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Register a new userPeriodic Cubic Hyperbolic Schrodinger equation on $T^2$
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Authors
Yuzhao Wang
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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.
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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.
We consider the nonlinear Schrodinger equation on the half-line with a given Dirichlet (Neumann) boundary datum which for large $t$ tends to the periodic function $g_0^b(t)$ ($g_1^b(t)$). Assuming that the unknown Neumann (Dirichlet) boundary value tends for large $t$ to a periodic function $g_1^b(t)$ ($g_0^b(t)$), we derive an easily verifiable condition that the functions $g_0^b(t)$ and $g_1^b(t)$ must satisfy. Furthermore, we introduce two different methods, one based on the formulation of a Riemann-Hilbert problem, and one based on a perturbative approach, for constructing $g_1^b(t)$ ($g_0^b(t)$) in terms of $g_0^b(t)$ ($g_1^b(t)$).
We consider the nonlinear Schrodinger equation on the half-line with a given Dirichlet boundary datum which for large $t$ tends to a periodic function. We assume that this function is sufficiently small, namely that it can be expressed in the form $alpha g_0^b(t)$, where $alpha$ is a small constant. Assuming that the Neumann boundary value tends for large $t$ to the periodic function $g_1^b(t)$, we show that $g_1^b(t)$ can be expressed in terms of a perturbation series in $alpha$ which can be constructed explicitly to any desired order. As an illustration, we compute $g_1^b(t)$ to order $alpha^8$ for the particular case that $g_0^b(t)$ is the sum of two exponentials. We also show that there exist particular functions $g_0^b(t)$ for which the above series can be summed up, and therefore for these functions $g_1^b(t)$ can be obtained in closed form. The simplest such function is $exp(iomega t)$, where $omega$ is a real constant.
We consider solutions of the defocusing nonlinear Schrodinger (NLS) equation on the half-line whose Dirichlet and Neumann boundary values become periodic for sufficiently large $t$. We prove a theorem which, modulo certain assumptions, characterizes the pairs of periodic functions which can arise as Dirichlet and Neumann values for large $t$ in this way. The theorem also provides a constructive way of determining explicit solutions with the given periodic boundary values. Hence our approach leads to a class of new exact solutions of the defocusing NLS equation on the half-line.
In this note, we prove the profile decomposition for hyperbolic Schrodinger (or mixed signature) equations on $mathbb{R}^2$ in two cases, one mass-supercritical and one mass-critical. First, as a warm up, we show that the profile decomposition works for the ${dot H}^{frac12}$ critical problem, which gives a simple generalization of for instance one of the results in Fanelli-Visciglia (2013). Then, we give the derivation of the profile decomposition in the mass-critical case by proving an improved Strichartz estimate. We will use a very similar approach to that laid out in the notes of Killip-Visan (2008), but we are forced to do a double Whitney decomposition to accommodate an extra scaling symmetry that arises in the problem with mixed signature.
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