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Continuing a line of investigation initiated in [11] 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 semi-separable integral kernels, which include in particular the general one-dimensional case, and for sums thereof, which latter possibility 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 [23] on successive Galerkin subspaces, giving a natural extension of the one-dimensional results of [11] and answering a question of [27] whether this sequence might possibly converge (in general, no, but with renormalization, yes). Convergence is useful in practice for numerical error control and acceleration.
We consider the stability problem for standing waves of nonlinear Dirac models. Under a suitable definition of linear stability, and under some restriction on the spectrum, we prove at the same time orbital and asymptotic stability. We are not able t
We consider dispersion generalized nonlinear Schrodinger equations (NLS) of the form $i partial_t u = P(D) u - |u|^{2 sigma} u$, where $P(D)$ denotes a (pseudo)-differential operator of arbitrary order. As a main result, we prove symmetry results for
We consider nonlinear half-wave equations with focusing power-type nonlinearity $$ i pt_t u = sqrt{-Delta} , u - |u|^{p-1} u, quad mbox{with $(t,x) in R times R^d$} $$ with exponents $1 < p < infty$ for $d=1$ and $1 < p < (d+1)/(d-1)$ for $d geq 2$.
In this paper we deal with two dimensional cubic Dirac equations appearing as effective model in gapped honeycomb structures. We give a formal derivation starting from cubic Schrodinger equations and prove the existence of standing waves bifurcating from one band-edge of the linear spectrum.
We consider random elliptic equations in dimension $dgeq 3$ at small ellipticity contrast. We derive the large-distance asymptotic expansion of the annealed Greens function up to order $4$ in $d=3$ and up to order $d+2$ for $dgeq 4$. We also derive a