Threshold solutions for the focusing 3d cubic Schroedinger equation
arXiv:0806.1752
Abstract
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.
References in corpus (5)
- A sharp condition for scattering of the radial 3d cubic nonlinear Schroedinger equation
- The mass-critical nonlinear Schrödinger equation with radial data in dimensions three and higher
- The cubic nonlinear Schrödinger equation in two dimensions with radial data
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Cited by in corpus (5)
- A Critical Centre-Stable Manifold for the Cubic Focusing Schroedinger Equation in Three Dimensions
- A Critical Centre-Stable Manifold for Schroedinger's Equation in R^3
- Construction of multi-soliton solutions for the L2-supercritical gKdV and NLS equations
- A class of solutions to the 3d cubic nonlinear Schroedinger equation that blow-up on a circle
- Multi-soliton solutions for the supercritical gKdV equations