Derivatives of (modified) Fredholm determinants and stability of standing and traveling waves

作者：Gesztesy Fritz^*; Latushkin Yuri; Zumbrun Kevin

来源：Journal de Mathematiques Pures et Appliquees, 2008, 90(2): 160-200.

DOI：10.1016/j.matpur.2008.04.001

• Evans functions
• stability of traveling waves
• infinite determinants
• Jost functions
• schrodinger operators

Continuing a line of investigation initiated in [F. Gesztesy, Y. Latushkin, K.A. Makarov, Evans functions, Jost functions, and Fredholm determinants, Arch. Rat. Mech. Anal. 186 (2007) 361-42 1 ] exploring the connections between Jost and Evans functions and (modified) Fredholm determinants of Birman-Schwinger type integral operators, we here examine the stability index, or sign of the first nonvanishing derivative at frequency zero of the characteristic determinant, an object that has found considerable use in the study by Evans function techniques of stability of standing and traveling wave solutions of partial differential equations (PDE) in one dimension. This leads us to the derivation of general perturbation expansions for analytically-varying modified Fredholm determinants of abstract operators. Our main conclusion, similarly in the analysis of the determinant itself, is that the derivative of the characteristic Fredholm determinant may be efficiently computed from first principles for integral operators with semiseparable integral kernels, which include in particular the general one-dimensional case, and for sums thereof, which appears to offer applications in the multi-dimensional case.
A second main result is to show that the multi-dimensional characteristic Fredholm determinant is the renormalized limit of a sequence of Evans functions defined in [G.J. Lord, D. Peterhof, B. Sandstede, A. Scheel, Numerical computation of solitary waves in infinite cylindrical domains, SIAM J. Numer. Anal. 37 (2000) 1420-1454] on successive Galerkin subspaces, giving a natural extension of the one-dimensional results of [F. Gesztesy, Y. Latushkin, K.A. Makarov, Evans functions, Jost functions, and Fredholm determinants, Arch. Rat. Mech. Anal. 186 (2007) 361-421] and answering a question of [J. Niesen, Evans function calculations for a two-dimensional system, presented talk, SIAM Conference on Applications of Dynamical Systems, Snowbird, UT, USA, May 2007] whether this sequence might possibly converge (in general, no, but with renormalization, yes). Convergence is useful in practice for numerical error control and acceleration.

• 出版日期2008-8