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Register a new userNormal form approach to the one-dimensional periodic cubic nonlinear Schrodinger equation in almost critical Fourier-Lebesgue spaces
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In this paper, we study the one-dimensional cubic nonlinear Schrodinger equation (NLS) on the circle. In particular, we develop a normal form approach to study NLS in almost critical Fourier-Lebesgue spaces. By applying an infinite iteration of normal form reductions introduced by the first author with Z. Guo and S. Kwon (2013), we derive a normal form equation which is equivalent to the renormalized cubic NLS for regular solutions. For rough functions, the normal form equation behaves better than the renormalized cubic NLS, thus providing a further renormalization of the cubic NLS. We then prove that this normal form equation is unconditionally globally well-posed in the Fourier-Lebesgue spaces $mathcal{F} L^p(mathbb{T})$, $1 leq p < infty$. By inverting the transformation, we conclude global well-posedness of the renormalized cubic NLS in almost critical Fourier-Lebesgue spaces in a suitable sense. This approach also allows us to prove unconditional uniqueness of the (renormalized) cubic NLS in $mathcal{F} L^p(mathbb{T})$ for $1leq p leq frac32$.
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In this paper we prove some multi-linear Strichartz estimates for solutions to the linear Schrodinger equations on torus $T^n$. Then we apply it to get some local well-posed results for nonlinear Schrodinger equation in critical $H^{s}(T^n)$ spaces. As by-products, the energy critical global well-posed results and energy subcritical global well-posed results with small initial data are also obtained.
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We consider the cubic Hyperbolic Schrodinger equation eqref{eq:nls} on torus $T^2$. We prove that sharp $L^4$ Strichartz estimate, which implies that eqref{eq:nls} is analytic locally well-posed in in $H^s(T^2)$ with $s>1/2$, meanwhile, the ill-posedness in $H^s(T^2)$ for $s<1/2$ is also obtained. The main difficulty comes from estimating the number of representations of an integer as a difference of squares.
In this paper, we consider the following three dimensional defocusing cubic nonlinear Schrodinger equation (NLS) with partial harmonic potential begin{equation*}tag{NLS} ipartial_t u + left(Delta_{mathbb{R}^3 }-x^2 right) u = |u|^2 u, quad u|_{t=0} = u_0. end{equation*} Our main result shows that the solution $u$ scatters for any given initial data $u_0$ with finite mass and energy. The main new ingredient in our approach is to approximate (NLS) in the large-scale case by a relevant dispersive continuous resonant (DCR) system. The proof of global well-posedness and scattering of the new (DCR) system is greatly inspired by the fundamental works of Dodson cite{D3,D1,D2} in his study of scattering for the mass-critical nonlinear Schrodinger equation. The analysis of (DCR) system allows us to utilize the additional regularity of the smooth nonlinear profile so that the celebrated concentration-compactness/rigidity argument of Kenig and Merle applies.
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