Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2610.24831

On the long wave limit of the good Boussinesq equation

Benjamin Harrop-Griffiths
View original source ↗Read source document ↗Export metadata ↓

Abstract

We consider the good Boussinesq equation on the line. Sufficiently regular solutions are known to converge in the long wave limit to a pair of counterpropagating solutions to the Korteweg--de Vries equation. We prove that this convergence holds for solutions in $H^{-\frac12}$, the largest Sobolev space in which the good Boussinesq equation is well-posed. As part of our analysis, we revisit the well-posedness theory, proving new a priori estimates and a corresponding small data global well-posedness result.

Explore related subjects

math.AP

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Benjamin Harrop-Griffiths. 2026-10-17. On the long wave limit of the good Boussinesq equation. https://arxiv.org/abs/2610.24831

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

On the dispersive estimates for the discrete Schrödinger equation on a honeycomb lattice

The discrete Schrödinger equation on a two-dimensional honeycomb lattice is a fundamental tight-binding approximation model that describes the propagation of waves on graphene. For free evolution, we first show that the degenerate frequencies of the dispersion relation are completely characterized by three symmetric periodic curves (Theorem 2.1), and that the three curves meet at Dirac points where conical singularities appear (see Figure 2.1). Based on this observation, we prove the $L^1\to L^\infty$ dispersion estimates for the linear flow depending on the frequency localization (Theorem 2.3). Collecting all, we obtain the dispersion estimate with $O(|t|^{-2/3})$ decay as well as Strichartz estimates. As an application, we prove small data scattering for a nonlinear model (Theorem 2.10). The proof of the key dispersion estimates is based on the associated oscillatory integral estimates with degenerate phases and conical singularities at Dirac points. Our proof is direct and uses only elementary methods.

math.AP↗
arXiv2026

Well-posedness and Stability of Thermoelastic Suspension Bridges with Cattaneo Heat Conduction: The Role of Viscoelastic Memory

Thermoelastic suspension bridges operate under strong thermo-mechanical coupling, and the classical Fourier law fails to capture the finite speed of thermal propagation observed in low-conductivity materials such as concrete. This paper investigates a suspension bridge model in which the heat flux is governed by the Cattaneo law with relaxation time $τ$, and in which the bridge deck may or may not possess viscoelastic memory. Using operator semigroup theory, we establish well-posedness and derive sharp decay rates for both regimes. For the memoryless system ($δ=0$), we show that the semigroup is polynomially stable with rate $t^{-1/2}$ when the structural parameters satisfy $\frac{ρ_1}κ = \frac{ρ_2}{b}$, and that the rate deteriorates to $t^{-1/4}$ otherwise; moreover, exponential stability fails whenever $χ_0 \neq 0$, or $χ_0 = 0$ and $χ_1 = 0$, where $χ_0$ and $χ_1$ are parameter-dependent coefficients. In sharp contrast, when viscoelastic memory is present ($δ=1$), the semigroup becomes exponentially stable, regardless of the structural parameter ratio. This reveals a regularizing role of viscoelastic memory: it converts polynomial decay into exponential decay and removes the parameter-ratio obstruction. The results provide a theoretical basis for exploiting inherent viscoelasticity in long-span bridges and for optimizing deck parameters in memoryless structures.

math.AP↗