Ronald M. Caplan

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We present a simple to use, yet powerful code package called NLSEmagic to numerically integrate the nonlinear Schrödinger equation in one, two, and three dimensions. NLSEmagic is a high-order finite-difference code package which utilizes graphic processing unit (GPU) parallel architectures. The codes running on the GPU are many times faster than their(More)
a r t i c l e i n f o a b s t r a c t Linearized numerical stability bounds for solving the nonlinear time-dependent Schrödinger equation (NLSE) using explicit finite-differencing are shown. The bounds are computed for the fourth-order Runge–Kutta scheme in time and both second-order and fourth-order central differencing in space. Results are given for(More)
An easy to implement modulus-squared Dirichlet (MSD) boundary condition is formulated for numerical simulations of time-dependent complex partial differential equations in multidi-mensional settings. The MSD boundary condition approximates a constant modulus-square value of the solution at the boundaries. Application of the MSD boundary condition to the(More)
We study the azimuthal modulational instability of vortices with different topological charges, in the focusing two-dimensional nonlinear Schrödinger (NLS) equation. The method of studying the stability relies on freezing the radial direction in the Lagrangian functional of the NLS in order to form a quasi-one-dimensional azimuthal equation of motion, and(More)
We describe and test an easy-to-implement two-step high-order compact (2SHOC) scheme for the Laplacian operator and its implementation into an explicit finite-difference scheme for simulating the nonlinear Schrödinger equation (NLSE). Our method relies on a compact 'double-differencing' which is shown to be computationally equivalent to standard(More)
The dynamics of vortex ring pairs in the homogeneous nonlinear Schrödinger equation is studied. The generation of numerically exact solutions of traveling vortex rings is described and their translational velocity compared to revised analytic approximations. The scattering behavior of co-axial vortex rings with opposite charge undergoing collision is(More)
An easy to implement modulus-squared Dirichlet (MSD) boundary condition is formulated for numerical simulations of time-dependent complex partial differential equations in multidimensional settings. The MSD boundary condition approximates a constant modulus-squared value of the solution at the boundaries and is defined as ∂Ψ ∂t | b ≈ i Im[ 1 Ψ b−1 ∂Ψ ∂t |(More)
We describe and test an easy-to-implement two-step high-order compact (2SHOC) scheme for the Laplacian operator and its implementation into an explicit finite-difference scheme for simulating the nonlinear Schrödinger equation (NLSE). Our method relies on a compact 'double-differencing' which is shown to be computationally equivalent to standard(More)