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Stability of travelling waves for Schrödinger’s equations of cubic-quintic type This work is devoted to understand some mathematical aspects of the following nonlinear models: the nonlinear Schrödinger equation with double power in its non-linearity, that is iut + uxx + u|u| + u|u| = 0, (3) and a perturbation of delta type of this model, namely iut + uxx + Zδ(x)u+ u|u| + u|u| = 0. (4) For the first model, by using the theory of Jacobi elliptic integrals and the implicit function theorem, we obtain a family of standing waves u(x, t) = eφw(x), where φw : R → R is a positive periodic function of period L > 0, known as the wave profile. When L → ∞, we show that the periodic standing waves converge uniformly on compact intervals to the solitary waves. Moreover, using an extension of the Angulo&Natali stability theory, as well as, the stability ideas developed by Weinstein, Bona, Grillakis, Shatah and Strauss, we show the orbital stability of the standing waves for perturbations of the same period of the wave profile. Finally, an orbital instability result by subharmonic perturbations is proved. For the second model, by using the existence of the solitary wave φw,0 in the case Z = 0, we obtain two families of solitary peaks. We observe that when Z → 0, we have that φw,Z → φw,0, where φw,0 denotes the solitary wave. Then, using the analytic perturbation theory of unbounded linear operators, we obtain an accurate result about orbital stability of solitary peaks. Furthermore, we give some natural problems that can be solved futurely. In particular, we propose a new approach to solve question of linear stability of equilibrium solutions for certain type of parabolic equations.

Estabilidade de ondas viajantes para equações de Schrodinger do tipo cúbica-quíntica