Stability of solutions to PDEs through the numerical evaluation of the Evans function

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Stability of solutions to PDEs through the
numerical evaluation of the Evans function
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Department of Mathematics

• S. Lafortune
• College of Charleston
• Collaborators J. Lega, S. Madrid-Jaramillo, S.
  Balasuriya, and J. Hornibrook

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Plan of Talk

• Toy example KdV
• First Model Kirchhoff rods.
• Existence analytic
• Stability Evans Function (numerical)
• Second model Combustion
• Existence and Stability Numerical

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Toy example KdV

• KdV
• Model for shallow water

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Toy example KdV

• KdV
• Traveling solution

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Toy example KdV

• Solution

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Toy example KdV

• Solution Perturbed

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Toy example KdV

• Solution Perturbation mode

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Toy example KdV

• Solution Perturbation mode

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Toy example KdV

• Solution Perturbation mode

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Toy example KdV

• Eqn for perturbation
• Plug in
• Into KdV

First order in w
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Toy example KdV

• Eigenvalue problem
• where

The solution is unstable if there is an
eigenvalue on the right side of the complex
plane
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Toy example KdV

• Eigenvalue problem turned into a dynamical system

The solution is unstable this system has a
bounded solution For ? positive
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Model Kirchhoff Rods

• Elastic rods
• One-dimensional elastic structure that offers
  resistance to bending and torsion. A rod can be
  twisted and/or bent.
• A description of a rod is obtained by specifying
• Ribbon geometry
• Mechanics
• Elasticity

Ref Antmans book (95)
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Coiling Bifurcation

• Amplitude equations For the inextensible,
  unshearable model.
• A Amplitude of deformation
• B Amplitude of twist
• A and B are coupled.

Ref Goriely and Tabor (96, 97, 98)
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Pulse Solutions Existence

• Form of solutions

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Coiling Bifurcation Pulses
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Coiling Bifurcation Pulses
Ref Numerics by Lega and Goriely (00)
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Evans Function

• Perturb Solution

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Evans Function
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Evans Function

• The asymptotic matrix
• Eigenvalues and eigenvectors known explicitly
• 3-dim stable space

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Evans Function
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Evans Function
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Evans Function
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Evans Function
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Evans Function Numerical Study
Values of E(?) on a closed contour
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Evans Function Numerical Study
Evans function on the real axis
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Evans Function Numerical Study

• For each value of ?, find numerically 3 solutions
  converging at 8 and 3 solutions
• at -8
• Calculate the determinant of the initial
  conditions
• Calculate E(?) on the boundary of a closed box
• Number of zeros in the box is given by

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Evans Function Analytical Results

• Solve the linearization at the origin using
  symmetries
• Expand the solutions of the linearization in ?
• Get the first nonzero derivative of E(?)
• Instability result using the behavior of the
• Evans function as ? approaches ?

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Evans Function Analytical Results
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Hamiltonian Formulation

• Recall
• Hamiltonian structure

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Hamiltonian Formulation Strategy

• Hamiltonian system
• Noether Theorem
• Lagrange multiplier problem

Ref Grillakis, Shatah and Strauss (87 and
90)
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Hamiltonian Formulation Strategy

• Infinite-dimensional Hessian
• Only one negative eigenvalue
• Continuous spectrum positive, bounded away from
  zero
• One-dimensional Kernel

Ref Grillakis, Shatah and Strauss (87 and
90)
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Stability Condition
Ref Grillakis, Shatah and Strauss (87 and
90)
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Infinite-dimensional Hessian
Fundamental step Infinite-dimensional Hessian

• 2-dim Kernel generated by generators of Lie
  algebra

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Infinite-dimensional Hessian

• One negative eigenvalue
• Reduction of the operator, symmetry arguments
  and Sturm-Liouville theory
• But continuous spectrum touches the origin
• Theorems of Grillakis, Shatah, Strauss extended
  to include this fact
• ? Spectral stability only

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Theorems
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Spectral Stability Criterion
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Conclusions

• Study of amplitude equations coupled
  Klein-Gordon equations
• Explicit conditions for stability of pulses
• Numerical Evans

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Beyond

• This technique can be applied to generalizations
  with tension mode and extensibility (work in
  progress with Tabor and Goriely)
• Use same technique for Kirchhoff

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Evans Function

• The Evans function vanishes on the point spectrum
  of a linear operator.
• Stability results for the FitzHugh-Nagumo
  equations, the generalized KDV,
  Benjamin-Bona-Mahoey equation, the Boussinesq,
  the MKDV, the complex Ginzburg-Landau equation.
• Our point of view Evans function defined as a
  determinant

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Evans Function

• Consider a Linear ODE
• A value of ? is an eigenvalue if there exists a
  solution f such that
• f is an eigenvector

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Evans Function
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Evans Function
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Evans Function
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Evans Function
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Evans Function
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Evans Function Numerical Study
Values of E(?) on a closed contour
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Evans Function Numerical Study
Evans function on the real axis
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Conclusions

• Hamiltonian method gave a stability criterion
• The Evans function method gave precise info on
  the mechanism by which instabilities appear
• The numerical method presented here can be
  applied to other cases. It presents several
  advantages w/r to other more traditional methods

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Evans Function Numerical Study

• For each value of ?, find numerically 3 solutions
  converging at 8 and 3 solutions at -8
• Calculate the determinant of the initial
  conditions
• Calculate E(?) on the boundary of a closed box
• Number of zeros in the box is given by

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Evans Function Numerical Study
Some numerical difficulties to overcome

• Integration of equation at 8
• Evans function can be numerically zero everywhere
• E(0)0

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Stability Condition
Function of 2 variables
Condition on the determinant
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Hamiltonian Symmetries
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Coiling Bifurcation Stability

• Stability Do solutions survive under small
  perturbations?
• Spectral instability Evans function method
  reveals zone of instability (work by S.L. and
  Lega, 03).
• Study of the spectrum of Linear operator
• Evans function Determinant of solutions that
  vanishes whenever ? is an eigenvalue.
• Evans function is a completely general method
  that establishes instability.

Ref J.W. Evans (75)
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Coiling Bifurcation Stability

• Stability Do solutions survive under small
  perturbations?
• Hamiltonian formalism The conservation laws for
  the amplitude equations can be used to prove
  spectral stability (S.L. and Lega, preprint 04).
• Generalization Same method applies to most
  bifurcation and stability analysis follow as well
  (S.L., Goriely and Tabor, preprint 04).
• The Hamiltonian technique establishes stability
  as well but requires an Hamiltonian structure.

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Evans Function