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Smooth quasi-periodic solutions for the perturbed mKdV equation

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


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