arXiv subjects

Kaizhi Wang

Publications and source records attributed to Kaizhi Wang.

At least 19 recordsLinked to original sources

arXiv2026

Generalized Hamiltonian gradient flow of contact type I: regularity of the fundamental solutions

This paper studies generalized Hamiltonian gradient flows for contact-type Hamilton--Jacobi equations, adopting the variational framework of Herglotz's principle and its fundamental solution $h_L(t,x,y,u)$. Two main results are presented. First, precise first-order sensitivity relations are derived, linking derivatives of $h_L$ to dual arcs of minimizing trajectories. Second, quantitative second-order estimates show that $h_L$ is locally semiconcave and, over short time, semiconvex, indeed uniformly convex in certain variables. These regularity properties follow from a detailed variational analysis of minimizing trajectories. The work establishes a foundation for intrinsic methods in analyzing singularity propagation and generalized gradient flows in contact context, with implications for weak KAM theory, optimal transport, and regularity of viscosity solutions.

math.AP↗

arXiv2025

Periodic limit for non-autonomous Lagrangian systems and applications to a Kuramoto type model

This paper explores the asymptotic properties of non-autonomous Lagrangian systems, assuming that the associated Tonelli Lagrangian converges to a time-periodic function. Specifically, given a continuous initial condition, we provide a suitable construction of a Lax-Oleinik semigroup such that it converges toward a periodic solution of the equation. Moreover, the graph of its gradient converges as time tends to infinity to the graph of the gradient of the periodic limit function with respect to the Hausdorff distance. Finally, we apply this result to a Kuramoto-type model, proving the existence of an invariant torus given by the graph of the gradient of the limiting periodic solution of the Hamilton-Jacobi equation.

math.OC↗

arXiv2025

Discretization and Vanishing Discount Problems for First-order Mean Field Games

This article focuses two issues related to the first-order discounted mean field games system. The first is the time discretization problem. The time discretization approach enables us to prove the existence of solutions (u,m) of the system, where u is a viscosity solution of the discounted Hamilton-Jacobi equation and m is a projected minimizing measure satisfying the continuity equation in the sense of distributions. The second is the vanishing discount problems for both the discounted mean field games system and its discretized system. The methods we use primarily derive from weak KAM theory. Moreover, we provide an example demonstrating the non-uniqueness of solutions to the discounted mean field games system.

math.AP↗

arXiv2025

Persistence of Invariant Tori for Stochastic Nonlinear Schrödinger in the Sense of Most Probable Paths

This paper investigates the application of KAM theory to the stochastic nonlinear Schrödinger equation on infinite lattices, focusing on the stability of low-dimensional invariant tori in the sense of most probable paths. For generality, we provide an abstract proof within the framework of stochastic Hamiltonian systems on infinite lattices. We begin by constructing the Onsager-Machlup functional for these systems in a weighted infinite sequence space. Using the Euler-Lagrange equation, we identify the most probable transition path of the system's trajectory under stochastic perturbations. Additionally, we establish a large deviation principle for the system and derive a rate function that quantifies the deviation of the system's trajectory from the most probable path, especially in rare events. Combining this with classical KAM theory for the nonlinear Schrödinger equation, we demonstrate the persistence of low-dimensional invariant tori under small deterministic and stochastic perturbations. Furthermore, we prove that the probability of the system's trajectory deviating from these tori can be described by the derived rate function, providing a new probabilistic framework for understanding the stability of stochastic Hamiltonian systems on infinite lattices.

math.DS↗

arXiv2025

Quasi-periodic Swing via Weak KAM Theory

We investigate the dynamics of the quasi-periodic swing equations from the perspective of weak KAM theory. To this end, we firstly study a class of Hamiltonian systems. We obtain that the limit $u$, which derived from convergence of a sequence of functional minimizers, satisfies Hamilton-Jacobi equations in a weak sense. This is the so-called weak KAM solution. Meanwhile, we also get aminimal measures $μ$. Finally, we derive that the existence of weak KAM solutions for the swing equations.

math.DS↗

arXiv2025

Direct tensor processing with coherent light

Tensor processing is the cornerstone of modern technological advancements, powering critical applications in data analytics and artificial intelligence. While optical computing offers exceptional advantages in bandwidth, parallelism, and energy efficiency, existing methods optimized for scalar operations struggle to efficiently handle tensor-based tasks, limiting their applicability in complex applications, such as neural networks. Here, we report Parallel Optical Matrix Matrix Multiplication (POMMM), a novel paradigm that enables fully parallel tensor processing through a single coherent light propagation. This approach addresses key limitations of current optical methods, scaling the performance with data dimension, while improving theoretical computational power and efficiency. We demonstrate its high consistency with GPU based matrix matrix multiplication across both real-valued and complex valued domains. Moreover, we showcase its adaptability, scalability, and versatility in tensor processing applications such as convolutional and vision transformer neural networks. Furthermore, we analyse the theoretical compatibility and efficiency of POMMM in relation to existing optical computing paradigms, highlighting its potential to outperform current state-of-the-art methods. By enabling a variety of computational tasks and supporting multi2 wavelength and large-scale expansion, POMMM provides a scalable, high-efficient foundation for advancing next-generation optical computing.

physics.optics↗

arXiv2025

Relaxed Lagrangian Approach to First-Order Non-Convex Mean Field Type Control Problem

This paper addresses the existence of equilibria for Mean Field type Control problems of first-order with non-convex action functional. Introducing a relaxed Lagrangian approach on the Wasserstein space to handle the lack of convexity. we prove the existence of new relaxed Nash equilibria and we show that our existence result encompasses the classical Mean Field Control problem's existence result under convex data conditions.

math.OC↗

arXiv2025

High-rate self-referenced continuous-variable quantum key distribution over high-loss free-space channel

The advent of quantum computers has significantly challenged the security of traditional cryptographic systems, prompting a surge in research on quantum key distribution (QKD). Among various QKD approaches, continuous-variable QKD (CVQKD) offers superior resilience against background noise. However, the local local oscillator (LLO) CVQKD scheme faces substantial physical limitations in scenarios with high channel attenuation, and the large attenuation CVQKD remains unrealized. Bottleneck challenges include ensuring stable low-noise transmission and accurately estimating parameters under fluctuating channel conditions. In this paper, we introduce a continuous-time mode theory for high-precision estimation of time-varying parameters and design a free-space experimental system with a main quantum system and an auxiliary counterpart. We further develop advanced digital signal post-processing techniques for compensating time-varying frequency offset and phase noise under dynamic channel. Notably, the estimation of the time-varying free-space channel is achieved through the use of the auxiliary quantum system. Through experimental validation, we first demonstrate high-rate secure quantum key distribution over high-loss free-space channels. Specifically, we achieve asymptotic key rates of 76.366 kbps and 403.896 kbps in 25 dB attenuation free-space channels without turbulence and 21.5 dB average attenuation free-space channels with turbulence, respectively. Additionally, we confirm the feasibility of experiments on mildly turbulent atmospheric channels spanning at least 10.5 km using current equipments. Our scheme provides direct insight into constructing an integrated air-ground quantum communication network.

quant-ph↗

arXiv2025

Lyapunov stability and uniqueness problems for Hamilton-Jacobi equations without monotonicity

We consider the evolutionary Hamilton-Jacobi equation \begin{align*} w_t(x,t)+H(x,Dw(x,t),w(x,t))=0, \quad(x,t)\in M\times [0,+\infty), \end{align*} where $M$ is a compact manifold, $H:T^*M\times R\to R$, $H=H(x,p,u)$ satisfies Tonelli conditions in $p$ and the Lipschitz condition in $u$. This work mainly concerns with the Lyapunov stability (including asymptotic stability, and instability) and uniqueness of stationary viscosity solutions of the equation. A criterion for stability and a criterion for instability are given. We do not utilize auxiliary functions and thus our method is different from the classical Lyapunov's direct method. We also prove several uniqueness results for stationary viscosity solutions. The Hamiltonian $H$ has no concrete form and it may be non-monotonic in the argument $u$, where the situation is more complicated than the monotonic case. Several simple but nontrivial examples are provided, including the following equation on the unit circle \[ w_t(x,t)+\frac{1}{2}w^2_x(x,t)-a\cdot w_x(x,t)+(\sin x+b)\cdot w(x,t)=0,\quad x\in \mathbf{S}, \] where $a$, $b\in R$ are parameters. We analyze the stability, and instability of the stationary solution $w=0$ when parameters vary, and show that $w=0$ is the unique stationary solution when $a=0$, $b>1$ and $a\neq0$, $b\geqslant 1$. The sign of the integral of $\frac{\partial H}{\partial u}$ with respect to the Mather measure of the contact Hamiltonian system generated by $H$ plays an essential role in the proofs of aforementioned results. For this reason, we first develop the Mather and weak KAM theories for contact Hamiltonian systems in this non-monotonic setting. A decomposition theorem of the Mañé set is the main result of this part.

math.AP↗

arXiv2024

Variational construction of singular characteristics and propagation of singularities

On a smooth closed manifold $M$, we introduce a novel theory of maximal slope curves for any pair $(ϕ,H)$ with $ϕ$ a semiconcave function and $H$ a Hamiltonian. By using the notion of maximal slope curve from gradient flow theory, the intrinsic singular characteristics constructed in [Cannarsa, P.; Cheng, W., \textit{Generalized characteristics and Lax-Oleinik operators: global theory}. Calc. Var. Partial Differential Equations 56 (2017), no. 5, 56:12], the smooth approximation method developed in [Cannarsa, P.; Yu, Y. \textit{Singular dynamics for semiconcave functions}. J. Eur. Math. Soc. 11 (2009), no. 5, 999--1024], and the broken characteristics studied in [Khanin, K.; Sobolevski, A., \textit{On dynamics of Lagrangian trajectories for Hamilton-Jacobi equations}. Arch. Ration. Mech. Anal. 219 (2016), no. 2, 861--885], we prove the existence and stability of such maximal slope curves and discuss certain new weak KAM features. We also prove that maximal slope curves for any pair $(ϕ,H)$ are exactly broken characteristics which have right derivatives everywhere. Applying this theory, we establish a global variational construction of strict singular characteristics and broken characteristics. Moreover, we prove a result on the global propagation of cut points along generalized characteristics, as well as a result on the propagation of singular points along strict singular characteristics, for weak KAM solutions. We also obtain the continuity equation along strict singular characteristics which clarifies the mass transport nature in the problem of propagation of singularities.

math.AP↗

arXiv2023

Time periodic and almost periodic viscosity solutions of contact Hamilton-Jacobi equations on $\mathbb{T}^n$

This paper concerns with the time periodic viscosity solution problem for a class of evolutionary contact Hamilton-Jacobi equations with time independent Hamiltonians on the torus $\mathbb{T}^n$. Under certain suitable assumptions we show that the equation has a non-trivial $T$-periodic viscosity solution if and only if $T\in D$, where $D$ is a dense subset of $[0,+\infty)$. Moreover, we clarify the structure of $D$. As a consequence, we also study the existence of Bohr almost periodic viscosity solutions.

math.AP↗

arXiv2023

Fully parallel optical matrix-matrix multiplication

In recent years, with the rapid development of electro-optic modulators, optical computing has become a potential excellent candidate for various computing tasks. New structures and devices for optical computing are emerging one after another, but the computing method is still the optical vector-matrix multiplication method that was decades ago. Here, we propose a novel optical computing paradigm that can parallelly implement matrix-matrix multiplication operation, which can directly replace existing vector-matrix multiplication, greatly improving computational efficiency. This preprint presents theoretical analysis, and we will supplement experimental results and conclusions in the future.

physics.optics↗

arXiv2022

Ergodic problems for contact Hamilton-Jacobi equations

This paper deals with the generalized ergodic problem \[ H(x,u(x),Du(x))=c, \quad x\in M, \] where the unknown is a pair $(c,u)$ of a constant $c \in \mathbb{R}$ and a function $u$ on $M$ for which $u$ is a viscosity solution. We assume $H=H(x,u,p)$ satisfies Tonelli conditions in the argument $p\in T^*_xM$ and the Lipschitz condition in the argument $u\in\R$. For a given $c\in \R$, we first discuss necessary and sufficient conditions for the existence of viscosity solutions. Let $\mathfrak{C}$ denote the set of all real numbers $c$'s for which the above equation admits viscosity solutions. Then we show $\mathfrak{C}$ is an interval, whose endpoints $\x$, $\y$ with $\x\leqslant\y$ can be characterized by a min-max formula and a max-min formula, respectively. The most significant finding is that we figure out the structure of $\mathfrak{C}$ without monotonicity assumptions on $u$.

math.AP↗

arXiv2021

Time-periodic solutions of contact Hamilton-Jacobi equations on the circle

We are concerned with the existence and multiplicity of nontrivial time-periodic viscosity solutions to \[ \partial_t w(x,t) + H( x,\partial_x w(x,t),w(x,t) )=0,\quad (x,t)\in \mathbb{S} \times [0,+\infty). \] We find that there are infinitely many nontrivial time-periodic viscosity solutions with different periods when $\frac{\partial H}{\partial u}(x,p,u)\leqslant-δ<0$ by analyzing the asymptotic behavior of the dynamical system $(C(\mathbb{S} ,\mathbb{R}),\{T_t\}_{t\geqslant 0})$, where $\{T_t\}_{t\geqslant 0}$ was introduced in \cite{WWY1}. Moreover, in view of the convergence of $T_{t_n}φ$, we get the existence of nontrivial periodic points of $T_t$, where $φ$ are initial data satisfying certain properties. This is a long-time behavior result for the solution to the above equation with initial data $φ$. At last, as an application, we describe to readers a bifurcation phenomenon for \[ \partial_t w(x,t) + H( x,\partial_x w(x,t),λw(x,t) )=0,\quad (x,t)\in \mathbb{S} \times [0,+\infty), \] when the sign of the parameter $λ$ varies. The structure of the unit circle $\mathbb{S}$ plays an essential role here. The most important novelty is the discovery of the nontrivial recurrence of $(C(\mathbb{S} ,\mathbb{R}),\{T_t\}_{t\geqslant 0})$.

math.AP↗

arXiv2021

A semi-discrete approximation for first-order stationary mean field games

We provide an approximation scheme for first-order stationary mean field games with a separable Hamiltonian. First, we discretize Hamilton-Jacobi equations by discretizing in time, and then prove the existence of minimizing holonomic measures for mean field games. At last, we obtain two sequences of solutions $\{u_i\}$ of discrete Hamilton-Jacobi equations and minimizing holonomic measures $\{m_i\}$ for mean field games and show that $(u_i,m_i)$ converges to a solution of the stationary mean field games.

math.AP↗

arXiv2021

Hamilton-Jacobi equations with their Hamiltonians depending Lipschitz continuously on the unknown

We study the Hamilton-Jacobi equations $H(x,Du,u)=0$ in $M$ and $\partial u/\partial t +H(x,D_xu,u)=0$ in $M\times(0,\infty)$, where the Hamiltonian $H=H(x,p,u)$ depends Lipschitz continuously on the variable $u$. In the framework of the semicontinuous viscosity solutions due to Barron-Jensen, we establish the comparison principle, existence theorem, and representation formula as value functions for extended real-valued, lower semicontinuous solutions for the Cauchy problem. We also establish some results on the long-time behavior of solutions for the Cauchy problem and classification of solutions for the stationary problem.

math.AP↗

arXiv2021

Aubry-Mather theory for contact Hamiltonian systems II

In this paper, we continue to develop Aubry-Mather and weak KAM theories for contact Hamiltonian systems $H(x,u,p)$ with certain dependence on the contact variable $u$. For the Lipschitz dependence case, we obtain some properties of the Mañé set. For the non-decreasing case, we provide some information on the Aubry set, such as the comparison property, graph property and a partially ordered relation for the collection of all projected Aubry sets with respect to backward weak KAM solutions. Moreover, we find a new flow-invariant set $\tilde{\mathcal{S}}_s$ consists of strongly static orbits, which coincides with the Aubry set $\tilde{\mathcal{A}}$ in classical Hamiltonian systems. Nevertheless, a class of examples are constructed to show $\tilde{\mathcal{S}}_s\subsetneqq\tilde{\mathcal{A}}$ in the contact case. As their applications, we find some new phenomena appear even if the strictly increasing dependence of $H$ on $u$ fails at only one point, and we show that there is a difference for the vanishing discount problem from the negative direction between the minimal viscosity solution and non-minimal ones.

math.DS↗

arXiv2021

Existence of solutions to contact mean field games of first order

This paper deals with the existence of solutions of a class of contact mean field games systems of first order. Cardaliaguet \cite{CAR} found a link between the weak KAM theory for Hamiltonian systems and mean field games systems. We prove that there is still a connection between the weak KAM theory for contact Hamiltonian systems and contact mean field games systems. By the analysis of properties of the Mather set for contact Hamiltonian systems, we prove the main existence result.

math.AP↗