Mass Concentration Phenomenon for the Quintic Nonlinear Schrödinger Equation in One Dimension

Details Mass Concentration Phenomenon for the Quintic Nonlinear Schrödinger Equation in One Dimension Mass Concentration Phenomenon for the Quintic Nonlinear Schrödinger Equation in One Dimension

2006-03-29

arxiv.org/abs/math/0603692v1

Mathematics

Analysis of PDEs

23 pages

Scientific paper

We consider the $L^{2}$-critical quintic focusing nonlinear Schr\"odinger equation (NLS) on ${\bf R}$. It is well known that $H^{1}$ solutions of the aforementioned equation blow up in finite time. In higher dimensions, for $H^{1}$ spherically symmetric blow-up solutions of the $L^{2}$-critical focusing NLS, there is a minimal amount of concentration of the $L^{2}$-norm (the mass of the ground state) at the origin. In this paper we prove the existence of a similar phenomenon for the one-dimensional case and rougher initial data, $(u_{0}\in H^{s}, s<1)$, without any additional assumption.

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