Quasi-periodic Solutions of the Schrödinger Equation with Arbitrary Algebraic Nonlinearities
نویسنده
چکیده
We present a geometric formulation of existence of time quasi-periodic solutions. As an application, we prove the existence of quasi-periodic solutions of b frequencies, b ≤ d + 2, in arbitrary dimension d and for arbitrary non integrable algebraic nonlinearity p. This reflects the conservation of d momenta, energy and L norm. In 1d, we prove the existence of quasi-periodic solutions with arbitrary b and for arbitrary p, solving a problem that started Hamiltonian PDE.
منابع مشابه
Quasi-Periodic Solutions for 1D Schrödinger Equation with the Nonlinearity |u|2pu∗
In this paper, one-dimensional (1D) nonlinear Schrödinger equation iut − uxx + |u|2pu= 0, p ∈N, with periodic boundary conditions is considered. It is proved that the above equation admits small-amplitude quasi-periodic solutions corresponding to 2-dimensional invariant tori of an associated infinite-dimensional dynamical system. The proof is based on infinite-dimensional KAM theory, partial no...
متن کاملQuasi-periodic solutions in a nonlinear Schrödinger equation
In this paper, one-dimensional (1D) nonlinear Schrödinger equation iut − uxx +mu+ |u|4u= 0 with the periodic boundary condition is considered. It is proved that for each given constant potential m and each prescribed integer N > 1, the equation admits a Whitney smooth family of small amplitude, time quasi-periodic solutions with N Diophantine frequencies. The proof is based on a partial Birkhof...
متن کاملNonuniqueness of Weak Solutions of the Nonlinear Schrödinger Equation
Generalized solutions of the Cauchy problem for the one-dimensional periodic nonlinear Schrödinger equation, with cubic or quadratic nonlinearities, are not unique. For any s < 0 there exist nonzero generalized solutions varying continuously in the Sobolev space H, with identically vanishing initial data.
متن کاملQuasi-periodic Solutions of 1d Nonlinear Schrödinger Equation with a Multiplicative Potential
This paper deals with one-dimensional (1D) nonlinear Schrödinger equation with a multiplicative potential, subject to Dirichlet boundary conditions. It is proved that for each prescribed integer b > 1, the equation admits smallamplitude quasi-periodic solutions, whose b-dimensional frequencies are small dilation of a given Diophantine vector. The proof is based on a modified infinitedimensional...
متن کاملQuasi-periodic solutions of Schrödinger equations with quasi-periodic forcing in higher dimensional spaces
In this paper, d-dimensional (dD) quasi-periodically forced nonlinear Schrödinger equation with a general nonlinearity iut −∆u+Mξu+ εφ(t)(u+ h(|u| 2)u) = 0, x ∈ T, t ∈ R under periodic boundary conditions is studied, where Mξ is a real Fourier multiplier and ε is a small positive parameter, φ(t) is a real analytic quasi-periodic function in t with frequency vector ω = (ω1,ω2 . . . ,ωm), and h(|...
متن کامل