Singular Solutions of the Biharmonic Nonlinear Schrödinger Equation
- Creators
- Baruch, G.
- Fibich, G.
- Mandelbaum, E.
Abstract
We consider singular solutions of the L^2-critical biharmonic nonlinear Schrödinger equation. We prove that the blowup rate is bounded by a quartic-root, the solution approaches a quasi–self-similar profile, and a finite amount of L^2-norm, which is no less than the critical power, concentrates into the singularity. We also prove the existence of a ground-state solution. We use asymptotic analysis to show that the blowup rate of peak-type singular solutions is slightly faster than that of a quartic-root, and the self-similar profile is given by the ground-state standing wave. These findings are verified numerically (up to focusing levels of 10^8) using an adaptive grid method. We also use the spectral renormalization method to compute the ground state of the standing-wave equation, and the critical power for collapse, in one, two, and three dimensions.
Additional Information
© 2010 Society for Industrial and Applied Mathematics. Received by the editors January 27, 2010; accepted for publication (in revised form) September 29, 2010; published electronically December 22, 2010. This research was partially supported by grant 123/2008 from the Israel Science Foundation (ISF). We thank Nir Gavish and Steve Schochet for useful discussions. We thank an anonymous referee for pointing out the extension of Nawa's nonradial compactness lemma to the biharmonic NLS and its consequences (existence of ground state, nonradial power concentration), and we thank both referees for numerous useful comments.Attached Files
Published - Baruch2010p12458Siam_J_Appl_Math.pdf
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Additional details
- Eprint ID
- 21959
- Resolver ID
- CaltechAUTHORS:20110202-081454589
- 123/2008
- Israel Science Foundation (ISF)
- Created
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2011-02-14Created from EPrint's datestamp field
- Updated
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2021-11-09Created from EPrint's last_modified field