arXiv subjects

Jacky Jia Wei Chong

Publications and source records attributed to Jacky Jia Wei Chong.

3 recordsLinked to original sources

arXiv2020

Dynamical Hartree-Fock-Bogoliubov Approximation of Interacting Bosons

We consider a many-body Boson system with pairwise particle interaction given by $N^{3β-1}v(N^βx)$ for $0<β<1$ and $v$ a non-negative spherically-symmetric function. Our main result is the extension of the local-in-time Fock space (norm) approximation of the exact dynamics of squeezed states proved in Grillakis and Machedon, Comm. PDEs, (2017) for $0<β<\frac{2}{3}$ to a global-in-time norm approximation for $0<β<1$. Our work can also be viewed as a generalization of the results of Boccato, Cenatiempo, and Schlein, Ann. Henri Poincaré, (2017) to a more general set of initial data that includes coherent states along with an improved error estimate. The key ingredients in establishing the norm approximation are the work of Grillakis and Machedon on the local wellposedness theory, Grillakis and Machedon, Comm. PDEs, (2019), some recent established global estimates in Chong, Grillakis, Machedon, and Zhao, arXiv:2008.01753 (2020), and our quantitative result on the uniform in $N$ global wellposedness of the time-dependent Hartree-Fock-Bogoliubov (TBHFB) system.

math.AP↗

arXiv2020

Global estimates for the Hartree-Fock-Bogoliubov equations

We prove that certain Sobolev-type norms, slightly stronger than those given by energy conservation, stay bounded uniformly in time and $N$. This allows one to extend the local existence results of the second and third author globally in time. The proof is based on interaction Morawetz-type estimates and Strichartz estimates (including some new end-point results) for the equation $\{ \frac{1}{i}\partial_t-Δ_{x}-Δ_{y}+\frac{1}{N}V_N(x-y) \}Λ(t, x, y) =F$ in mixed coordinates such as $L^p(dt) L^q(dx) L^2(dy)$, $L^p(dt) L^q(dy) L^2(dx)$, $L^p(dt) L^q(d(x-y)) L^2(d(x+y))$. The main new technical ingredient is a dispersive estimate in mixed coordinates, which may be of interest in its own right.

math.AP↗

arXiv2020

Dynamics of Large Boson Systems with Attractive Interaction and a Derivation of the Cubic Focusing NLS in $\mathbb{R}^3$

We consider a system of $N$ bosons where the particles experience a short range two-body interaction given by $N^{-1}v_N(x)=N^{3β-1}v(N^βx)$ where $v \in C^\infty_c(\mathbb{R}^3)$, without a definite sign on $v$. We extend the results of M. Grillakis and M. Machedon, Comm. Math. Phys., $\textbf{324}$, 601(2013) and E. Kuz, Differ. Integral Equ., $\textbf{137}$, 1613(2015) regarding the second-order correction to the mean-field evolution of systems with repulsive interaction to systems with attractive interaction for $0<β<\frac{1}{2}$. Our extension allows for a more general set of initial data which includes coherent states. Inspired by the works of X. Chen and J. Holmer, Arch. Ration. Mech. Anal., $\textbf{221}$, 631(2016) and Int. Math. Res. Not., $\textbf{2017}$, 4173(2017), and P. T. Nam and M. Napiórkowski, Adv. Theor. Math. Phys., $\textbf{21}$, 683(2017), we also provide both a derivation of the focusing nonlinear Schrödinger equation (NLS) in $3$D from the many-body system and its rate of convergence toward mean-field for $0<β<\frac{1}{3}$. In particular, we give two derivations of the focusing NLS, one based on the $N$-norm approximation given in the work of Nam and Napiórkowski and the other via a method introduced in P. Pickl, J. Stat. Phys., $\textbf{140}$, 76(2010). The techniques used in this article are standard in the literature of dispersive PDEs. Nevertheless, the derivation of the focusing NLS had only previously been studied for the 1D \& 2D cases and conditionally answered for the 3D case for $0<β<\frac{1}{6}$.

math-ph↗