Thesis Summary:

The present thesis contains contributions associated to three different contexts. The first one proposes a finite difference scheme that solves a High Order Nonlinear Schrödinger equation (HNLS) in one dimension, scheme that also has conservation and stabilization properties, if a certain damping function is present. The second one deals with a Nonlinear Schrödinger equation (NLS) in two dimensions, where a finite volume scheme was used to approximate the solution when a localized damping function is present. The scheme replicates a stabilization result proved by Cavalcanti, Corrêa, Ozsari, Sepúlveda y Véjar-Asem. In the third contribution, a hanging bridge problem is solved numerically using a finite difference scheme. The scheme also manages to replicate a stabilization result proved in Domingos Cavalcanti, M. Cavalcanti, Corrêa, Hajjej, Sepúlveda, y Véjar Asem.

ISI Publications from the Thesis

Marcelo CAVALCANTI, Wellington CORREA, Andrei V. FAMINSKII, Mauricio SEPúLVEDA, Rodrigo VéJAR: Well-posedness and asymptotic behavior of a generalized higher order nonlinear Schrödinger equation with localized dissipation. Computers & Mathematics with Applications, vol. 96, pp. 188-208, (2021).

Marcelo CAVALCANTI, Wellington CORREA, Türker ÖZSARI, Mauricio SEPúLVEDA, Rodrigo VéJAR: Exponential stability for the nonlinear Schrödinger equation with locally distributed damping. Communications in Partial Differential Equations, vol. 45, 9, pp. 1134-1167, (2020).

Marcelo CAVALCANTI, Wellington CORREA, André DOMINGOS, Zaid HAJJEJ, Mauricio SEPúLVEDA, Rodrigo VéJAR: Uniform decay rates for a suspension bridge with locally distributed nonlinear damping. Journal of the Franklin Institute, vol. 357, 4, pp. 2388-2419, (2020).

Marcelo CAVALCANTI, Wellington CORREA, Mauricio SEPúLVEDA, Rodrigo VéJAR: Finite difference scheme for a higher order nonlinear Schrödinger equation. Calcolo, vol. 56, 4, article:40, (2019).

Marcelo CAVALCANTI, Wellington CORREA, Mauricio SEPúLVEDA, Rodrigo VéJAR: Finite difference scheme for a high order nonlinear Schrödinger equation with localized damping. Studia Universitatis Babeș-Bolyai Mathematica, vol. 64, 2, pp. 161-172, (2019).