A new fourth-order Fourier-Bessel split-step method for the extended nonlinear Schrödinger equation (Q2476871)

Fourier split-step techniques are often used to compute soliton-like numerical solutions of the nonlinear Schrödinger equation. Here, a new fourth-order implementation of the Fourier split-step algorithm is described for problems possessing azimuthal symmetry in \(3 + 1\)-dimensions. This implementation is based, in part, on a finite difference approximation \(\Delta^{\text{FDA}}_\perp\) of \(\frac1r\,\frac{\partial}{\partial r}\,r\,\frac{\partial}{\partial r}\) that possesses an associated exact unitary representation of \(e^{\frac i2\lambda\Delta^{\text{FDA}}_\perp}\). The matrix elements of this unitary matrix are given by special functions known as the associated Bessel functions. Hence the attribute Fourier-Bessel for the method. The Fourier-Bessel algorithm is shown to be unitary and unconditionally stable. The Fourier-Bessel algorithm is employed to simulate the propagation of a periodic series of short laser pulses through a nonlinear medium. This numerical simulation calculates waveform intensity profiles in a sequence of planes that are transverse to the general propagation direction, and labeled by the cylindrical coordinate \(z\). These profiles exhibit a series of isolated pulses that are offset from the time origin by characteristic times, and provide evidence for a physical effect that may be loosely termed normal mode condensation. Normal mode condensation is consistent with experimentally observed pulse filamentation into a packet of short bursts, which may occur as a result of short, intense irradiation of a medium.