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\ Department of Mathematics Stability of solutions to PDEs through the numerical evaluation of the Evans function S. Lafortune College of Charleston Collaborators: J. Lega, S. Madrid-Jaramillo, S. Balasuriya, and J. Hornibrook Plan of Talk Toy example: KdV First Model : Kirchhoff rods.

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stability of solutions to pdes through the numerical evaluation of the evans function

\

Department of Mathematics

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

S. Lafortune

College of Charleston

Collaborators: J. Lega, S. Madrid-Jaramillo, S. Balasuriya, and J. Hornibrook

plan of talk
Plan of Talk
  • Toy example: KdV
  • First Model: Kirchhoff rods.
  • Existence: analytic
  • Stability: Evans Function (numerical)
  • Second model: Combustion
  • Existence and Stability: Numerical
toy example kdv
Toy example: KdV
  • KdV
  • Model for shallow water:
toy example kdv4
Toy example: KdV
  • KdV
  • Traveling solution
toy example kdv6
Toy example: KdV
  • Solution: Perturbed
toy example kdv7
Toy example: KdV
  • Solution: Perturbation mode
toy example kdv8
Toy example: KdV
  • Solution: Perturbation mode
toy example kdv9
Toy example: KdV
  • Solution: Perturbation mode
toy example kdv10
Toy example: KdV
  • Eqn for perturbation

Plug in

Into KdV

First order in w

toy example kdv11
Toy example: KdV
  • Eigenvalue problem

where

The solution is unstable if there is an eigenvalue on the right side of

the complex plane

toy example kdv12
Toy example: KdV
  • Eigenvalue problem turned into a dynamical system

The solution is unstable this system has a bounded solution

For  positive

model kirchhoff rods
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: Antman‘s book(‘95)

coiling bifurcation
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)

coiling bifurcation pulses17
Coiling Bifurcation: Pulses

Ref: Numerics by Lega and Goriely (‘00)

evans function
Evans Function
  • Perturb Solution
evans function20
Evans Function
  • The asymptotic matrix
  • Eigenvalues and eigenvectors known explicitly
  • 3-dim stable space
evans function numerical study
Evans Function: Numerical Study

Values of E() on a closed contour

evans function numerical study26
Evans Function: Numerical Study

Evans function on the real axis

evans function numerical study27
Evans Function: Numerical Study
  • For each value of , find numerically 3 solutions converging at +∞ and 3 solutions

at -∞

  • 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
evans function analytical results
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 

hamiltonian formulation
Hamiltonian Formulation
  • Recall
  • Hamiltonian structure
hamiltonian formulation strategy
Hamiltonian Formulation: Strategy
  • Hamiltonian system:
  • Noether Theorem:
  • Lagrange multiplier problem:

Ref: Grillakis, Shatah and Strauss (’87 and ‘90)

hamiltonian formulation strategy32
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)

stability condition
Stability Condition

Ref: Grillakis, Shatah and Strauss (’87 and ‘90)

infinite dimensional hessian
Infinite-dimensional Hessian

Fundamental step: ‘‘Infinite-dimensional Hessian’’

  • 2-dim Kernel generated by generators of Lie algebra
infinite dimensional hessian35
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

conclusions
Conclusions
  • Study of amplitude equations: coupled Klein-Gordon equations
  • Explicit conditions for stability of pulses
  • Numerical Evans
beyond
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
evans function40
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
evans function41
Evans Function
  • Consider a Linear ODE
  • A value of λ is an eigenvalue if there exists a solution φ such that
  • φis an eigenvector
evans function numerical study47
Evans Function: Numerical Study

Values of E(λ) on a closed contour

evans function numerical study48
Evans Function: Numerical Study

Evans function on the real axis

conclusions49
Conclusions
  • Hamiltonian methodgave 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
evans function numerical study50
Evans Function: Numerical Study
  • For each value of , find numerically 3 solutions converging at +∞ and 3 solutions at -∞
  • 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
evans function numerical study51
Evans Function: Numerical Study

Some numerical difficulties to overcome

  • Integration of equation at ∞
  • Evans function can be numerically zero everywhere
  • E(0)=0
stability condition52
Stability Condition

Function of 2 variables

Condition on the determinant

coiling bifurcation stability
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 λ is an eigenvalue.
  • Evans function is a completely general method that establishes instability.

Ref:J.W. Evans(‘75)

coiling bifurcation stability55
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.