Smooth quasi-periodic solutions for the perturbed mKdV equation

Siqi  Xu; Dongfeng Yan

doi:10.3934/cpaa.2016019

Article Contents

2016, Volume 15, Issue 5: 1857-1869. Doi: 10.3934/cpaa.2016019

This issue Previous Article Positive solutions of Kirchhoff type problem with singular and critical nonlinearities in dimension four Next Article Improved higher order poincaré inequalities on the hyperbolic space via Hardy-type remainder terms

Smooth quasi-periodic solutions for the perturbed mKdV equation

Siqi Xu^1, and
Dongfeng Yan^1,

1.
School of Mathematics and Statistics, Zhengzhou University, Zhengzhou, Henan, 450001

Received: October 31, 2015

Revised: February 29, 2016

Published: August 2016

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Abstract

This paper aims to study the time quasi-periodic solutions for one dimensional modified KdV (mKdV, for short) equation with perturbation \begin{eqnarray} u_t=-u_{x x x}-6 u^{2}u_x+perturbation ,x\in \mathbb{T}. \end{eqnarray} We show that, for any $n \in \mathbb{N}$ and a subset of $\mathbb{Z} \backslash \{0\}$ like $\{j_1 < j_2 < \cdots < j_n\}$, this equation admits a large amount of smooth n-dimensional invariant tori, along which exists a quantity of smooth quasi-periodic solutions. The proof is based on partial Birkhoff normal form and an unbounded KAM theorem established by Liu-Yuan in [Commun. Math. Phys., 307 (2011), 629-673].

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Mathematics Subject Classification: O175.14, O175.29.

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