Guy Baruch

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 Schrodinger equation i ψ t ( t , x ) + Δ ψ + | ψ | 2 σ ψ = 0 with σ > 2 , the biharmonic nonlinear Schrodinger equation i ψ t ( t , x ) − Δ 2 ψ + | ψ | 2 σ ψ = 0 with σ > 4 , the nonlinear heat equation ψ t ( t , x ) − Δ ψ − | ψ | 2 σ ψ = 0 with σ > 0 , and the nonlinear biharmonic heat equation ψ t ( t , x ) + Δ 2 ψ − | ψ | 2 σ ψ = 0 with σ > 0 .

Singular solutions of the L2-supercritical biharmonic nonlinear Schrödinger equation

We use asymptotic analysis and numerical simulations to study peak-type singular solutions of the supercritical biharmonic nonlinear Schrodinger equation. These solutions have a quartic-root blowup rate, and collapse with a quasi-self-similar universal profile, which is a zero-Hamiltonian solution of a fourth-order nonlinear eigenvalue problem.

Singular Solutions of the Biharmonic Nonlinear Schrödinger Equation

We consider singular solutions of the

Simulations of the nonlinear Helmholtz equation: arrest of beam collapse, nonparaxial solitons and counter-propagating beams

L^2

High-order numerical method for the nonlinear Helmholtz equation with material discontinuities in one space dimension

-critical biharmonic nonlinear Schrodinger 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

Ring-type singular solutions of the biharmonic nonlinear Schrödinger equation

L^2

Fourth Order Schemes for Time-Harmonic Wave Equations with Discontinuous Coefficients

-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

A high-order numerical method for the nonlinear Helmholtz equation in multidimensional layered media

10^8

High-Order Numerical Method for the Nonlinear Helmholtz Equation with Material Discontinuities

) 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.

High-order numerical solution of the nonlinear Helmholtz equation with axial symmetry

We solve the (2+1)D nonlinear Helmholtz equation (NLH) for input beams that collapse in the simpler NLS model. Thereby, we provide the first ever numerical evidence that nonparaxiality and backscattering can arrest the collapse. We also solve the (1+1)D NLH and show that solitons with radius of only half the wavelength can propagate over forty diffraction lengths with no distortions. In both cases we calculate the backscattered field, which has not been done previously. Finally, we compute the dynamics of counter-propagating solitons using the NLH model, which is more comprehensive than the previously used coupled NLS model.