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

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 △⊥FDA of 1r∂∂rr∂∂r that possesses an associated exact unitary representation of ei2λ△⊥FDA. 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.