Svetlana Roudenko

Threshold solutions for the focusing 3D cubic Schrödinger equation

We study the focusing 3d cubic NLS equation with H^1 data at the mass-energy threshold, namely, when M[u_0]E[u_0] = M[Q]E[Q]. In earlier works of Holmer-Roudenko and Duyckaerts-Holmer-Roudenko, the behavior of solutions (i.e., scattering and blow up in finite time) is classified when M[u_0]E[u_0] < M[Q]E[Q]. In this paper, we first exhibit 3 special solutions: e^{it}Q and Q^+, Q^-; here Q is the ground state, and Q^+, Q^- exponentially approach the ground state solution in the positive time direction, meanwhile Q^+ having finite time blow up and Q^- scattering in the negative time direction. Secondly, we classify solutions at this threshold and obtain that up to \dot{H}^{1/2} symmetries, they behave exactly as the above three special solutions, or scatter and blow up in both time directions as the solutions below the mass-energy threshold. These results are obtained by studying the spectral properties of the linearized Schroedinger operator in this mass-supercritical case, establishing relevant modulational stability and careful analysis of the exponentially decaying solutions to the linearized equation.

Divergence of Infinite-Variance Nonradial Solutions to the 3D NLS Equation

We consider solutions u(t) to the 3d NLS equation i∂ t u + Δu + |u|2 u = 0 such that ‖xu(t)‖ L 2 = ∞ and u(t) is nonradial. Denoting by M[u] and E[u], the mass and energy, respectively, of a solution u, and by Q(x) the ground state solution to −Q + ΔQ + |Q|2 Q = 0, we prove the following: if M[u]E[u] < M[Q]E[Q] and ‖u 0‖ L 2 ‖∇u 0‖ L 2 > ‖Q‖ L 2 ‖∇Q‖ L 2 , then either u(t) blows-up in finite positive time or u(t) exists globally for all positive time and there exists a sequence of times t n → + ∞ such that ‖∇u(t n )‖ L 2 → ∞. Similar statements hold for negative time.

Blow-up criteria for the 3D cubic nonlinear Schrödinger equation

We consider solutions u to the 3D nonlinear Schrodinger equation i∂tu + Δu + |u|2u = 0. In particular, we are interested in finding criteria on the initial data u0 that predict the asymptotic behaviour of u(t), e.g., whether u(t) blows up in finite time, exists globally in time but behaves like a linear solution for large times (scatters), or exists globally in time but does not scatter. This question has been resolved (at least for H1 data) (Duyckaerts–Holmer–Roudenko) if M[u]E[u] ≤ M[Q]E[Q], where M[u] and E[u] denote the mass and energy of u and Q denotes the ground state solution to −Q + ΔQ + |Q|2Q = 0. Here we consider the complementary case M[u]E[u] > M[Q]E[Q]. In the first (analytical) part of the paper, we present a result due to Lushnikov, based on the virial identity and the generalized uncertainty principle, giving a sufficient condition for blow-up. By replacing the uncertainty principle in his argument with an interpolation-type inequality, we obtain a new blow-up condition that in some cases improves upon Lushnikovs condition. Our approach also allows for an adaptation to radial infinite-variance initial data that has a conceptual interpretation: for real-valued initial data, if a certain fraction of the mass is contained within the ball of radius M[u], then blow up occurs. We also show analytically (if one takes the numerically computed value of ) that there exist Gaussian initial data u0 with negative quadratic phase such that but the solution u(t) blows up. In the second (numerical) part of the paper, we examine several different classes of initial data—Gaussian, super Gaussian, off-centred Gaussian, and oscillatory Gaussian—and for each class give the theoretical predictions for scattering or blow-up provided by the above theorems as well as the results of numerical simulation. We find that depending upon the form of the initial conditions, any of the three analytical criteria for blow-up can be optimal. We formulate several conjectures, among them that for real initial data, the quantity provides the threshold for scattering.

Efficient Deterministic Compressed Sensing for Images with Chirps and Reed-Muller Codes

A recent approach to compressed sensing using deterministic sensing matrices formed from discrete frequency-modulated chirps or from Reed-Muller codes is extended to support efficient deterministic reconstruction of signals that are much less sparse than envisioned in the original work. In particular, this allows the application of this approach in imaging. The reconstruction algorithm developed for images incorporates several new elements to improve computational complexity and reconstruction fidelity in this application regime.

Going Beyond the Threshold: Scattering and Blow-up in the Focusing NLS Equation

We study the focusing nonlinear Schrödinger equation

Special Issue on Mathematical Methods in Medical Imaging

{i\partial_t u +\Delta u + |u|^{p-1}u=0}

Numerical Characterization of Thresholds for the Focusing 1d Nonlinear Schrödinger Equation

i∂tu+Δu+|u|p-1u=0,