Instability of local deformations of an elastic filament

1
Instability of local deformations of an elastic
filament

• with
• S. Lafortune S. Madrid-Jaramillo
• Department of Mathematics
• University of Arizona

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
University of Arizona
Tucson
AZ
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 functionS.
  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
Reconstructionof elastic filament
Near-thresholddynamics
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 equationsA
  complex envelope of helical modeB axial
  twistA. 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 functionS.
  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 x
• Define an eigenfunction of L with eigenvalue l as
  a solution Y of L Y l Y such that
  (u,u,w)?(H1H1L?)?(C1C1C1) The equation L Y
  l Y can be writtenas a six-dimensional system of
  the form

13
The Evans function

• An eigenfunction exists if the two vector spaces
  of solutions of X A(x, l) X that are bounded
  at ? and ? intersect non-trivially
• Typically, this only happens for particular
  values of l
• The Evans function E(l) is an analytic function
  of l, which is real-valued for l real and which
  vanishes on the point spectrum of L
• It can be viewed as the Wronskian, calculated for
  instance at x 0, of linearly independent
  solutions of X A(x, l) X that converge at ?
  and linearly independent solutions of X
  A(x, l) X that converge 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 x ? ?, A(x, l) ? A0(l), where the matrix
  A0(l) has constant coefficients
• If the deviator
  converges and if the asymptotic matrix A0(l) is
  diagonalizable, then for each (wi,ni) such that
  A0(l) wi ni wi , there is a solution Fi(x,l) to
  X A(x, l) X such that
  
• Moreover, if the convergence of the deviator is
  uniform in l on compact subsets of the complex
  plane, one can find solutions Fi(x,l) which are
  analytic in l 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 l gt 0, the asymptotic matrix A0(l) has 3
  positive and 3 negative eigenvalues
• We define the Evans function E(l) bywhere
• We look for a criterion, involving the speed of
  the traveling pulse, which guarantees that E(l)
  vanishes on the real axis. This is done by
  comparing the sign of E(l) near the origin with
  its sign for large positive values of the
  spectral parameter l

17
Behavior of E(l) near the origin

• The Evans function, as well as its first four
  derivatives, vanish at l 0
• The fifth derivative has to be computed by
  finding expansions of the Fi(x,l) in powers of l
• Because of the symmetry B ? B constant of the
  original nonlinear Klein-Gordon equations, two of
  the Fi(x,l) are bounded at l 0, and singular
  expansions of the form are needed in this
  particular case
• The fifth derivative of the Evans function at l
  0 is given by

18
Behavior of E(l) at infinity

• The change of variable z l x, turns the system
  X A(x, l) X into X Ã(z, l) X
• As l ? ?, Ã(z, l) converges uniformly toward a
  matrix Ã0, which has constant coefficients
• Because X Ã(z, l) X has an exponential
  dichotomy for l large, its solutions are
  uniformly close to solutions of X Ã0 X for l
  large enough
• The sign of E(l) for l large is therefore the
  same as the sign of the Evans function
  associated with X Ã0 X, provided the basis
  vectors are arranged in the same order
• One finds

19
Instability criterion

• For l positive and small,
• For m gt 0, the asymptotic state of the pulse is
  unstable to plane waves perturbations, so we
  assume m lt 0. Then, E(l) is negative for l
  positive and large
• Therefore, E(l) vanishes for a real positive
  value of l if
  
  , which can be
  rewritten as
• In the numerical simulation shown before, T
  27.078 lt 0

20
Numerical evaluation of the Evans function

• One can numerically evaluate the Evans function
  for l on the real axis and see how E(l) changes
  as c2 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 functionS.
  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 v0 corresponds to a pulse solution, then the
  linearization of the Hamiltonian system about v0
  is of the formwhere Hc,w is self-adjoint
• The idea is to use the fact that the spectrum of
  Hc,w is relatively simpler to analyze, to get
  information on the spectrum of J Hc,w
• 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
Hamiltonian formalism

• Assume that the continuous spectrum of Hc,w 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 Hc,w.
• Let d(c,w) E(v0) cQ1(v0)
  wQ2(v0),where v0 corresponds to the pulse
  solution, Q1(v0) is the conserved quantity
  associated with translational invariance and
  Q2(v0) is the conserved quantity associated with
  gauge invariance.
• Let d?(c,w) be non-singular, n(Hc,w) be the
  dimension of the negative subspace of Hc,w, and
  p(d?) be the number of positive eigenvalues of
  the Hessian of d.

26
Hamiltonian formalism

• Theorem (M. Grillakis, J. Shatah and W. Strauss)
• If n(Hc,w) p(d?) is odd, then J Hc,w has at
  least one pair of real non-zero eigenvalues.
• If n(Hc,w) 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)

27
Hamiltonian formalism

• Here, one can prove that n(Hc,w) 1
• But the continuous spectrum of Hc,w touches the
  origin, so the above theorem is not directly
  applicable
• However, one has the following result
• Let d?(c,w) 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 w 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 c2 is less than K ? c02
• 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 l 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 w 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).