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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 Toy example: KdV First Model : Kirchhoff rods.

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

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### Presentation Transcript

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

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