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Yifei Wu

Publications and source records attributed to Yifei Wu.

At least 55 records · Page 3Linked to original sources
arXiv2023

Non-relativistic limit for the cubic nonlinear Klein-Gordon equations

We investigate the non-relativistic limit of the Cauchy problem for the defocusing cubic nonlinear Klein-Gordon equations whose initial velocity contains a factor of $c^2$, with $c$ being the light speed. While the classical WKB expansion is applied to approximate these solutions, the modulated profiles can be chosen as solutions to either a Schrödinger-wave equation or a Schrödinger equation. We show that, as the light speed tends to infinity, the error function is bounded by, (1) in the case of 2D and modulated Schrödinger-wave profiles, $Cc^{-2}$ with $C$ being a generic constant uniformly for all time, under $H^2$ initial data; (2) in the case of both 2D and 3D and modulated Schrödinger profiles, $c^{-2} +(c^{-2}t)^{α/4}$ multiplied by a generic constant uniformly for all time, under $H^α$ initial data with $2 \leq α\leq 4$. We also show the sharpness of the upper bounds in (1) and (2), and the required minimal regularity on the initial data in (2). One of the main tools is an improvement of the well-known result of Machihara, Nakanishi, and Ozawa in \cite{MaNaOz-KG-Limits} which may be of interest by itself. The proof also relies on \textit{a fantastic complex expansion} of the Klein-Gordon equation, \textit{introducing the leftward wave and exploring its enhanced performance} and a \textit{regularity gain mechanism} through a high-low decomposition.

math.AP↗
arXiv2023

Instability of the solitary waves for the Generalized Benjamin-Bona-Mahony Equation

In this work, we consider the generalized Benjamin-Bona-Mahony equation $$\partial_t u+\partial_x u+\partial_x( |u|^pu)-\partial_t \partial_x^{2}u=0, \quad(t,x) \in \mathbb{R} \times \mathbb{R}, $$ with $p>4$. This equation has the traveling wave solutions $ϕ_{c}(x-ct), $ for any frequency $c>1.$ It has been proved by Souganidis and Strauss \cite{Strauss-1990} that, there exists a number $c_{0}(p)>1$, such that solitary waves $ϕ_{c}(x-ct)$ with $1 c_{0}(p), $ $ϕ_{c}(x-ct)$ is orbitally stable. The linear exponential instability in the former case was further proved by Pego and Weinstein \cite{Pego-1991-eigenvalue}. In this paper, we prove the orbital instability in the critical case $c=c_{0}(p)$.

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arXiv2023

Movable Antenna-Enhanced Multiuser Communication: Optimal Discrete Antenna Positioning and Beamforming

Movable antennas (MAs) are a promising paradigm to enhance the spatial degrees of freedom of conventional multi-antenna systems by flexibly adapting the positions of the antenna elements within a given transmit area. In this paper, we model the motion of the MA elements as discrete movements and study the corresponding resource allocation problem for MA-enabled multiuser multiple-input single-output (MISO) communication systems. Specifically, we jointly optimize the beamforming and the MA positions at the base station (BS) for the minimization of the total transmit power while guaranteeing the minimum required signal-to-interference-plus-noise ratio (SINR) of each individual user. To obtain the globally optimal solution to the formulated resource allocation problem, we develop an iterative algorithm capitalizing on the generalized Bender's decomposition with guaranteed convergence. Our numerical results demonstrate that the proposed MA-enabled communication system can significantly reduce the BS transmit power and the number of antenna elements needed to achieve a desired performance compared to state-of-the-art techniques, such as antenna selection. Furthermore, we observe that refining the step size of the MA motion driver improves performance at the expense of a higher computational complexity.

cs.IT↗
arXiv2023

Globally Optimal Resource Allocation Design for IRS-Assisted Multiuser Networks with Discrete Phase Shifts

Intelligent reflecting surfaces (IRSs) are envisioned as a low-cost solution to achieve high spectral and energy efficiency in future communication systems due to their ability to customize wireless propagation environments. Although resource allocation design for IRS-assisted multiuser wireless communication systems has been exhaustively investigated in the literature, the optimal design and performance of such systems are still not well understood. To fill this gap, in this paper, we study optimal resource allocation for IRS-assisted multiuser multiple-input single-output (MISO) systems. In particular, we jointly optimize the beamforming at the base station (BS) and the discrete IRS phase shifts to minimize the total transmit power. For attaining the globally optimal solution of the formulated non-convex combinatorial optimization problem, we develop a resource allocation algorithm with guaranteed convergence based on Schur's complement and the generalized Bender's decomposition. Our numerical results reveal that the proposed algorithm can significantly reduce the BS transmit power compared to the state-of-the-art suboptimal alternating optimization-based approach, especially for moderate-to-large numbers of IRS elements.

cs.IT↗
arXiv2022

A modified splitting method for the cubic nonlinear Schrödinger equation

As a classical time-stepping method, it is well-known that the Strang splitting method reaches the first-order accuracy by losing two spatial derivatives. In this paper, we propose a modified splitting method for the 1D cubic nonlinear Schrödinger equation: \begin{align*} u^{n+1}=\mathrm{e}^{i\frac\tau2\partial_x^2}{\mathcal N}_τ\left[\mathrm{e}^{i\frac\tau2\partial_x^2}\big(Π_τ+\mathrm{e}^{-2πiλM_0τ}Π^τ\big)u^n\right], \end{align*} with ${\mathcal N}_t(ϕ)=\mathrm{e}^{-iλt|Π_τϕ|^2}ϕ,$ and $M_0$ is the mass of the initial data. Suitably choosing the filters $Π_τ$ and $Π^τ$, it is shown rigorously that it reaches the first-order accuracy by only losing $\frac32$-spatial derivatives. Moreover, if $γ\in (0,1)$, the new method presents the convergence rate of $τ^{\frac{4γ}{4+γ}}$ in $L^2$-norm for the $H^γ$-data; if $γ\in [1,2]$, it presents the convergence rate of $τ^{\frac25(1+γ)-}$ in $L^2$-norm for the $H^γ$-data. %In particular, the regularity requirement of the initial data for the first-order convergence in $L^2$-norm is only $H^{\frac32+}$. These results are better than the expected ones for the standard (filtered) Strang splitting methods. Moreover, the mass is conserved: $$\frac1{2π}\int_{\mathbb T} |u^n(x)|^2\,d x\equiv M_0, \quad n=0,1,\ldots, L . $$ The key idea is based on the observation that the low frequency and high frequency components of solutions are almost separated (up to some smooth components). Then the algorithm is constructed by tracking the solution behavior at the low and high frequency components separately.

math.NA↗
arXiv2022

Almost sure well-posedness and scattering of the 3D cubic nonlinear Schrödinger equation

We study the random data problem for 3D, defocusing, cubic nonlinear Schrödinger equation in $H_x^s(\mathbb{R}^3)$ with $s<\frac 12$. First, we prove that the almost sure local well-posedness holds when $\frac{1}{6}\leqslant s<\frac 12$ in the sense that the Duhamel term belongs to $H_x^{1/2}(\mathbb{R}^3)$. Furthermore, we prove that the global well-posedness and scattering hold for randomized, radial, large data $f\in H_x^{s}(\mathbb{R}^3)$ when $\frac{17}{40}< s<\frac 12$. The key ingredient is to control the energy increment including the terms where the first order derivative acts on the linear flow, and our argument can lower down the order of derivative more than $\frac12$. To our best knowledge, this is the first almost sure large data global result for this model.

math.AP↗
arXiv2022

An unfiltered low-regularity integrator for the KdV equation with solutions below ${\bf H^1}$

This article is concerned with the construction and analysis of new time discretizations for the KdV equation on a torus for low-regularity solutions below $H^1$. New harmonic analysis tools, including new averaging approximations to the exponential phase functions, new frequency decomposition techniques, and new trilinear estimates of the KdV operator, are established for the construction and analysis of time discretizations with higher convergence orders under low-regularity conditions. In addition, new techniques are introduced to establish stability estimates of time discretizations under low-regularity conditions without using filters when the energy techniques fail. The proposed method is proved to be convergent with order $γ$ (up to a logarithmic factor) in $L^2$ under the regularity condition $u\in C([0,T];H^γ)$ for $γ\in(0,1]$.

math.NA↗
arXiv2022

Designing IRS-Aided MIMO Systems for Secrecy Enhancement

Intelligent reflecting surfaces (IRSs) enable multiple-input multiple-output (MIMO) transmitters to modify the communication channels between the transmitters and receivers. In the presence of eavesdropping terminals, this degree of freedom can be used to effectively suppress the information leakage towards such malicious terminals. This leads to significant potential secrecy gains in IRS-aided MIMO systems. This work exploits these gains via a tractable joint design of downlink beamformers and IRS phase-shifts. In this respect, we consider a generic IRS-aided MIMO wiretap setting and invoke fractional programming and alternating optimization techniques to iteratively find the beamformers and phase-shifts that maximize the achievable weighted secrecy sum-rate. Our design concludes two low-complexity algorithms for joint beamforming and phase-shift tuning. Performance of the proposed algorithms are numerically evaluated and compared to the benchmark. The results reveal that integrating IRSs into MIMO systems not only boosts the secrecy performance of the system, but also improves the robustness against passive eavesdropping.

cs.IT↗
arXiv2021

Large global solutions for energy-critical nonlinear Schrödinger equation

In this work, we consider the 3D defocusing energy-critical nonlinear Schrödinger equation $i\partial_t u+Δu =|u|^4 u,\quad (t,x)\in \mathbb{R}\times \mathbb{R}^3$. Applying the outgoing and incoming decomposition presented in the recent work \cite{BECEANU-DENG-SOFFER-WU-2021}, we prove that any radial function $f$ with $χ_{\leq1}f\in H^1$ and $χ_{\geq1}f\in H^{s_0}$ with $\frac{5}{6}<s_0<1$, there exists an outgoing component $f_+$ (or incoming component $f_-$) of $f$, such that when the initial data is $f_+$, then the corresponding solution is globally well-posed and scatters forward in time; when the initial data is $f_-$, then the corresponding solution is globally well-posed and scatters backward in time.

math.AP↗
arXiv2021

A second order low-regularity integrator for the nonlinear Schrödinger equation

In this paper, we analyse a new exponential-type integrator for the nonlinear cubic Schrödinger equation on the $d$ dimensional torus $\mathbb T^d$. The scheme has recently also been derived in a wider context of decorated trees in [Y. Bruned and K. Schratz, arXiv:2005.01649]. It is explicit and efficient to implement. Here, we present an alternative derivation, and we give a rigorous error analysis. In particular, we prove second-order convergence in $H^γ(\mathbb T^d)$ for initial data in $H^{γ+2}(\mathbb T^d)$ for any $γ> d/2$. This improves the previous work in [Knöller, A. Ostermann, and K. Schratz, SIAM J. Numer. Anal. 57 (2019), 1967-1986]. The design of the scheme is based on a new method to approximate the nonlinear frequency interaction. This allows us to deal with the complex resonance structure in arbitrary dimensions. Numerical experiments that are in line with the theoretical result complement this work.

math.NA↗
arXiv2021

Large global solutions for nonlinear Schrödinger equations I, mass-subcritical cases

In this paper, we consider the nonlinear Schrödinger equation, $$ i\partial_{t}u+Δu= μ|u|^p u, \quad (t,x)\in \mathbb{R}^{d+1}, $$ with $μ=\pm1, p>0$. In this work, we consider the mass-subcritical cases, that is, $p\in (0,\frac4d)$. We prove that under some restrictions on $d,p$, any radial initial data in the critical space $\dot H^{s_c}(\mathbb{R}^d)$ with compact support, implies global well-posedness.

math.AP↗
arXiv2021

Large global solutions for nonlinear Schrödinger equations II, mass-supercritical, energy-subcritical cases

In this paper, we consider the defocusing mass-supercritical, energy-subcritical nonlinear Schrödinger equation, $$ i\partial_{t}u+Δu= |u|^p u, \quad (t,x)\in \mathbb R^{d+1}, $$ with $p\in (\frac4d,\frac4{d-2})$. We prove that under some restrictions on $d,p$, any radial function in the rough space $H^{s_0}(\mathbb R^d),\textit{for some } s_0<s_c$ with the support away from the origin, there exists an incoming/outgoing decomposition, such that the initial data in the outgoing part leads to the global well-posedness and scattering forward in time; while the initial data in the incoming part leads to the global well-posedness and scattering backward in time. The proof is based on Phase-Space analysis of the nonlinear dynamics.

math.AP↗
arXiv2021

A fully discrete low-regularity integrator for the 1D periodic cubic nonlinear Schrödinger equation

A fully discrete and fully explicit low-regularity integrator is constructed for the one-dimensional periodic cubic nonlinear Schrödinger equation. The method can be implemented by using fast Fourier transform with $O(N\ln N)$ operations at every time level, and is proved to have an $L^2$-norm error bound of $O(τ\sqrt{\ln(1/τ)}+N^{-1})$ for $H^1$ initial data, without requiring any CFL condition, where $τ$ and $N$ denote the temporal stepsize and the degree of freedoms in the spatial discretisation, respectively.

math.NA↗
arXiv2020

Global existences and asymptotic behavior for semilinear heat equation

In this paper, we consider the global Cauchy problem for the $L^2$-critical semilinear heat equations $ \partial_t h=Δh\pm |h|^{\frac4d}h, $ with $h(0,x)=h_0$, where $h$ is an unknown real function defined on $ \R^+\times\R^d$. In most of the studies on this subject, the initial data $h_0$ belongs to Lebesgue spaces $L^p(\R^d)$ for some $p\ge 2$ or to subcritical Sobolev space $H^{s}(\R^d)$ with $s>0$. {\it First,} we prove that there exists some positive constant $γ_0$ depending on $d$, such that the Cauchy problem is locally and globally well-posed for any initial data $h_0$ which is radial, supported away from the origin and in the negative Sobolev space $\dot H^{-γ_0}(\R^d)$. In particular, it leads to local and global existences of the solutions to Cauchy problem considered above for the initial data in a proper subspace of $L^p(\R^d)$ with some $p<2$. {\it Secondly,} the sharp asymptotic behavior of the solutions ( i.e. $L^2$-decay estimates ) as $t\to +\infty$ are obtained with arbitrary large initial data $h_0\in \dot H^{-γ_0}(\R^d)$ in the defocusing case and in the focusing case with suitably small initial data $h_0$.

math.AP↗
arXiv2020

Instability of the solitary waves for the generalized Boussinesq equations

In this work, we consider the following generalized Boussinesq equation \begin{align*} \partial_{t}^2u-\partial_{x}^2u+\partial_{x}^2(\partial_{x}^2u+|u|^{p}u)=0,\qquad (t,x)\in\mathbb R\times \mathbb R, \end{align*} with $0<p<\infty$. This equation has the traveling wave solutions $ϕ_ω(x-ωt)$, with the frequency $ω\in (-1,1)$ and $ϕ_ω$ satisfying \begin{align*} -\partial_{xx}ϕ_ω+(1-{ω^2})ϕ_ω-ϕ_ω^{p+1}=0. \end{align*} Bona and Sachs (1988) proved that the traveling wave $ϕ_ω(x-ωt)$ is orbitally stable when $0<p<4,$ $\frac p4<ω^2<1$. Liu (1993) proved the orbital instability under the conditions $0<p<4,$ $ω^2<\frac p4$ or $p\ge 4,$ $ω^2<1$. In this paper, we prove the orbital instability in the degenerate case $0<p<4,ω^2=\frac p4$ .

math.AP↗
arXiv2020

A first-order Fourier integrator for the nonlinear Schrödinger equation on $\mathbb T$ without loss of regularity

In this paper, we propose a first-order Fourier integrator for solving the cubic nonlinear Schrödinger equation in one dimension. The scheme is explicit and can be implemented using the fast Fourier transform. By a rigorous analysis, we prove that the new scheme provides the first order accuracy in $H^γ$ for any initial data belonging to $H^γ$, for any $γ>\frac32$. That is, up to some fixed time $T$, there exists some constant $C=C(\|u\|_{L^\infty([0,T]; H^γ)})>0$, such that $$ \|u^n-u(t_n)\|_{H^γ(\mathbb T)}\le C τ, $$ where $u^n$ denotes the numerical solution at $t_n=nτ$. Moreover, the mass of the numerical solution $M(u^n)$ verifies $$ \left|M(u^n)-M(u_0)\right|\le Cτ^5. $$ In particular, our scheme dose not cost any additional derivative for the first-order convergence and the numerical solution obeys the almost mass conservation law. Furthermore, if $u_0\in H^1(\mathbb T)$, we rigorously prove that $$ \|u^n-u(t_n)\|_{H^1(\mathbb T)}\le Cτ^{\frac12-}, $$ where $C= C(\|u_0\|_{H^1(\mathbb T)})>0$.

math.NA↗
arXiv2020

Embedded exponential-type low-regularity integrators for KdV equation under rough data

In this paper, we introduce a novel class of embedded exponential-type low-regularity integrators (ELRIs) for solving the KdV equation and establish their optimal convergence results under rough initial data. The schemes are explicit and efficient to implement. By rigorous error analysis, we first show that the ELRI scheme provides the first order accuracy in $H^γ$ for initial data in $H^{γ+1}$ for $γ>\frac12$. Moreover, by adding two more correction terms to the first order scheme, we show a second order ELRI that provides the second order accuracy in $H^γ$ for initial data in $H^{γ+3}$ for $γ\ge0$. The proposed ELRIs further reduce the regularity requirement of existing methods so far for optimal convergence. The theoretical results are confirmed by numerical experiments, and comparisons with existing methods illustrate the efficiency of the new methods.

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