arXiv subjects

Mitchell Curran

Publications and source records attributed to Mitchell Curran.

3 recordsLinked to original sources

arXiv2026

Hadamard-type formulas for real eigenvalues of canonically symplectic operators

We give first-order asymptotic expansions for the resolvent and Hadamard-type formulas for the eigenvalue curves of one-parameter families of canonically symplectic operators. We allow for parameter dependence in the boundary conditions, bounded perturbations and trace operators associated with each off-diagonal operator, and give formulas for derivatives of eigenvalue curves emanating from the discrete eigenvalue of the unperturbed operator in terms of Maslov crossing forms. We derive the Hadamard-type formulas using two different methods: via a symplectic resolvent difference formula and asymptotic expansions of the resolvent, and using Lyapunov-Schmidt reduction and the implicit function theorem. The latter approach facilitates derivative formulas when the eigenvalue curves are viewed as functions of the spectral parameter. We apply our abstract results to derive a spectral index theorem for the linearised operator associated with a standing wave in the nonlinear Schr\"odinger equation on a compact star graph.

math.SP↗

arXiv2024

Detecting eigenvalues in a fourth-order nonlinear Schr\"odinger equation with a non-regular Maslov box

We use the Maslov index to study the eigenvalue problem arising from the linearisation about solitons in the fourth-order cubic nonlinear Schr\"odinger equation (NLSE). Our analysis is motivated by recent work by Bandara et al., in which the fourth-order cubic NLSE was shown to support infinite families of multipulse solitons. Using a homotopy argument, we prove that the Morse indices of two selfadjoint fourth-order operators appearing in the linearisation may be computed by counting conjugate points, as well as a lower bound for the number of real unstable eigenvalues of the linearisation. We also give a Vakhitov-Kolokolov type stability criterion. The interesting aspects of this problem as an application of the Maslov index are the instances of non-regular crossings, which feature crossing forms with varying ranks of degeneracy. We handle such degeneracies directly via higher order crossing forms, using a definition of the Maslov index developed by Piccione and Tausk.

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arXiv2022

Hamiltonian spectral flows, the Maslov index, and the stability of standing waves in the nonlinear Schr\"{o}dinger equation

We use the Maslov index to study the spectrum of a class of linear Hamiltonian differential operators. We provide a lower bound on the number of positive real eigenvalues, which includes a contribution to the Maslov index from a non-regular crossing. A close study of the eigenvalue curves, which represent the evolution of the eigenvalues as the domain is shrunk or expanded, yields formulas for their concavity at the non-regular crossing in terms of the corresponding Jordan chains. This, along with homotopy techniques, enables the computation of the Maslov index at such a crossing. We apply our theory to study the spectral (in)stability of standing waves in the nonlinear Schr\"odinger equation on a compact spatial interval. We derive new stability results in the spirit of the Jones--Grillakis instability theorem and the Vakhitov--Kolokolov criterion, both originally formulated on the real line. A fundamental difference upon passing from the real line to the compact interval is the loss of translational invariance, in which case the zero eigenvalue of the linearised operator is geometrically simple. Consequently, the stability results differ depending on the boundary conditions satisfied by the wave. We compare our lower bound to existing results involving constrained eigenvalue counts, finding a direct relationship between the correction factors found therein and the objects of our analysis, including the second-order Maslov crossing form.

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