arXiv subjects

Zhaojie Yang

Publications and source records attributed to Zhaojie Yang.

9 recordsLinked to original sources

arXiv2026

On the internal modes of the ground state for 3D cubic--quintic equations

Motivated by the problem of asymptotic stability for ground states of the Klein--Gordon and Schrödinger equations in three spatial dimensions in the presence of internal modes, we study the corresponding models with cubic--quintic nonlinearities. These equations arise in several physical contexts; in particular, the cubic--quintic Klein--Gordon equation appears naturally in the study of spin-$0$ particles in quantum field theory. More precisely, let $Q_ω$ be the positive radial ground state of the three-dimensional cubic--quintic elliptic equation \[ -ΔQ_ω+ωQ_ω-Q_ω^3+Q_ω^5=0. \] We study two related, but logically distinct, spectral problems as $ω$ approaches the endpoint $3/16$ of the ground-state branch. First, we determine the complete discrete spectrum of the scalar Hessian \[ L_+=-Δ+ω-3Q_ω^2+5Q_ω^4. \] This operator is also the exact linearized spatial operator around the static state $Q_ω$ for the real scalar cubic--quintic Klein--Gordon equation with mass parameter $ω$. We prove that the number of its discrete angular-momentum sectors tends to infinity, and obtain sharp asymptotics for their locations and multiplicities. Second, for the cubic--quintic nonlinear Schrödinger equation, we analyze the full Hamiltonian matrix linearization and prove the existence of internal modes in precisely an interval of angular-momentum sectors of length comparable to $(3/16-ω)^{-1}$. %The rotating complex Klein--Gordon problem leads %to a different gyroscopically coupled pencil; since that pencil is not %analyzed here, no identification of its internal spectrum is asserted. These results provide a rigorous linear foundation for future nonlinear stability and radiation-damping analysis in three-dimensional cubic--quintic models.

math.AP↗

arXiv2024

Global well-posedness and uniform-in-time vanishing damping limit for the inviscid Oldroyd-B model

In this paper, we consider global strong solutions and uniform-in-time vanishing damping limit for the inviscid Oldroyd-B model in R^d, where d=2 and 3. The well-recognized problem of the global existence of smooth solutions for the 2D inviscid Oldroyd-B model without smallness assumptions is open due to the complex structure of Q. Therefore improving the smallness assumptions, especially in lower regularity class, is the core question in the area of fluid models. On the other hand, long-time behaviors of solutions including temporal decay and uniform-in-time damping stability are also of deep significance. These problems have been widely studied, however, the existing results are not regularity critical and the (uniform) vanishing damping limit has not been discussed. The goal of this work is to dig deeper in this direction. In this work we first establish the local well-posedness in the sense of Hadamard with critical regularity. Then, by virtue of the sharp commutator estimate for Calderon-Zygmund operator, we establish the global existence of solutions for d=2 with damping in the low regularity class, which to our best knowledge, is novel in the literature. Furthermore, in both 2D and 3D cases, we prove the global existence of the solutions to the inviscid Oldroyd-B model independent of the damping parameters. In addition, we obtain the optimal temporal decay rates and time integrability by improving the existing Fourier splitting method and developing a novel decomposition strategy. One of the major contributions of the presenting paper is to prove the uniform-in-time vanishing damping limit for the inviscid Oldroyd-B model and discover the correlation between sharp vanishing damping rate and the temporal decay rate. Finally, we will support our findings by providing numerical evidence regarding the vanishing damping limit in the periodic domain T^d.

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arXiv2023

A Precoding for ORIS-Assisted MIMO Multi-User VLC System

In this paper, we study a multi-user visible light communication (VLC) system assisted with optical reflecting intelligent surface (ORIS). Joint precoding and alignment matrices are designed to maximize the average signal-to-interference plus noise ratio (SINR) criteria. Considering the constraints of the constant mean transmission power of LEDs and the power associated with all users, an optimization problem is proposed. To solve this problem, we utilize an alternating optimization algorithm to optimize the precoding and alignment matrices. The simulation results demonstrate that the resultant SINR of the proposed method outperforms ZF and MMSE precoding algorithms.

cs.IT↗

arXiv2023

Backward Uniqueness for 3D Navier-Stokes Equations with Non-trivial Final Data and Applications

Presented is a backward uniqueness result of bounded mild solutions of 3D Navier-Stokes Equations in the whole space with non-trivial final data. A direct consequence is that a solution must be axi-symmetric in $[0, T]$ if it is so at time $T$. The proof is based on a new weighted estimate which enables to treat terms involving Calderon-Zygmund operators. The new weighted estimate is expected to have certain applications in control theory when classical Carleman-type inequality is not applicable.

math.AP↗

arXiv2023

A Public Information Precoding for MIMO Visible Light Communication System Based on Manifold Optimization

Visible light communication (VLC) is an attractive subset of optical communication that provides a high data rate in the access layer of the network. The combination of multiple inputmultiple output (MIMO) with a VLC system leads to a higher speed of data transmission named as MIMO-VLC system. In multi-user (MU) MIMO-VLC, a LED array transmits signals for users. These signals are categorized as signals of private information for each user and signals of public information for all users. The main idea of this paper is to design an omnidirectional precoding to transmit the signals of public information in the MUMIMO-VLC network. To this end, we propose to maximize the achievable rate which leads to maximizing the received mean power at the possible location of the users. Besides maximizing the achievable rate, we consider equal mean transmission power constraint in all LEDs to achieve higher power efficiency of the power amplifiers used in the LED array. Based on this we formulate an optimization problem in which the constraint is in the form of a manifold and utilize a gradient method projected on the manifold to solve the problem. Simulation results indicate that the proposed omnidirectional precoding can achieve superior received mean power and bit error rate with respect to the classical form without precoding utilization.

cs.IT↗

arXiv2023

Energy transfer and radiation in Hamiltonian nonlinear Klein-Gordon equations: general case

In this paper, we consider Klein-Gordon equations with cubic nonlinearity in three spatial dimensions, which are Hamiltonian perturbations of the linear one with potential. It is assumed that the corresponding Klein-Gordon operator $B = \sqrt{-Δ+ V(x) + m^2} $ admits an arbitrary number of possibly degenerate eigenvalues in $(0, m)$, and hence the unperturbed linear equation has multiple time-periodic solutions known as bound states. In \cite{SW1999}, Soffer and Weinstein discovered a mechanism called Fermi's Golden Rule for this nonlinear system in the case of one simple but relatively large eigenvalue $Ω\in (\frac{m}{3}, m)$, by which energy is transferred from discrete to continuum modes and the solution still decays in time. In particular, the exact energy transfer rate is given. In \cite{LLY22}, we solved the general one simple eigenvalue case. In this paper, we solve this problem in full generality: multiple and simple or degenerate eigenvalues in $(0, m)$. The proof is based on a kind of pseudo-one-dimensional cancellation structure in each eigenspace, a renormalized damping mechanism, and an enhanced damping effect. It also relies on a refined Birkhoff normal form transformation and an accurate generalized Fermi's Golden Rule over those of Bambusi--Cuccagna \cite{BC}.

math.AP↗

arXiv2022

Energy transfer, weak resonance, and Fermi's golden rule in Hamiltonian nonlinear Klein-Gordon equations

This paper focuses on a class of nonlinear Klein-Gordon equations in three dimensions, which are Hamiltonian perturbations of the linear Klein-Gordon equation with potential. The unperturbed dynamical system has a bound state with frequency $ω$, a spatially localized and time periodic solution. In quantum mechanics, metastable states, which last longer than expected, have been observed. These metastable states are a consequence of the instability of the bound state under the nonlinear Fermi's Golden Rule. In this study, we explore the underlying mathematical instability mechanism from the bound state to these metastable states. Besides, we derive the sharp energy transfer rate from discrete to continuum modes, when the discrete spectrum was not close to the continuous spectrum of the Schördinger operator $H= -Δ+ V + m^2$, i.e. weak resonance regime $ σ_c(\sqrt{H}) = [m, \infty)$, $0< 3ω< m$. This extends the work of Soffer and Weinstein \cite{SW1999} for resonance regime $3ω> m$ and confirms their conjecture in \cite{SW1999}. Our proof relies on a more refined version of normal form transformation of Bambusi and Cuccagna \cite{BC}, the generalized Fermi's Golden Rule, as well as certain weighted dispersive estimates.

math.AP↗

arXiv2022

Radiation of the energy-critical wave equation with compact support

We prove exterior energy lower bounds for (nonradial) solutions to the energy-critical nonlinear wave equation in space dimensions $3 \le d \le 5$, with compactly supported initial data. In particular, it is shown that nontrivial global solutions with compact spatial support must be radiative in the sense that at least one of the following is true: (1) $\int_{|x|> |t|} \left( |\partial_t u|^2 + |\nabla u|^2 \right) \mathrm{d}x \ge η_1(u) > 0, \ \mathrm{for} \ \mathrm{all} \ t \ge 0 \ \mathrm{or} \ \mathrm{all} \ t \le 0,$ (2) $\int_{|x|> -\varepsilon +|t|} \left( |\partial_t u|^2 + |\nabla u|^2 \right) \mathrm{d}x \ge η_2(\varepsilon, u) > 0, \ \mathrm{for} \ \mathrm{all} \ t \in \mathbb{R}, \varepsilon > 0.$ In space dimensions 3 and 4, a nontrivial soliton background is also considered. As an application, we obtain partial results on the rigidity conjecture concerning solutions with the compactness property, including a new proof for the global existence of such solutions.

math.AP↗

arXiv2019

Analysis and Optimization of Tail-Biting Spatially Coupled Protograph LDPC Codes for BICM-ID Systems

As a typical example of bandwidth-efficient techniques, bit-interleaved coded modulation with iterative decoding (BICM-ID) provides desirable spectral efficiencies in various wireless communication scenarios. In this paper, we carry out a comprehensive investigation on tail-biting (TB) spatially coupled protograph (SCP) low-density parity-check (LDPC) codes in BICM-ID systems. Specifically, we first develop a two-step design method to formulate a novel type of constellation mappers, referred to as labeling-bit-partial-match (LBPM) constellation mappers, for SC-P-based BICM-ID systems. The LBPM constellation mappers can be seamlessly combined with high-order modulations, such as M-ary phase-shift keying (PSK) and M-ary quadrature amplitude modulation (QAM). Furthermore, we conceive a new bit-level interleaving scheme, referred to as variable node matched mapping (VNMM) scheme, which can substantially exploit the structure feature of SC-P codes and the unequal protection-degree property of labeling bits to trigger the wave-like convergence for TB-SC-P codes. In addition, we propose a hierarchical extrinsic information transfer (EXIT) algorithm to predict the convergence performance (i.e., decoding thresholds) of the proposed SC-P-based BICM-ID systems. Theoretical analyses and simulation results illustrate that the LBPM-mapped SC-P-based BICM-ID systems are remarkably superior to the state-of-the-art mapped counterparts. Moreover, the proposed SC-P-based BICM-ID systems can achieve even better error performance with the aid of the VNMM scheme. As a consequence, the proposed LBPM constellation mappers and VNMM scheme make the SC-P-based BICM-ID systems a favorable choice for the future-generation wireless communication systems.

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