Singular standing-ring solutions of nonlinear partial differential equations

Abstract

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 Schrödinger equation iψ t (t, x) + ∆ψ + |ψ| 2σ ψ = 0 with σ > 2, the biharmonic nonlinear Schrödinger equation iψ t (t, x) − ∆ 2 ψ + |ψ| 2σ ψ = 0 with σ > 4, the nonlinear heat equation ψ t (t, x) − ∆ψ − |ψ| 2σ ψ = 0 with σ > 0, and the nonlinear biharmonic heat equation ψ t (t, x) + ∆ 2 ψ − |ψ| 2σ ψ = 0 with σ > 0.

Singular solutions of theL2-supercritical biharmonic nonlinear Schrödinger equation

“…We find that their properties mirror those of the supercritical NLS. Ring-type singular solutions of the supercritical BNLS were studied in [BFG10,BFM10a].…”

Positive and necklace solitary waves on bounded domains

“…Since the addition of a symmetric perturbation to an anti-symmetric profile lowers one of the peaks while increasing the other, the instability evolves as power flows from one pearl to the other. has an unstable eigenvalue Ω 1 (µ) of multiplicity 2 for µ cr < µ < ∞, whose value is the same as for ψ (2×1) sw , and an additional simple eigenvalue Ω 2 (µ) for µ (2) cr < µ < ∞, where µ cr < µ of Ω 2 .…”

“…This does not lead to a contradiction, however, since the VK condition applies to positive solutions, which is not the case for multi-pearl necklaces. P necklace th (n = 2) = P (R (2) µcr ) ≈ 0.12P cr ,…”

New solutions in the theory of self-focusing with saturating nonlinearity

“…Note that the forma tion of radially nonmonotonic distributions of beam intensity (from the initially monotonically decreasing radial distribution of intensity) during the propagation of a beam in a nonlinear medium has been observed in many numerical experiments, including media with cubic (nonsaturating) nonlinearity [12]. The appear ance of this nonmonotonicity depends on a number of factors such as the type of the nonlinearity, radial decrease rate of intensity (for example, for super Gaussian beams), and the nonsmoothness of a beam associated, for example, with small scale perturba tions in the distribution of the beam intensity on the boundary of a nonlinear medium (a plasma) [22][23][24].…”