TY - JOUR

T1 - Singular standing-ring solutions of nonlinear partial differential equations

AU - Baruch, Guy

AU - Fibich, Gadi

AU - Gavish, Nir

N1 - Funding Information:
This research was partially supported by the Israel Science Foundation (ISF grant No. 123/08 ). The research of Nir Gavish was also partially supported by the Israel Ministry of Science Culture and Sports .

PY - 2010/10/15

Y1 - 2010/10/15

N2 - We present a general framework for constructing singular solutions of nonlinear evolution equations that become singular on a d-dimensional sphere, where d>1. The asymptotic profile and blowup rate of these solutions are the same as those of solutions of the corresponding one-dimensional equation that become singular at a point. We provide a detailed numerical investigation of these new singular solutions for the following equations: The nonlinear Schrdinger equation iψt(t,x)+Δψ+|ψ|2σψ=0 with σ>2, the biharmonic nonlinear Schrdinger equation iψt(t,x)- Δ2ψ+|ψ|2σψ=0 with σ>4, the nonlinear heat equation ψt(t,x)-Δψ-|ψ|2σψ=0 with σ>0, and the nonlinear biharmonic heat equation ψt(t,x)+Δ2ψ- |ψ|2σψ=0 with σ>0.

AB - We present a general framework for constructing singular solutions of nonlinear evolution equations that become singular on a d-dimensional sphere, where d>1. The asymptotic profile and blowup rate of these solutions are the same as those of solutions of the corresponding one-dimensional equation that become singular at a point. We provide a detailed numerical investigation of these new singular solutions for the following equations: The nonlinear Schrdinger equation iψt(t,x)+Δψ+|ψ|2σψ=0 with σ>2, the biharmonic nonlinear Schrdinger equation iψt(t,x)- Δ2ψ+|ψ|2σψ=0 with σ>4, the nonlinear heat equation ψt(t,x)-Δψ-|ψ|2σψ=0 with σ>0, and the nonlinear biharmonic heat equation ψt(t,x)+Δ2ψ- |ψ|2σψ=0 with σ>0.

KW - Biharmonic nonlinear Schrdinger equation

KW - Biharmonic nonlinear heat equation

KW - Blowup

KW - Nonlinear Schrdinger equation

KW - Nonlinear heat equation

KW - Ring

UR - http://www.scopus.com/inward/record.url?scp=77956095269&partnerID=8YFLogxK

U2 - 10.1016/j.physd.2010.07.009

DO - 10.1016/j.physd.2010.07.009

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AN - SCOPUS:77956095269

SN - 0167-2789

VL - 239

SP - 1968

EP - 1983

JO - Physica D: Nonlinear Phenomena

JF - Physica D: Nonlinear Phenomena

IS - 20-22

ER -