In this paper, we investigate how the method of (Poincar$\acute{e}$-Dulac) normal form can be applied to the theory of dispersive partial differential equations. In particular, we prove unconditional well-posedness of canonical dispersive equations on the real line. More precisely, we implement an infinite iteration scheme of normal form reductions to the cubic nonlinear Schr$\ddot{o}$dinger equation (NLS) and the modified Korteweg-de Vries equation (mKdV) on the real line and prove unconditional well-posedness in $H^s(\R)$ with (i) $s\geq\frac 16$ for the cubic NLS and (ii) $s>\frac 14$ for the mKdV. One novelty of this paper is that we present normal form reductions in an abstract form, reducing multilinear estimates of arbitrarily high degrees to successive applications to trilinear localized modulation estimates.
In Chapter 1, we briefly introduce the idea of (Poincar$\acute{e}$-Dulac) normal form approach to dispersive equations, and surmmarize some relevant previous researches. In Chapter 2, we establish crucial trilinear estimates (localized modulation estimates) for the cubic NLS and the mKdV. In Chapter 3, we perform an infinite iteration of normal form reductions and derive the normal form equation. We carry out the computation in Chapter 3 at a formal level. In Chapter 4, we justify the formal computation in Chapter 3 and then prove our main result.

- Advisors
- Kwon, Soon Sik
*researcher*; 권순식*researcher*

- Description
- 한국과학기술원 :수리과학과,

- Publisher
- 한국과학기술원

- Issue Date
- 2017

- Identifier
- 325007

- Language
- eng

- Description
학위논문(박사) - 한국과학기술원 : 수리과학과, 2017.8,[i, 57 p. :]

- Keywords
(Poincar$\acute{e}$-Dulac) Normal form reduction▼aModified KdV equation▼aNonlinear Schr$\ddot{o}dinger equation$▼aUnconditional Well-posedness; (푸앵카레-뒬락) 표준형▼a변형된 콜테베그-데브리스 방정식▼a비선형 슈뢰딩거 방정식▼a해의 존재성▼a(보조 공간에 의존하지 않는) 해의 존재성과 유일성

- Appears in Collection
- MA-Theses_Ph.D.(박사논문)

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