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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