TY - JOUR

T1 - New singular solutions of the nonlinear Schrödinger equation

AU - Fibich, Gadi

AU - Gavish, Nir

AU - Wang, Xiao Ping

N1 - Funding Information:
Partially supported by grant 2000311 from the United States–Israel Binational Science Foundation, Jerusalem, Israel.

PY - 2005/11/15

Y1 - 2005/11/15

N2 - We present numerical simulations of a new type of singular solutions of the critical nonlinear Schrödinger equation (NLS), that collapse with a quasi self-similar ring profile at a square root blowup rate. We find and analyze the equation of the ring profile. We observe that the self-similar ring profile is an attractor for a large class of radially-symmetric initial conditions, but is unstable under symmetry-breaking perturbations. The equation for the ring profile admits also multi-ring solutions that give rise to collapsing self-similar multi-ring solutions, but these solutions are unstable even in the radially-symmetric case, and eventually collapse with a single ring profile. Collapsing ring solutions are also observed in the supercritical NLS.

AB - We present numerical simulations of a new type of singular solutions of the critical nonlinear Schrödinger equation (NLS), that collapse with a quasi self-similar ring profile at a square root blowup rate. We find and analyze the equation of the ring profile. We observe that the self-similar ring profile is an attractor for a large class of radially-symmetric initial conditions, but is unstable under symmetry-breaking perturbations. The equation for the ring profile admits also multi-ring solutions that give rise to collapsing self-similar multi-ring solutions, but these solutions are unstable even in the radially-symmetric case, and eventually collapse with a single ring profile. Collapsing ring solutions are also observed in the supercritical NLS.

KW - Blowup rate

KW - Collapse

KW - Nonlinear Schrödinger equation

KW - Ring profile

KW - Self-similar solution

KW - Singularity

UR - http://www.scopus.com/inward/record.url?scp=27644512464&partnerID=8YFLogxK

U2 - 10.1016/j.physd.2005.08.007

DO - 10.1016/j.physd.2005.08.007

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AN - SCOPUS:27644512464

VL - 211

SP - 193

EP - 220

JO - Physica D: Nonlinear Phenomena

JF - Physica D: Nonlinear Phenomena

SN - 0167-2789

IS - 3-4

ER -