TY - JOUR

T1 - Singular solutions of the biharmonic nonlinear Schrödinger equation

AU - Baruch, G.

AU - Fibich, G.

AU - Mandelbaum, E.

PY - 2010

Y1 - 2010

N2 - We consider singular solutions of the L2-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 L2-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 108) 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.

AB - We consider singular solutions of the L2-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 L2-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 108) 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.

KW - Biharmonic

KW - Blowup

KW - High-order dispersion

KW - NLS

KW - Nonlinear Schrödinger

KW - Self-similar solutions

UR - http://www.scopus.com/inward/record.url?scp=78751564006&partnerID=8YFLogxK

U2 - 10.1137/100784199

DO - 10.1137/100784199

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AN - SCOPUS:78751564006

SN - 0036-1399

VL - 70

SP - 3319

EP - 3341

JO - SIAM Journal on Applied Mathematics

JF - SIAM Journal on Applied Mathematics

IS - 8

ER -