arXiv subjects

Boyang Wu

Publications and source records attributed to Boyang Wu.

4 recordsLinked to original sources

arXiv2026

Rigorous Derivation of the Wave Kinetic Equation for full $\beta$-FPUT System

The Fermi--Pasta--Ulam--Tsingou (FPUT) system, describing the evolution of $N$ coupled harmonic oscillators, has been the subject of much attention since the 1950's when experiments which contradicted predictions of thermalization of the system. A full explanation of this behavior is still not fully known. Here, we rigorously derive the corresponding wave kinetic equation, which provides a precise evolution of the statistics for the FPUT system and demonstrates thermalization in an appropriate regime. In particular, we justify the kinetic equation for the 4-wave $\beta$-FPUT system in the kinetic limit $N \to \infty$ and $\beta \to 0$ for weakly nonlinear scaling laws $\beta \sim N^{-\gamma}$, reaching times up to $T_{\mathrm{kin}}^{2/3}$, where $T_{\mathrm{kin}}$ represents the kinetic (thermalization) timescale. While we use a typical diagrammatic expansion to derive the kinetic equations, few works have dealt with nonlinearities with non-resonant terms, which are not part of the kinetic equation, which is the major novelty of this work. The only other such work \cite{DIP25} made use of a normal form method to push the non-resonant terms to higher order nonlinearities. Here, we directly incorporate the non-resonant terms into the diagrammatic expansion and demonstrate corresponding gains. This method can be adapted to other 4-wave non-resonant nonlinearities.

math.AP↗

arXiv2025

Validity condition of normal form transformation for the $\beta$-FPUT system

In this work, we provide a validity condition for the normal form transformation to remove the non-resonant cubic terms in the $\beta$-FPUT system. We show that for a wave field with random phases, the normal form transformation is valid by dominant probability if $\beta \ll 1/N^{1+\epsilon}$, with $N$ the number of masses and $\epsilon$ an arbitrarily small constant. To obtain this condition, a bound is needed for a summation in the transformation equation, which we prove rigorously in the paper. The condition also suggests that the importance of the non-resonant terms in the evolution equation is governed by the parameter $\beta N$. We design numerical experiments to demonstrate that this is indeed the case for spectra at both thermal-equilibrium and out-of-equilibrium conditions. The methodology developed in this paper is applicable to other Hamiltonian systems where a normal form transformation needs to be applied.

math-ph↗

arXiv2025

Rigorous Derivation of the Wave Kinetic Equation for $\beta$-FPUT System

Wave kinetic theory has been suggested as a way to understand the longtime statistical behavior of the Fermi-Pasta-Ulam-Tsingou (FPUT) system, with the aim of determining the thermalization time scale. The latter has been a major problem since the model was introduced in the 1950s. In this thesis we establish the wave kinetic equation for a reduced evolution equation obtained from the $\beta$-FPUT system by removing the non-resonant terms. We work in the kinetic limit $N\to \infty$ and $\beta\to 0$ under the scaling laws $\beta=N^{-\gamma}$ with $0<\gamma<1$. The result holds up to the sub-kinetic time scale $T=N^{-\epsilon}\min\bigl(N,N^{5\gamma/4}\bigr)=N^{-\epsilon}T_{\mathrm{kin}}^{5/8}$ for $\epsilon\ll 1$, where $T_{\mathrm{kin}}$ represents the kinetic (thermalization) timescale. The novelties of this work include the treatment of non-polynomial dispersion relations, and the introduction of a robust phase renormalization argument to cancel dangerous divergent interactions.

math.AP↗

arXiv2022

A simple suboptimal moving horizon estimation scheme with guaranteed robust stability

We propose a suboptimal moving horizon estimation (MHE) scheme for a general class of nonlinear systems. To this end, we consider an MHE formulation that optimizes over the trajectory of a robustly stable observer. Assuming that the observer admits a Lyapunov function, we show that this function is an M-step Lyapunov function for suboptimal MHE. The presented sufficient conditions can be easily verified in practice. We illustrate the practicability of the proposed suboptimal MHE scheme with a standard nonlinear benchmark example. Here, performing a single iteration is sufficient to significantly improve the observer's estimation results under valid theoretical guarantees.

eess.SY↗