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Derivatives of (modified) Fredholm determinants and stability of standing and traveling waves. (English) Zbl 1161.47058

The paper continues investigations initiated in [F.Gesztesy, Yu.Latushkin and K.A.Makarov, Arch.Ration.Mech.Anal.186, No.3, 361–421 (2007; Zbl 1134.34004)] exploring the connections between Jost and Evans functions and (modified) Fredholm determinants of Birman–Schwinger type integral operators. The authors 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 in one dimension. This leads to the derivation of general perturbation expansions for analytically-varying modified Fredholm determinants of abstract operators. 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 and A.Scheel, SIAM J. Numer.Anal.37, No.5, 1420–1454 (2000; Zbl 0956.65108)].

MSC:

47N20 Applications of operator theory to differential and integral equations
47A55 Perturbation theory of linear operators
35B35 Stability in context of PDEs
35K57 Reaction-diffusion equations
47B10 Linear operators belonging to operator ideals (nuclear, \(p\)-summing, in the Schatten-von Neumann classes, etc.)
35J10 Schrödinger operator, Schrödinger equation
35P05 General topics in linear spectral theory for PDEs
45P05 Integral operators
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