Envelope equations

5. The coupled Klein-Gordon equations They describe the near-threshold dynamics of an elastic filament subject to sufficiently high constant twist Dimensionless form of the equations A: complex envelope of helical mode B: axial twist A. Goriely & M. Tabor, Nonlinear dynamics of filaments II: Nonlinear analysis, Physica D 105, 45-61 (1997).

6. Some special solutions Traveling holes Traveling fronts Periodic solutions Traveling pulses J.L. & A. Goriely, Pulses, fronts and oscillations of an elastic rod, Physica D 132, 373-391 (1999).

7. Traveling pulse solutions Analytic expression J.L. & A. Goriely, Pulses, fronts and oscillations of an elastic rod, Physica D 132, 373-391 (1999).

8. Numerical simulations of pulse solutions Numerical simulation – Non-reflecting boundary conditions – Compact finite differences + Runge-Kutta in time J.L. & A. Goriely, Pulses, fronts and oscillations of an elastic rod, Physica D 132, 373-391 (1999).

9. Question Numerical simulations indicate that some pulses are stable and some are not Can one find a criterion which guarantees the instability of the pulse solutions? If so, this condition should depend on the speed of propagation c of each pulse Below, we present two complementary ways - Evans function methods and Hamiltonian techniques - of answering this question

10. Outline Motivations – Dynamics of an elastic filament – The coupled Klein-Gordon equations – Special solutions and numerical simulations of pulse solutions Evans function methods [S. Lafortune & J.L., Physica D 182, 103-124 (2003)] – Linearization about pulse solutions – Behavior of the Evans function at the origin – Behavior of the Evans function at infinity – Instability criterion – Numerical evaluation of the Evans function [S. Lafortune, J.L. & S. Madrid-Jaramillo, submitted to Chaos] Hamiltonian formalism [S. Lafortune & J.L, submitted to SIMA] – Hamiltonian form of the coupled Klein-Gordon equations – Spectral stability criterion Conclusions

11. Linear stability analysis of the pulse solutions Coupled Klein-Gordon equations Pulse solutions

12. Linearization about the pulse solutions Write where a and b correspond to a pulse solution Obtain a linearized system of the form where is a six-dimensional vector and L is a differential operator in  Define an eigenfunction of L with eigenvalue  as a solution Y of L Y = Y such that (u,ū,w)  (H 1 ×H 1 ×L  )  (C 1 ×C 1 ×C 1 ) The equation L Y = Y can be written as a six-dimensional system of the form

13. An eigenfunction exists if the two vector spaces of solutions of X’ = A ( , ) X that are bounded at +  and –  intersect non- trivially Typically, this only happens for particular values of The Evans function E( ) is an analytic function of, which is real-valued for real and which vanishes on the point spectrum of L It can be viewed as the Wronskian, calculated for instance at  = 0, of linearly independent solutions of X’ = A ( , ) X that converge at +  and linearly independent solutions of X’ = A ( , ) X that converge at –  The Evans function Solution bounded at +  Solution bounded at – 

14. The Evans function References: – J.W. Evans, Nerve axon equations. IV. The stable and unstable impulse, Indiana Univ. Math. J. 24, 1169-1190 (1975). – C.K.R.T. Jones, Stability of the travelling wave solution to the FitzHugh-Nagumo equation, Trans. AMS 286, 431-469 (1984). – E. Yanagida, Stability of fast traveling pulse solutions to the FitzHugh-Nagumo equations, J. Math. Biol. 22, 81-104 (1985). – J. Alexander, R. Gardner & C. Jones, A topological invariant arising in the stability analysis of travelling waves, J. Reine Angew. Math. 410, 167-212 (1990). – R. Pego & M. Weinstein, Eigenvalues, and instabilities of solitary waves, Phil. Trans. R. Soc. London A 340, 47-94 (1992). – R.A. Gardner & K. Zumbrun, The Gap Lemma and geometric criteria for instability of viscous shock profiles, Commun. Pure Appl. Math. LI, 0797-0855 (1998). – T. Kapitula & B. Sandstede, Stability of bright solitary-wave solutions to perturbed nonlinear Schrödinger equations, Physica D 124, 58-103 (1998). – B. Sandstede, Stability of travelling waves, in Handbook of Dynamical Systems II: Towards Applications, pp. 983-1055, B. Fielder Ed., Elsevier, 2002.

15. The Evans function As   ± , A ( , )  A 0 ( ), where the matrix A 0 ( ) has constant coefficients If the “deviator” converges and if the asymptotic matrix A 0 ( ) is diagonalizable, then for each (w i, i ) such that A 0 ( ) w i = i w i, there is a solution  i  to X’ = A ( , ) X such that Moreover, if the convergence of the deviator is uniform in on compact subsets of the complex plane, one can find solutions  i  which are analytic in in some subset of the complex plane E. Coddington & N. Levinson, Theory of Ordinary Differential Equations, McGraw-Hill, New-York (1955).

16. The Evans function When  the asymptotic matrix  A 0 ( ) has 3 positive and 3 negative eigenvalues We define the Evans function E( ) by where We look for a criterion, involving the speed of the traveling pulse, which guarantees that E( ) vanishes on the real axis. This is done by comparing the sign of E( ) near the origin with its sign for large positive values of the spectral parameter

17. Behavior of E( ) near the origin The Evans function, as well as its first four derivatives, vanish at = 0 The fifth derivative has to be computed by finding expansions of the  i  in powers of Because of the symmetry B  B + constant of the original nonlinear Klein-Gordon equations, two of the  i  are bounded at = 0, and singular expansions of the form are needed in this particular case The fifth derivative of the Evans function at = 0 is given by

18. Behavior of E( ) at infinity The change of variable  = , turns the system X’ = A ( , ) X into X’ = à ( , ) X As  + , à ( , ) converges uniformly toward a matrix à 0, which has constant coefficients Because X’ = à ( , ) X has an exponential dichotomy for large, its solutions are uniformly close to solutions of X’ = à 0 X for large enough The sign of E( ) for large is therefore the same as the sign of the Evans function associated with X’ = à 0 X, provided the basis vectors are arranged in the same order One finds

19. Instability criterion For positive and small, For  > 0, the asymptotic state of the pulse is unstable to plane waves perturbations, so we assume  < 0. Then, E( ) is negative for positive and large Therefore, E( ) vanishes for a real positive value of if, which can be rewritten as In the numerical simulation shown before, T = – 27.078 < 0

20. Numerical evaluation of the Evans function One can numerically evaluate the Evans function for on the real axis and see how E( ) changes as c 2 gets close to K Below, S. Lafortune, J.L. & S. Madrid-Jaramillo, Instability of local deformations of an elastic rod: numerical evaluation of the Evans function, submitted to Chaos.

21. Numerical evaluation of the Evans function Contour integration in the complex plane can also be used to detect the zeros of the Evans function in a region near the origin Here, the winding number is equal to 6 for S. Lafortune, J.L. & S. Madrid-Jaramillo, Instability of local deformations of an elastic rod: numerical evaluation of the Evans function, submitted to Chaos.

22. Outline Motivations – Dynamics of an elastic filament – The coupled Klein-Gordon equations – Special solutions and numerical simulations of pulse solutions Evans function methods [S. Lafortune & J.L., Physica D 182, 103-124 (2003)] – Linearization about pulse solutions – Behavior of the Evans function at the origin – Behavior of the Evans function at infinity – Instability criterion – Numerical evaluation of the Evans function [S. Lafortune, J.L. & S. Madrid-Jaramillo, submitted to Chaos] Hamiltonian formalism [S. Lafortune & J.L, submitted to SIMA] – Hamiltonian form of the coupled Klein-Gordon equations – Spectral stability criterion Conclusions

23. Hamiltonian formalism The coupled nonlinear Klein-Gordon equations form a Hamiltonian system J.L. & A. Goriely, Pulses, fronts and oscillations of an elastic rod, Physica D 132, 373-391 (1999)

24. Hamiltonian formalism If v 0 corresponds to a pulse solution, then the linearization of the Hamiltonian system about v 0 is of the form where H c,  is self-adjoint The idea is to use the fact that the spectrum of H c,  is relatively simpler to analyze, to get information on the spectrum of J H c,  M. Grillakis, J. Shatah and W. Strauss, Stability theory of solitary waves in the presence of symmetry, I, J. Functional Analysis 74, 160-197 (1987) M. Grillakis, J. Shatah and W. Strauss, Stability theory of solitary waves in the presence of symmetry, II, J. Functional Analysis 94, 308-348 (1990)

25. Assume that the continuous spectrum of H c,  is positive and bounded away from the origin, and that the Hilbert space in which perturbations live is the direct sum of the kernel, positive and negative subspaces of H c,   Let d(c,  ) = E(v 0 ) – cQ 1 (v 0 ) –  Q 2 (v 0 ), where v 0 corresponds to the pulse solution, Q 1 (v 0 ) is the conserved quantity associated with translational invariance and Q 2 (v 0 ) is the conserved quantity associated with gauge invariance. Let d″(c,  ) be non-singular, n(H c,  ) be the dimension of the negative subspace of H c, , and p(d″) be the number of positive eigenvalues of the Hessian of d. Hamiltonian formalism

26. Theorem (M. Grillakis, J. Shatah and W. Strauss) – If n(H c,  ) – p(d″) is odd, then J H c,  has at least one pair of real non-zero eigenvalues. – If n(H c,  ) = p(d″), then pulse solutions are (orbitally) stable. M. Grillakis, J. Shatah and W. Strauss, Stability theory of solitary waves in the presence of symmetry, II, J. Functional Analysis 94, 308-348 (1990) Hamiltonian formalism

27. Here, one can prove that n(H c,  ) = 1 But the continuous spectrum of H c,  touches the origin, so the above theorem is not directly applicable However, one has the following result: Let d″(c,  ) be non-singular. Then, pulses are spectrally stable if and only if S. Lafortune & J.L., Spectral stability of local deformations of an elastic rod: Hamiltonian formalism, submitted to SIMA

28. Hamiltonian formalism This condition reads In the case  = 0, this is consistent with the instability criterion obtained by means of Evans function techniques

29. Summary: spectrum of linearized operator

30. Conclusions Instability of pulse solutions to the coupled nonlinear Klein- Gordon equations – Numerical simulations show that both stable and unstable pulses exist – This analysis indicates that pulses propagating at the speed c are spectrally stable if and only if c 2 is less than K  c 0 2 Evans function techniques – The Evans function as well as its first 4 derivatives vanish at the origin – The spaces of solutions which are bounded at plus infinity and at minus infinity are 3-dimensional

31. Conclusions Evans function techniques – Regular and singular perturbation expansions of the solutions in powers of are needed to calculate the fifth derivative of the Evans function at the origin – The sign of the Evans function for large and positive values of the spectral parameter is found by calculating the Evans function of a system with constant coefficients (obtained after a suitable change of variable) – Analytical results can be complemented by a numerical evaluation of the Evans function

32. Conclusions Hamiltonian formalism – Because the coupled Klein-Gordon form a Hamiltonian system, it is possible to obtain a necessary and sufficient condition for spectral stability – Classical Hamiltonian methods have to be adapted to this case because the continuous spectrum of the modified Hamiltonian touches the origin – It may be possible to obtain strong stability results provided the perturbations live in a different space

33. Conclusions Hamiltonian formalism – Because the coupled Klein-Gordon form a Hamiltonian system, it is possible to obtain a necessary and sufficient condition for spectral stability – Classical Hamiltonian methods have to be adapted to this case because the continuous spectrum of the modified Hamiltonian touches the origin – It may be possible to obtain strong stability results provided the perturbations live in a different space

34. Symmetries The coupled Klein-Gordon equations are – Invariant under space translations – Gauge invariant – Invariant under B  B + constant Two-parameter family pulse of solutions – The speed c is associated with the space translation invariance – The frequency  is associated with the gauge invariance

35. Traveling pulse solutions Analytic expression J.L. & A. Goriely, Pulses, fronts and oscillations of an elastic rod, Physica D 132, 373-391 (1999).