Near soliton dynamics and singularity formation for \( L^2\) critical problems (Q2875027)

The paper under review presents an overview of recent progress over the past ten years in the question of singularity formation for two canonical dispersive problems: the \(L^2\) critical non-linear Schrödinger equation NEWLINE\[NEWLINE (NLS)\qquad \qquad i\partial_t +\Delta u + u|u|^{4/d} = 0,\quad (t,x)\in \mathbb{R}\times\mathbb{R}^d,\;\;u\in \mathbb{C}, NEWLINE\]NEWLINE and the \(L^2\) critical one-dimensional generalized Korteweg--de Vriez equation NEWLINE\[NEWLINE (gKdV)\qquad \qquad \partial_t + ( u_{xx} + u^5)_x = 0,\quad (t,x)\in \mathbb{R}\times\mathbb{R}^d,\;\;u\in\mathbb{R}. NEWLINE\]NEWLINE It is well known that the Cauchy problems for both equations are locally well-posed in \(H^1(\mathbb{R}^d)\) and \(H^1(\mathbb{R})\), respectively. That is, for any initial data \(u_0\in H^1\) there is a unique solution \(u(t)\) which is either globally defined, \(u(t)\in C([0,\infty),H^1)\), or blows up in a finite time (i.e., there is \(T\in (0,\infty)\) such that \(u(t)\in C([0,T),H^1)\) and \(\|\nabla u(t)\|_{L^2}\to \infty\) as \(t\uparrow T\) for some \(T<\infty\)). Traveling wave solutions (also called solitons) play a distinguished role in this analysis. In particular, if the initial profile \(u_0\in H^1\) is such that NEWLINE\[NEWLINE \|u_0\|_{L^2} < \|Q\|_{L^2}, NEWLINE\]NEWLINE where \(Q\) stands for the ground state solitary wave solution, then both (NLS) and (gKdV) generate a unique global solution. A complete description of the nonlinear flow near the ground state solitary wave has become one of the main questions in the area. The first part of the paper (Section 2) summarizes the results on blow-up solutions near \(Q\) obtained for (NLS) by the authors in their previous work (this includes minimal-mass blowup, log-log blowup, threshold dynamics etc.). The second part of the paper (Sections 3 and 4) reviews the state of the art in the problem of formation of singularities for the gKdV flow.