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.


35Q55 NLS equations (nonlinear Schrödinger equations)
35Q51 Soliton equations
35B40 Asymptotic behavior of solutions to PDEs
35B45 A priori estimates in context of PDEs


Zbl 1007.35087
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[1] Agmon, S.: Lectures on Exponential Decay of Solutions of Second-Order Elliptic Equations: Bounds on Eigenfunctions of N -body Schrödinger Operators. Math. Notes 29, Princeton Univ. Press, Princeton, NJ, University of Tokyo Press, Tokyo (1982) · Zbl 0503.35001
[2] Berestycki, H., Cazenave, T.: Instabilité des états stationnaires dans les équations de Schrödinger et de Klein-Gordon non linéaires. C. R. Acad. Sci. Paris Sér. I Math. 293, 489-492 (1981) · Zbl 0492.35010
[3] Bourgain, J.: Global Solutions of Nonlinear Schrödinger Equations. Amer. Math. Soc. Colloq. Publ. 46, Amer. Math. Soc. (1999) · Zbl 0933.35178
[4] Bourgain, J., Wang, W.: Construction of blowup solutions for the nonlinear Schrödinger equation with critical nonlinearity. Ann. Scuola Norm. Sup. Pisa (4) 25, 197-215 (1997) · Zbl 1043.35137
[5] Buslaev, V. S., Perelman, G. S.: Scattering for the nonlinear Schrödinger equation: states that are close to a soliton. Algebra i Analiz 4, no. 6, 63-102 (1992) (in Rus- sian); English transl.: St. Petersburg Math. J. 4, 1111-1142 (1993) · Zbl 0853.35112
[6] Cazenave, T., Lions, P.-L.: Orbital stability of standing waves for some nonlinear Schrödinger equations. Comm. Math. Phys. 85, 549-561 (1982) · Zbl 0513.35007
[7] Christ, M., Kiselev, A.: Maximal functions associated with filtrations. J. Funct. Anal. 179, 409-425 (2001) · Zbl 0974.47025
[8] Cuccagna, S.: Stabilization of solutions to nonlinear Schrödinger equations. Comm. Pure Appl. Math. 54, no. 9, 1110-1145 (2001) · Zbl 1031.35129
[9] Flügge, S.: Practical Quantum Mechanics. Reprinting in one volume of Vols. I, II. Springer, New York (1974) · Zbl 0934.81001
[10] Fröhlich, J., Gustafson, S., Jonsson, B. L. G., Sigal, I. M.: Solitary wave dynamics in an external potential. Comm. Math. Phys. 250, 613-642 (2004) · Zbl 1075.35075
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