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

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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  1. \ 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

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

  3. Toy example: KdV • KdV • Model for shallow water:

  4. Toy example: KdV • KdV • Traveling solution

  5. Toy example: KdV • Solution

  6. Toy example: KdV • Solution: Perturbed

  7. Toy example: KdV • Solution: Perturbation mode

  8. Toy example: KdV • Solution: Perturbation mode

  9. Toy example: KdV • Solution: Perturbation mode

  10. Toy example: KdV • Eqn for perturbation Plug in Into KdV First order in w

  11. Toy example: KdV • Eigenvalue problem where The solution is unstable if there is an eigenvalue on the right side of the complex plane

  12. Toy example: KdV • Eigenvalue problem turned into a dynamical system The solution is unstable this system has a bounded solution For  positive

  13. 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)

  14. 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)

  15. Pulse Solutions: Existence • Form of solutions

  16. Coiling Bifurcation: Pulses

  17. Coiling Bifurcation: Pulses Ref: Numerics by Lega and Goriely (‘00)

  18. Evans Function • Perturb Solution

  19. Evans Function

  20. Evans Function • The asymptotic matrix • Eigenvalues and eigenvectors known explicitly • 3-dim stable space

  21. Evans Function

  22. Evans Function

  23. Evans Function

  24. Evans Function

  25. Evans Function: Numerical Study Values of E() on a closed contour

  26. Evans Function: Numerical Study Evans function on the real axis

  27. 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

  28. 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 

  29. Evans Function: Analytical Results

  30. Hamiltonian Formulation • Recall • Hamiltonian structure

  31. Hamiltonian Formulation: Strategy • Hamiltonian system: • Noether Theorem: • Lagrange multiplier problem: Ref: Grillakis, Shatah and Strauss (’87 and ‘90)

  32. 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)

  33. Stability Condition Ref: Grillakis, Shatah and Strauss (’87 and ‘90)

  34. Infinite-dimensional Hessian Fundamental step: ‘‘Infinite-dimensional Hessian’’ • 2-dim Kernel generated by generators of Lie algebra

  35. 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

  36. Theorems

  37. Spectral Stability Criterion

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

  39. 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

  40. 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

  41. Evans Function • Consider a Linear ODE • A value of λ is an eigenvalue if there exists a solution φ such that • φis an eigenvector

  42. Evans Function

  43. Evans Function

  44. Evans Function

  45. Evans Function

  46. Evans Function

  47. Evans Function: Numerical Study Values of E(λ) on a closed contour

  48. Evans Function: Numerical Study Evans function on the real axis

  49. 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

  50. 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