##
**Non-generic blow-up solutions for the critical focusing NLS in 1-D.**
*(English)*
Zbl 1163.35035

The paper aims to establish generic properties of collapse in the one-dimensional nonlinear Schrödinger equation with the self-focusing quintic nonlinearity,
\[
i\psi_t + \psi_{xx} + |\psi|^4\psi = 0.
\]
It is commonly known that this equation has a family of soliton solutions generated by initial configuration
\[
\psi_{\text{sol}} = (3/2)^{1/4}a[\cosh(a^2x/2)]^{-1/2},
\]
with arbitrary scale parameter \(a\). The solitons are unstable because of the possibility of the critical collapse in this equation. In fact, an exact solution for a collapsing pulse is known in this case. Indeed, the equation maintains the pseudo-conformal invariance, which means that any solution \(\psi(x,t)\) can be transformed into a new solution, as follows:
\[
\tilde{\psi}(x,t) = (a + bt)^{-1/2} \exp(\frac{ibx^2}{4(a + bt)})\psi(\frac{c + dt}{a + bt},\frac{x}{a + bt}),
\]
where the rows \((a,b)\) and \((c,d)\) form an \(SL(2)\) matrix with real elements. The application of this transformation to the soliton generates a new solution, which features the collapse at \(t = -a/b\), due to the presence of the singular factor, \((a + bt)^{-1/2}\). Nevertheless, it is known from the work of G. Perelman [Ann. Henri Poincaré 2, No. 4, 605–673 (2001; Zbl 1007.35087)] and subsequent works that this exact solution represents a nongeneric collapsing solution, the generic one featuring a different type of the near-collapse singular behavior, with the amplitude growing as
\[
A \sim \sqrt{\frac{\ln|\ln(T-t)|}{T - t}}
\]
(\(T\) is the collapse time). The present work produces a rigorous proof of the fact that, while the nongeneric collapse regime is unstable (otherwise, it would be generic), one can identify a manifold of codimension 1 in a space of initial perturbations around the exact soliton, with some reasonably defined norm, such that the nongeneric collapse is stable on that manifold. The proof is based on a specially developed a priori estimate which allows one to split the initial perturbation into a dispersive (radiation) part, and one accounting for a modification of the soliton’s parameters. Subsequently, a strong local estimate is obtained for the rate of the decay of the dispersive part. This analysis makes it possible to identify a restriction on the perturbation that maintains the nongeneric regime of the collapse, and thus determines the above-mentioned codimension-1 manifold on which this regime occurs.

Reviewer: Boris A. Malomed (Tel Aviv)

### MSC:

35Q55 | NLS equations (nonlinear Schrödinger equations) |

35Q51 | Soliton equations |

35B40 | Asymptotic behavior of solutions to PDEs |

35B45 | A priori estimates in context of PDEs |

### Keywords:

nonlinear Schrödinger equation; collapse; conformal invariance; symplectic structure; root space### Citations:

Zbl 1007.35087
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\textit{J. Krieger} and \textit{W. Schlag}, J. Eur. Math. Soc. (JEMS) 11, No. 1, 1--125 (2009; Zbl 1163.35035)

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