Escorcia-Tafur, Jose M.

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Soluciones para la ecuación de Schrodinger no lineal con coeficientes variables: Existencia de solitones y su dinámica
(2014-08) Escorcia-Tafur, Jose M.; Suazo, Erwin; College of Arts and Sciences - Sciences; Castillo, Paul; Rios Soto, Karen; Department of Mathematics; Serrano, Guillermo
The study of non-linear Schr¨odinger (NLS) equation with time dependent coefficients (non-autonomous) is of increasing interest for the different applications in non-linear optics, Bose-Einstein condensates and water waves. In this master thesis we impose a balance between the time-dependent coefficients; they are part of a nonlinear coupled Riccati system. We will use solutions of this Riccati system to study the dynamics of soliton solutions for the non-autonomous NLS, for this end we will use multiparameter solutions for this system proposed by Suslov and Suazo in 2009. As a contribution, in this thesis we present a modification of the method presented by Suslov in 2011 that allow us to construct global solutions. In particular, we are able to construct bright, dark and Peregrine type solitons for the non-autonomous NLS. Further, we are able to manipulate the parameters in order to produce solitons with bending. Finally, using a classical uniqueness result for the autonomous NLS in the space of functions L ∞ t L q x , with q = 2,∞, we prove uniqueness of the Cauchy initial value problem for the non-autonomous NLS.