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Open AccessArticle

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

1
Department of Mathematical Sciences, University of Puerto Rico at Mayagüez, Mayagüez, Puerto Rico, PR 00681-9018, USA
2
School of Mathematical and Statistical Sciences, University of Texas at Rio Grande Valley, Edinburg, TX 78539-2999, USA
*
Author to whom correspondence should be addressed.
Academic Editor: Young Suh Kim
Symmetry 2016, 8(6), 38; https://doi.org/10.3390/sym8060038
Received: 2 March 2016 / Revised: 27 April 2016 / Accepted: 6 May 2016 / Published: 28 May 2016
(This article belongs to the Special Issue Harmonic Oscillators In Modern Physics)
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. View Full-Text
Keywords: generalized harmonic oscillator; paraxial wave equation; nonlinear schrödinger-type equations; riccati systems; solitons; J0101 generalized harmonic oscillator; paraxial wave equation; nonlinear schrödinger-type equations; riccati systems; solitons; J0101
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MDPI and ACS Style

Amador, G.; Colon, K.; Luna, N.; Mercado, G.; Pereira, E.; Suazo, E. On Solutions for Linear and Nonlinear Schrödinger Equations with Variable Coefficients: A Computational Approach. Symmetry 2016, 8, 38.

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