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

Journal Article · Thu Jan 10 04:00:00 EST 2008 · Journal of Computational Physics

DOI:https://doi.org/10.1016/j.jcp.2007.10.012· OSTI ID:21028302

Fourier split-step techniques are often used to compute soliton-like numerical solutions of the nonlinear Schroedinger 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}{sub perpendicular} {sup FDA} of 1/r ({partial_derivative})/({partial_derivative}r) r({partial_derivative})/({partial_derivative}r) that possesses an associated exact unitary representation of e{sup i/2{lambda}}{sup {delta}{sub perpendicular}{sup 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.

• OSTI ID: 21028302
• Journal Information: Journal of Computational Physics, Journal Name: Journal of Computational Physics Journal Issue: 3 Vol. 227; ISSN JCTPAH; ISSN 0021-9991
• Country of Publication: United States
• Language: English

Related Subjects

71 CLASSICAL AND QUANTUM MECHANICS
GENERAL PHYSICS
ALGORITHMS
APPROXIMATIONS
BESSEL FUNCTIONS
LASERS
MATRICES
MATRIX ELEMENTS
NONLINEAR PROBLEMS
NUMERICAL SOLUTION
PERIODICITY
PULSES
SCHROEDINGER EQUATION
SIMULATION
SOLITONS
SYMMETRY
WAVE FORMS