# Quasi-Periodic Solutions in a Nonlinear Schrödinger Equation

@inproceedings{Geng2007QuasiPeriodicSI, title={Quasi-Periodic Solutions in a Nonlinear Schr{\"o}dinger Equation}, author={Jiansheng Geng and Yingfei Yi}, year={2007} }

- Published 2007
DOI:10.1016/j.jde.2006.07.027

Abstract In this paper, one-dimensional (1D) nonlinear Schrodinger equation i u t − u x x + m u + | u | 4 u = 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 Birkhoff normal form reduction and an improved KAM method.

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## QUASI-PERIODIC SOLUTIONS OF 1D NONLINEAR SCHRÖDINGER EQUATION WITH A MULTIPLICATIVE POTENTIAL

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CITES METHODS

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## Quasi-Periodic Solutions for 1 D Schrödinger Equation with the Nonlinearity | u | 2 p u ∗

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## Quasi-periodic solutions for a Schrödinger equation with a quintic nonlinear term depending on the time and space variables

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#### References

##### Publications referenced by this paper.

SHOWING 1-10 OF 36 REFERENCES

## Green's function estimates for lattice Schrodinger operators and applications (AM-158)

VIEW 10 EXCERPTS

HIGHLY INFLUENTIAL

## On Elliptic Lower Dimensional Tori in Hamiltonian Systems

VIEW 11 EXCERPTS

HIGHLY INFLUENTIAL

## A KAM-theorem for equations of the Korteweg-de Vries type

VIEW 8 EXCERPTS

HIGHLY INFLUENTIAL

## A KAM-theorem for some nonlinear partial differential equations

VIEW 12 EXCERPTS

HIGHLY INFLUENTIAL

## Quasi-periodic solutions for a nonlinear wave equation

VIEW 12 EXCERPTS

HIGHLY INFLUENTIAL

## Construction of periodic solutions of nonlinear wave equations in higher dimension

VIEW 10 EXCERPTS

HIGHLY INFLUENTIAL

## Construction of quasi-periodic solutions for Hamiltonian perturbations of linear equations and applications to nonlinear PDE

VIEW 10 EXCERPTS

HIGHLY INFLUENTIAL

## Nearly integrable infinite-dimensional Hamiltonian systems

VIEW 9 EXCERPTS

HIGHLY INFLUENTIAL

## Newton's method and periodic solutions of nonlinear wave equations

VIEW 3 EXCERPTS

HIGHLY INFLUENTIAL

## Periodic and quasi-periodic solutions of nonlinear wave equations via KAM theory

VIEW 7 EXCERPTS

HIGHLY INFLUENTIAL