A spectral analysis of the nonlinear Schrödinger equation in the co-exploding frame

The nonlinear Schrödinger 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.