1 / 56

560 likes | 938 Views

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.

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

**An Image/Link below is provided (as is) to download presentation**
Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.
Content is provided to you AS IS for your information and personal use only.
Download presentation by click this link.
While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.
During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

**\**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. • Existence: analytic • Stability: Evans Function (numerical) • Second model: Combustion • Existence and Stability: Numerical**Toy example: KdV**• KdV • Model for shallow water:**Toy example: KdV**• KdV • Traveling solution**Toy example: KdV**• Solution**Toy example: KdV**• Solution: Perturbed**Toy example: KdV**• Solution: Perturbation mode**Toy example: KdV**• Solution: Perturbation mode**Toy example: KdV**• Solution: Perturbation mode**Toy example: KdV**• Eqn for perturbation Plug in Into KdV First order in w**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: KdV**• Eigenvalue problem turned into a dynamical system The solution is unstable this system has a bounded solution For positive**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**• 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)**Pulse Solutions: Existence**• Form of solutions**Coiling Bifurcation: Pulses**Ref: Numerics by Lega and Goriely (‘00)**Evans Function**• Perturb Solution**Evans Function**• The asymptotic matrix • Eigenvalues and eigenvectors known explicitly • 3-dim stable space**Evans Function: Numerical Study**Values of E() on a closed contour**Evans Function: Numerical Study**Evans function on the real axis**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**• 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**• Recall • Hamiltonian structure**Hamiltonian Formulation: Strategy**• Hamiltonian system: • Noether Theorem: • Lagrange multiplier problem: Ref: Grillakis, Shatah and Strauss (’87 and ‘90)**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**Ref: Grillakis, Shatah and Strauss (’87 and ‘90)**Infinite-dimensional Hessian**Fundamental step: ‘‘Infinite-dimensional Hessian’’ • 2-dim Kernel generated by generators of Lie algebra**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**• Study of amplitude equations: coupled Klein-Gordon equations • Explicit conditions for stability of pulses • Numerical Evans**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 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 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 Study**Values of E(λ) on a closed contour**Evans Function: Numerical Study**Evans function on the real axis**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 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

More Related