Differential and Integral Equations

On dispersive blow-ups for the nonlinear Schrödinger equation

Younghun Hong and Maja Taskovic

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Abstract

In this article, we provide a simple method for constructing dispersive blow-up solutions to the nonlinear Schrödinger equation. Our construction mainly follows the approach in Bona, Ponce, Saut and Sparber [2]. However, we make use of the dispersive estimate to enjoy the smoothing effect of the Schrödinger propagator in the integral term appearing in Duhamel's formula. In this way, not only do we simplify the argument, but we also reduce the regularity requirement to construct dispersive blow-ups. In addition, we provide more examples of dispersive blow-ups by constructing solutions that blow up on a straight line and on a sphere.

Article information

Source
Differential Integral Equations Volume 29, Number 9/10 (2016), 875-888.

Dates
First available in Project Euclid: 14 June 2016

Permanent link to this document
https://projecteuclid.org/euclid.die/1465912607

Mathematical Reviews number (MathSciNet)
MR3513584

Subjects
Primary: 35Q55: NLS-like equations (nonlinear Schrödinger) [See also 37K10] 35B44: Blow-up 35L67: Shocks and singularities [See also 58Kxx, 76L05] 35L70: Nonlinear second-order hyperbolic equations

Citation

Hong, Younghun; Taskovic, Maja. On dispersive blow-ups for the nonlinear Schrödinger equation. Differential Integral Equations 29 (2016), no. 9/10, 875--888. https://projecteuclid.org/euclid.die/1465912607.


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