The purpose of this work is to provide a broader framework for analyzing the inhomogeneous nonlinear Schr\"odinger equation (INLS) \[iu_t + u_{xx} \pm |x|^{-b}|u|^{\alpha-1}u=0, \quad 1<\alpha<5-2b\; \text{and}\; 0<b\leq 1/4.\] Specifically, we establish low-regularity well-posedness in the Fourier-Lebesgue $\widehat{L}^{p}$ spaces for $4/3<p<8$. The analysis is carried out differently for the cases $p<2$ and $p>2$. Primarily, in both cases, we prove global well-posedness for arbitrarily large initial data via the data decomposition method adapted for the Fourier-Lebesgue spaces. Furthermore, we obtain analogous results for INLS in modulation spaces $M^{p,p'}$ for $4/3<p<2$.
We study positive solutions of the partially confined stationary nonlinear Schr\"odinger equation $$-\Delta u+|y|^2u+\lambda u=g(u),\quad (y,z)\in\mathbb{R}^d\times\mathbb{R}^{m},\quad 1\leq d<N,\quad m:=N-d.$$ The spectrum of $-\Delta + |y|^2$ is given by $[d,\infty)$. We prove that $\lambda\geq-d$ is necessary for po...
Zhijie Chen, Youjun Wang, Jian-Jun Zhang et al.· 0 citations
We study nonlinear Schr\"odinger equations with nonlinear Stratonovich noise \begin{equation*} \mathrm{d} u\,=\, i\bigl[ \Delta u \,+\, \lambda|u|^{p-1}u\bigr] \, \mathrm{d} t \,+\,i|u|^{(q-1)/2}u\circ \mathrm{d} {W}, \end{equation*} in their energy space $H^1(\mathbb R^d;\mathbb C)$. By combining the stochastic Strich...
We investigate the maximization problem for the family of mixed-norm Strichartz inequalities for the Schr\"odinger equation, $\|e^{-it\Delta/2}f\|_{L_t^qL_{\boldsymbol{x}}^r(\mathbb{R}^{1+d})}\le C_{q,r}\lVert f\rVert_{L^2(\mathbb{R}^d)}$, with $2/q+d/r=d/2$, $q,r\geq 2$, and thus $r\leq 2d/(d-2)$ if $d\geq 3$. We show...
Felipe Gonçalves, Giuseppe Negro, D. O. E. Silva· 0 citations
We construct modified wave operators for the nonlinear Schr\"odinger equation $i\partial_tu+\frac12\Delta u=|u|^pu$ in the full subcritical long-range case $0<p<2/d$, with small, nonvanishing, analytic final data $W(x)$ with bounded logarithmic gradients. Previous results established large-time asymptotics for selected...
This paper is concerned with the following planar Schrodinger-Poisson system: $\begin{cases} -\Delta u +V(x)u+\lambda\phi u=u\ln u^2,&\text{in }{R}^2,\\ \Delta\phi=u^2,&\text{in }{R}^2. \end{cases}$ where $\lambda\in{R}$ is a parameter and $V\in C^1({R}^2,R}^{+})$ is a coercive potential. Due to the presence of the log...
We prove dispersive estimates for the three-dimensional defocusing energy-critical nonlinear Schr\"odinger equation associated with $\mathcal L_a=-\Delta+a|x|^{-2}$. For nonnegative potentials, we extend the known finite-$p$ theory to the endpoint $L^1\to L^\infty$. For negative potentials in the global well-posedness...
Tie-Song Jiang, Kexue Li· 0 citations
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