{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,28]],"date-time":"2026-02-28T15:46:47Z","timestamp":1772293607202,"version":"3.50.1"},"reference-count":43,"publisher":"MDPI AG","issue":"6","license":[{"start":{"date-parts":[[2016,5,28]],"date-time":"2016-05-28T00:00:00Z","timestamp":1464393600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/100000001","name":"National Science Foundation","doi-asserted-by":"publisher","award":["DMS-1359016"],"award-info":[{"award-number":["DMS-1359016"]}],"id":[{"id":"10.13039\/100000001","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>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 Schr\u00f6dinger (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.<\/jats:p>","DOI":"10.3390\/sym8060038","type":"journal-article","created":{"date-parts":[[2016,5,30]],"date-time":"2016-05-30T10:26:58Z","timestamp":1464604018000},"page":"38","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":11,"title":["On Solutions for Linear and Nonlinear Schr\u00f6dinger Equations with Variable Coefficients: A Computational Approach"],"prefix":"10.3390","volume":"8","author":[{"given":"Gabriel","family":"Amador","sequence":"first","affiliation":[{"name":"Department of Mathematical Sciences, University of Puerto Rico at Mayag\u00fcez, Mayag\u00fcez, Puerto Rico, PR 00681-9018, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Kiara","family":"Colon","sequence":"additional","affiliation":[{"name":"Department of Mathematical Sciences, University of Puerto Rico at Mayag\u00fcez, Mayag\u00fcez, Puerto Rico, PR 00681-9018, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Nathalie","family":"Luna","sequence":"additional","affiliation":[{"name":"Department of Mathematical Sciences, University of Puerto Rico at Mayag\u00fcez, Mayag\u00fcez, Puerto Rico, PR 00681-9018, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Gerardo","family":"Mercado","sequence":"additional","affiliation":[{"name":"Department of Mathematical Sciences, University of Puerto Rico at Mayag\u00fcez, Mayag\u00fcez, Puerto Rico, PR 00681-9018, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Enrique","family":"Pereira","sequence":"additional","affiliation":[{"name":"Department of Mathematical Sciences, University of Puerto Rico at Mayag\u00fcez, Mayag\u00fcez, Puerto Rico, PR 00681-9018, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-1507-3697","authenticated-orcid":false,"given":"Erwin","family":"Suazo","sequence":"additional","affiliation":[{"name":"School of Mathematical and Statistical Sciences, University of Texas at Rio Grande Valley, Edinburg, TX 78539-2999, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2016,5,28]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"366","DOI":"10.1016\/j.physd.2009.12.005","article-title":"Bright and dark solitons in a quasi-1D Bose\u2013Einstein condensates modelled by 1D Gross\u2013Pitaevskii equation with time-dependent parameters","volume":"239","author":"Rajendran","year":"2010","journal-title":"Phys. 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