On Solutions for Linear and Nonlinear Schrödinger Equations with Variable Coefficients: A Computational Approach

In this work, after reviewing two different ways to solve Riccati systems, we are able to present an extensive list of families of integrable nonlinear Schrödinger (NLS) equations with variable coefficients. Using Riccati equations and similarity transformations, we are able to reduce them to the standard NLS models. Consequently, we can construct bright-, dark- and Peregrine-type soliton solutions for NLS with variable coefficients. As an important application of solutions for the Riccati equation with parameters, by means of computer algebra systems, it is shown that the parameters change the dynamics of the solutions. Finally, we test numerical approximations for the inhomogeneous paraxial wave equation by the Crank-Nicolson scheme with analytical solutions found using Riccati systems. These solutions include oscillating laser beams and Laguerre and Gaussian beams