A spectral analysis of the nonlinear Schr?dinger equation in the co-exploding frame

The nonlinear Schrodinger model is a prototypical dispersive wave equation that features finite time blowup, either for supercritical exponents (for fixed dimension) or for supercritical dimensions (for fixed nonlinearity exponent). Upon identifying the self-similar solutions in the so-called co-exploding frame, a dynamical systems analysis of their stability is natural, yet is complicated by the mixed Hamiltonian-dissipative character of the relevant frame. In the present work, we study the spectral picture of the relevant linearized problem. We examine the point spectrum of 3 eigenvalue pairs associated with translation, U(1) and conformal invariances, as well as the continuous spectrum. We find that two eigenvalues become positive, yet are attributed to symmetries and are thus not associated with instabilities. In addition to a vanishing eigenvalue, 3 more are found to be negative and real, while the continuous spectrum is nearly vertical and on the left-half (spectral) plane. The eigenfunctions and eigenvalues are approximated both asymptotically and numerically, with good agreement between the two approaches. The non-Hamiltonian nature of the co-exploding system results in the 3 eigenvalue pairs failing to be equal-and-opposite by an exponentially small amount. A projection method is used to evaluate this small correction, and at the same time explains the subtle effects of finite boundaries and their role in the observed weak eigenvalue oscillations. (c) 2022 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).

Automated summary

The present study examines the spectral characteristics of the linearized problem associated with the nonlinear Schrodinger model. The study identifies the eigenvalue pairs related to symmetries and the properties of the continuous spectrum, and approximates their values through asymptotic and numerical approaches. The findings highlight the physical significance of these eigenvalues.