اشترك بالحزمة الذهبية واحصل على وصول غير محدود شمرا أكاديميا
تسجيل مستخدم جديدThe $1$d nonlinear Schrodinger equation with a weighted $L^1$ potential
96
0
0.0
(
0
)
اسأل ChatGPT حول البحث
ﻻ يوجد ملخص باللغة العربية
We consider the $1d$ cubic nonlinear Schrodinger equation with a large external potential $V$ with no bound states. We prove global regularity and quantitative bounds for small solutions under mild assumptions on $V$. In particular, we do not require any differentiability of $V$, and make spatial decay assumptions that are weaker than those found in the literature (see for example cite{Del,N,GPR}). We treat both the case of generic and non-generic potentials, with some additional symmetry assumptions in the latter case. Our approach is based on the combination of three main ingredients: the Fourier transform adapted to the Schrodinger operator, basic bounds on pseudo-differential operators that exploit the structure of the Jost function, and improved local decay and smoothing-type estimates. An interesting aspect of the proof is an approximate commutation identity for a suitable notion of a vectorfield, which allows us to simplify the previous approaches and extend the known results to a larger class of potentials. Finally, under our weak assumptions we can include the interesting physical case of a barrier potential as well as recover the result of cite{MMS} for a delta potential.
قيم البحث
اقرأ أيضاً
In this paper, we show the scattering of the solution for the focusing inhomogenous nonlinear Schrodinger equation with a potential begin{align*} ipartial_t u+Delta u- Vu=-|x|^{-b}|u|^{p-1}u end{align*} in the energy space $H^1(mathbb R^3)$. We pro
ve a scattering criterion, and then we use it together with Morawetz estimate to show the scattering theory.
In this paper, we are going to investigate Cauchy problem for nonlocal nonlinear Schrodinger equation with the initial potential $q_0(x)$ in weighted sobolev space $H^{1,1}(mathbb{R})$, begin{align*} iq_t(x,t)&+q_{xx}(x,t)+2sigma q^2(x,t)bar q(-x,t)=
0,quadsigma=pm1, q(x,0)&=q_0(x). end{align*} We show that the solution can be represented by the solution of a Riemann-Hilbert problem (RH problem), and assuming no discrete spectrum, we majorly apply $barpartial$-steepest cescent descent method on analyzing the long-time asymptotic behavior of it.
Consider the focusing nonlinear Schrodinger equation with a potential with a single negative eigenvalue. It has solitons with negative small energy, which are asymptotically stable, and solitons with positive large energy, which are unstable. We clas
sify the global dynamics into 9 sets of solutions in the phase space including both solitons, restricted by small mass, radial symmetry, and an energy bound slightly above the second lowest one of solitons. The classification includes a stable set of solutions which start near the first excited solitons, approach the ground states locally in space for large time with large radiation to the spatial infinity, and blow up in negative finite time.
We study the instability of standing-wave solutions $e^{iomega t}phi_{omega}(x)$ to the inhomogeneous nonlinear Schr{o}dinger equation $$iphi_t=-trianglephi+|x|^2phi-|x|^b|phi|^{p-1}phi, qquad inmathbb{R}^N, $$ where $ b > 0 $ and $ phi_{omega} $ is
a ground-state solution. The results of the instability of standing-wave solutions reveal a balance between the frequency $omega $ of wave and the power of nonlinearity $p $ for any fixed $ b > 0. $
In this paper we deal with the heat equation with drift in $L_{d+1}$. Basically, we prove that, if the free term is in $L_{q}$ with high enough $q$, then the equation is uniquely solvable in a rather unusual class of functions such that $partial_{t}u, D^{2}uin L_{p}$ with $p<d+1$ and $Duin L_{q}$.
سجل دخول لتتمكن من نشر تعليقات
التعليقات
جاري جلب التعليقات
سجل دخول لتتمكن من متابعة معايير البحث التي قمت باختيارها
