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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.

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

Citations:

Zbl 1007.35087
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References:

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