Modelos de Schrödinger não lineares quase integráveis e infinitas cargas exatamente conservadas e sólitons brilhantes

Abstract

In this thesis we present the results published in JHEP 05 (2017) 106, in which we study the deformed non-linear Schrodinger ¨ model (1+1) dimensions.Deformations on the focusing non-linear Schrodinger ¨ model (NLS) are considered in the context of the quasiintegrability concept. We strenghen the results of JHEP 09 (2012) 13 for bright soliton collisions.We addressed the focusing NLS as a complement to the one in JHEP 03 (2016) 005, in which the modifield defocusing NLS models wiih dark solitons where show to exihibit an infinite tower of exactly conserved charges. We consider a parameterization with deformed NLS-type field and anomalies β (j) defined in the context of infinite quasiconservation laws. We show, by means of analytical and numerical methods, that for certain two-brightsolitons solutions, in which the modulus and phase of the complex modified NLS field exhibit even parities under a space-reflection symmetry, the first four and the sequensce of even order charges are exactly conserverd during the scattering process of the solitons (that is,β (j) = 0, j = 1, 2, 3, 4,, they are identically satisfied).Extensive numerical simulations were performed and bright solitons were considered for a deformed potential, with deformation parameter . The potencial = 0 contains the usual NLS integrable model, with bright soliton solutions for the coupling η < 0 (for η > 0 NLS displays dark solitons). for of two-soliton field componentes without definite parity we also show numericallly the vanishing the first non-trivial(β (4) = 0) anomaly and the exactly conservation of the corresponding charge is satisfied (Q(4) = constante),were the parity symmetry seems to be sufficient but not a necessary condition for the existence of the infinite tower of conserved charges. The models supports elastic scattering of solitons for a wide range of values of the amplitudes and velocities and the set η, , since the NLS equation i.s ubiquitous, our results may find potencial applications in several areas of non-linear science.