相关论文: Modified Scattering for the Time-Dependent Kohn--S…
In this paper, we study the global well-posedness and scattering of 3D defocusing, cubic Schr\"odinger equation. Recently, Dodson [arXiv:2004.09618] studied the global well-posedness in a critical Sobolev space $\dot{W}^{11/7,7/6}$. In this…
In this paper, we prove the decay and scattering in the energy space for nonlinear Schr\"odinger equations with regular potentials in $\Bbb R^d$ namely, $i{\partial _t}u + \Delta u - V(x)u + \lambda |u|^{p - 1}u = 0$. We will prove decay…
We consider the dispersion managed nonlinear Schr\"dinger equations with quintic and cubic nonlinearities in one and two dimensions, respectively. We prove the global well-posedness and scattering in $L_x^2$ for small initial data employing…
The purpose of this paper is to illustrate the I-method by studying low-regularity solutions of the nonlinear Schr\'[o]dinger equation in two space dimensions. By applying this method, together with the interaction Morawetz estimate, (see…
In this paper, we study the Cauchy problem of 2d Dirac equation with Hartree type nonlinearity $c(|\cdot|^{-\gamma} * \langle \psi, \beta \psi\rangle)\beta\psi$ with $c\in \mathbb R\setminus\{0\} $, $0 < \gamma < 2$. Our aim is to show the…
In the present paper, we consider the Cauchy problem of fourth order nonlinear Schr\"odinger type equations with a derivative nonlinearity. In one dimensional case, we prove that the fourth order nonlinear Schr\"odinger equation with the…
We consider both the defocusing and focusing cubic nonlinear Klein--Gordon equations $$ u_{tt} - \Delta u + u \pm u^3 =0 $$ in two space dimensions for real-valued initial data $u(0)\in H^1_x$ and $u_t(0)\in L^2_x$. We show that in the…
We consider the linear wave equation with the time-dependent scale-invariant damping and mass. We also treat the corresponding equation with the energy critical nonlinearity. Our aim is to show that the solution scatters to a modified…
Many integral equation-based methods are available for problems of time-harmonic electromagnetic scattering from perfect electric conductors. Among the many challenges that arise in such calculations are the avoidance of spurious…
The solution of the scattering problem based on the Lippmann-Schwinger equation requires in many cases a discretization of the spectrum in the continuum which does not respect the unitary equivalence of the S-matrix on the finite grid. We…
We extend the scattering result for the radial defocusing-focusing mass-energy double critical nonlinear Schr\"odinger equation in $d\leq 4$ given by Cheng et al. to the case $d\geq 5$. The main ingredient is a suitable long time…
The transfer matrix ${\mathbf{M}}$ of a short-range potential may be expressed in terms of the time-evolution operator for an effective two-level quantum system with a time-dependent non-Hermitian Hamiltonian. This leads to a dynamical…
The aim of the lecture is to briefly describe the mathematical background of scattering theory for two- and three-particle quantum systems. We discuss basic objects of the theory: wave and scattering operators and the corresponding…
We study the Schr\"odinger-Debye system over $\mathbb{R}^d$ iu_t+\frac 12\Delta u=uv,\quad \mu v_t+v=\lambda |u|^2 and establish the global existence and scattering of small solutions for initial data in several function spaces in…
We consider the Cauchy problem for the fourth order nonlinear Schr\"{o}dinger equation with derivative nonlinearity $(i\partial _t + \Delta ^2) u= \pm \partial (|u|^2u)$ on $\mathbb{R} ^d$, $d \ge 3$, with random initial data, where…
We construct (modified) scattering operators for the Vlasov-Poisson system in three dimensions, mapping small asymptotic dynamics as $t\to -\infty$ to asymptotic dynamics as $t\to +\infty$. The main novelty is the construction of modified…
We study the theory of scattering for a Schr"odinger equation in an external time dependent magnetic field in the Coulomb gauge, in space dimension 3. The magnetic vector potential is assumed to satisfy decay properties in time that are…
We prove scattering for the radial nonlinear Klein-Gordon equation $ \partial_{tt} u - \Delta u + u = -|u|^{p-1} u $ with $5 > p >3$ and data $ (u_{0}, u_{1}) \in H^{s} \times H^{s-1} $, $ 1 > s > 1- \frac{(5-p)(p-3)}{2(p-1)(p-2)} $ if $ 4…
We investigate the scattering matrix in mass-deformed N>=4 Chern-Simons models including as special cases the BLG and ABJM theories of multiple M2 branes. Curiously the structure of this scattering matrix in three spacetime dimensions is…
We consider the following wave guide nonlinear Schr\"odinger equation, \begin{equation} (i\partial \_t+\partial \_{xx}-\vert D\_y\vert )U=\vert U\vert ^2U\ \tag{WS} \end{equation} on the spatial cylinder $\mathbb{R} \_x\times \mathbb{T}…