In this paper we consider a class of fully nonlinear forced and reversible Schrödinger equations and prove existence and stability of quasi-periodic solutions. We use a Nash-Moser algorithm together with a reducibility theorem on the linearized operator in a neighborhood of zero. Due to the presence of the highest order derivatives in the non-linearity the classic KAM-reducibility argument fails and one needs to use a wider class of changes of variables such as diffeomorphisms of the torus and pseudo-differential operators. This procedure automatically produces a change of variables, well defined on the phase space of the equation, which diagonalizes the operator linearized at the solution. This gives the linear stability.

Feola, R., Procesi, M. (2015). Quasi-periodic solutions for fully nonlinear forced reversible Schrödinger equations. JOURNAL OF DIFFERENTIAL EQUATIONS, 259(7), 3389-3447 [10.1016/j.jde.2015.04.025].

Quasi-periodic solutions for fully nonlinear forced reversible Schrödinger equations

Feola, Roberto;PROCESI, MICHELA
2015

Abstract

In this paper we consider a class of fully nonlinear forced and reversible Schrödinger equations and prove existence and stability of quasi-periodic solutions. We use a Nash-Moser algorithm together with a reducibility theorem on the linearized operator in a neighborhood of zero. Due to the presence of the highest order derivatives in the non-linearity the classic KAM-reducibility argument fails and one needs to use a wider class of changes of variables such as diffeomorphisms of the torus and pseudo-differential operators. This procedure automatically produces a change of variables, well defined on the phase space of the equation, which diagonalizes the operator linearized at the solution. This gives the linear stability.
Feola, R., Procesi, M. (2015). Quasi-periodic solutions for fully nonlinear forced reversible Schrödinger equations. JOURNAL OF DIFFERENTIAL EQUATIONS, 259(7), 3389-3447 [10.1016/j.jde.2015.04.025].
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11590/301863
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