Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2009.04260

Long-time asymptotics and stability for the sine-Gordon equation

Gong Chen·Jiaqi Liu·Bingying Lu
View original source ↗Read source document ↗Export metadata ↓

Abstract

In this paper, we study the long-time dynamics and stability properties of the sine-Gordon equation $$f_{tt}-f_{xx}+\sin f=0.$$ Firstly, we use the nonlinear steepest descent for Riemann-Hilbert problems to compute the long-time asymptotics of the solutions to the sine-Gordon equation whose initial condition belongs to some weighted Sobolev spaces. Secondly, we study the asymptotic stability of the sine-Gordon equation. It is known that the obstruction to the asymptotic stability of the sine-Gordon equation in the energy space is the existence of small breathers which is also closely related to the emergence of wobbling kinks. Combining the long-time asymptotics and a refined approximation argument, we analyze the asymptotic stability properties of the sine-Gordon equation in weighted energy spaces. Our stability analysis gives a criterion for the weight which is sharp up to the endpoint so that the asymptotic stability holds.

Explore related subjects

math.AP

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Gong Chen, Jiaqi Liu, Bingying Lu. 2022-02-15. Long-time asymptotics and stability for the sine-Gordon equation. https://arxiv.org/abs/2009.04260

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

KEEP EXPLORING

Related papers

arXiv2026

Large friction limit of compressible Navier--Stokes equations with Navier boundary conditions in a half-space

We study the large-friction limit for the three-dimensional barotropic compressible Navier-Stokes equations in a half-space. The velocity satisfies a Navier boundary condition with friction coefficient $α>0$, while the limiting problem satisfies the no-slip boundary condition. We establish estimates for local-in-time smooth solutions that are uniform in $α$ and prove strong convergence of the density and velocity as $α\to\infty$. For weak solutions, we use the Lagrangian flow maps associated with the two velocities to compare the densities and construct suitable transported test functions. This yields weak convergence to the no-slip solution. Our results provide a compressible counterpart of the large-friction limit for incompressible flows.

math.AP↗
arXiv2026

Bochner-Riesz means for critical magnetic Schrödinger operators in ${\mathbb R^2}$

We study $L^p$-boundedness of the Bochner-Riesz means for critical magnetic Schrödinger operators $\LL_{\A}$ in ${\mathbb R^2}$, which involve the {physical} Aharonov-Bohm potential. We show that for $1\leq p\leq +\infty$ and $p\not= 2$, the Bochner-Riesz operator $S_λ^δ(\LL_{\A})$ of order $δ$ is bounded on $L^p(\R^2)$ if and only if $δ>\max\big\{0, 2\big|1/2-1/p\big|-1/2\big\}$. The new ingredient {in} the proof is to obtain the localized $L^4(\R^2)$ estimate of $S_λ^δ(\LL_{\A})$, whose kernel is heavily affected by the physical magnetic diffraction, and more singular than the classical Bochner-Riesz means $S_λ^δ(Δ)$ for the Laplacian $Δ$ in ${\mathbb R}^2$.

math.AP↗
arXiv2026

Solitons, scattering and blow-up for the nonlinear Schrödinger equation with combined power-type nonlinearities on $\mathbb{R}^d\times\mathbb{T}$

We investigate the long time dynamics of the nonlinear Schrödinger equation (NLS) with combined powers on the waveguide manifold $\mathbb{R}^d\times\mathbb{T}$. Different from the previously studied NLS-models with single power on the waveguide manifolds, where the non-scale-invariance is mainly due to the mixed nature of the underlying domain, the non-scale-invariance of the present model is both geometrical and structural. By considering different combinations of the nonlinearities, we establish both qualitative and quantitative properties of the soliton, scattering and blow-up solutions. As one of the main novelties of the paper compared to the previous results for the NLS with single power, we particularly construct two different rescaled families of variational problems, which leads to an NLS with single power in different limiting profiles respectively, to establish the periodic dependence results.

math.AP↗