The following is a list of my publications.

Papers:

5. A Modulus-Squared Dirichlet Boundary Condition for Time-Dependent Complex Partial Differential Equations and its Application to the Nonlinear Schrödinger Equation.
   R.M. Caplan and R. Carretero-González.
   Submitted to Journal of Computational and Applied Mathematics. Abstract. PDF.
   
   
4. A Two-Step High-Order Compact Scheme for the Laplacian Operator and its Implementation in an Explicit Method for Integrating the Nonlinear Schrödinger Equation.
   R.M. Caplan and R. Carretero-González.
   Submitted to Journal of Computational and Applied Mathematics. Abstract. PDF.
   
   
3. Numerical Stability of Explicit Runge-Kutta Finite-Difference Schemes for the Nonlinear Schrödinger Equation.
   R.M. Caplan and R. Carretero-González.
   Submitted to Journal of Numerical Analysis, Industrial and Applied Mathematics. Abstract. PDF.
   
   
2. Existence, Stability, and Scattering of Bright Vortices in the Cubic-Quintic Nonlinear Schrödinger Equation.
   R.M. Caplan, R. Carretero-González, P.G. Kevrekidis, and B.A. Malomed.
   To appear in Mathematics and Computers in Simulation. Abstract. PDF.
   
   
1. Azimuthal Modulational Instability of Vortices in the Nonlinear Schrödinger Equation.
   R.M. Caplan, Q.E. Hoq, R. Carretero-González, and P.G. Kevrekidis.
   Optics Communications. 282 (2009) 1399-1405. Abstract. PDF.

Masters Thesis:

Azimuthal Modulational Instability of Vortex Solutions to the Two Dimensional Nonlinear Schrödinger Equation
Masters Thesis in Computational Science. 
Thesis chair: Professor Ricardo Carretero.    
Abstract: We study the azimuthal modulational instability (MI) of vortices with different topological charges, in the focusing two-dimensional nonlinear Schrödinger (NLS) equation.  This setting has direct application in the realm of Bose-Einstein condensates and light propagation in nonlinear crystals.  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 then applying a stability analysis in Fourier space of the azimuthal modes. We formulate predictions of growth rates of individual modes and find that vortices are unstable below a critical azimuthal wave number.  Steady state vortex solutions are found by first using a variational approach to obtain an asymptotic analytical ansatz, and then using it as an initial condition to a nonlinear equation numerical optimization routine. The stability analysis predictions are corroborated by direct numerical simulations of the NLS performed on a polar coordinate finite-difference scheme.  We briefly show how to extend the method to encompass nonlocal nonlinearities that tend to stabilize solutions.