Quasi-Periodic Solutions for 1D Schrödinger Equation with the Nonlinearity |u|2pu∗
نویسنده
چکیده
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 normal form and scaling skills. © 2008 Elsevier Inc. All rights reserved. MSC: 37K55; 35B10; 35J10; 35Q40; 35Q55
منابع مشابه
Quasi-periodic Solutions of the Schrödinger Equation with Arbitrary Algebraic Nonlinearities
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 arbit...
متن کاملA KAM theorem for one dimensional Schrödinger equation with periodic boundary conditions
In this paper, one-dimensional (1D) nonlinear Schrödinger equation iut − uxx +mu+ g(u, ū) ū = 0, with Periodic Boundary Conditions is considered; m / ∈ 1 12Z is a real parameter and the nonlinearity g(u, ū)= ∑ j,l,j+l 4 ajlu j ū , aj l = alj ∈ R, a22 = 0 is a real analytic function in a neighborhood of the origin. The KAM machinery is adapted to fit the above equation so as to construct small-a...
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 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 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 کامل