Stability of traveling waves: dichotomies and eigenvalue conditions on finite intervals

Abstract

If a traveling wave is stable or unstable depends essentially on the spectrum of a differential operator P obtained by linearization. We investigate how spectral properties of the all-line operator P are related to eigenvalues of finite-interval . Here R is a. linear boundary operator, for which we will derive determinant conditions, and the x-interval is assumed to be sufficiently large.

Under suitable assumptions, we show (a) resolvent estimates for large s; (b) if s is in the resolvent of the all-line operator P, then s is also in the resolvent for finite-interval BVPs; (c) eigenvalues of P lead to approximating eigenvalues on finite intervals. These results allow to study the stability question for traveling waves by investigating eigenvalues of finite-interval problems. We give applications to the FitzHugh-Nagumo system with small diffusion and to the complex Ginzburg-Landau equations.

Keywords:

• Traveling waves
• stability
• exponential dichotomies
• eigenvalues
• finite boundary
• value problems
• 34L05
• 34A50
• 65L10

^∗Supported by DFG-Schwerpunktprogramm ‘Ergodentheorie, Analysis und effiziente Simulation dynamischer Systerae’ and SFB 343 ‘Diskrete Strukturen in der Mathematik’.

^∗Supported by DFG-Schwerpunktprogramm ‘Ergodentheorie, Analysis und effiziente Simulation dynamischer Systerae’ and SFB 343 ‘Diskrete Strukturen in der Mathematik’.

Notes

^∗Supported by DFG-Schwerpunktprogramm ‘Ergodentheorie, Analysis und effiziente Simulation dynamischer Systerae’ and SFB 343 ‘Diskrete Strukturen in der Mathematik’.