首页/论文/2012.12371

arXiv 2012.123712020-12math-phmath.MPnlin.SI

How Discrete Spectrum and Resonances Influence the Asymptotics of the Toda Shock Wave

Iryna Egorova, Johanna Michor

发表信息：: SIGMA 17 (2021), 045, 32 pages
DOI：: 10.3842/SIGMA.2021.045

英文摘要：

We rigorously derive the long-time asymptotics of the Toda shock wave in a middle region where the solution is asymptotically finite gap. In particular, we describe the influence of the discrete spectrum in the spectral gap on the shift of the phase in the theta-function representation for this solution. We also study the effect of possible resonances at the endpoints of the gap on this phase. This paper is a continuation of research started in [arXiv:2001.05184].

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

阅读原文

阅读 PDF ↗

arXiv 页面 ↗HTML 原文 ↗

本文内容

英文摘要相关论文

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

继续探索

相关论文

Long-time asymptotics for Toda shock waves in the modulation region

We show that Toda shock waves are asymptotically close to a modulated finite gap solution in the region separating the soliton and the elliptic wave regions. We previously derived formulas for the leading terms of the asymptotic expansion of these shock waves in all principal regions and conjectured that in the modulation region the next term is of order $O(t^{-1})$. In the present paper we prove this fact and investigate how resonances and eigenvalues influence the leading asymptotic behaviour. Our main contribution is the solution of the local parametrix Riemann-Hilbert problems and a rigoro…

2021-02-19math-phmath.MPnlin.SI

Long-Time Asymptotics for the Toda Shock Problem: Non-Overlapping Spectra

We derive the long-time asymptotics for the Toda shock problem using the nonlinear steepest descent analysis for oscillatory Riemann--Hilbert factorization problems. We show that the half plane of space/time variables splits into five main regions: The two regions far outside where the solution is close to free backgrounds. The middle region, where the solution can be asymptotically described by a two band solution, and two regions separating them, where the solution is asymptotically given by a slowly modulated two band solution. In particular, the form of this solution in the separating regi…

2019-02-01math-phmath.MPnlin.SI

Numerical study of the long wavelength limit of the Toda lattice

We present the first detailed numerical study of the Toda equations in $2+1$ dimensions in the limit of long wavelengths, both for the hyperbolic and elliptic case. We first study the formal dispersionless limit of the Toda equations and solve initial value problems for the resulting system up to the point of gradient catastrophe. It is shown that the break-up of the solution in the hyperbolic case is similar to the shock formation in the Hopf equation, a $1+1$ dimensional singularity. In the elliptic case, it is found that the break-up is given by a cusp as for the semiclassical system of the…

2014-04-09math-phmath.MPnlin.SI

Resonance and web structure in discrete soliton systems: the two-dimensional Toda lattice and its fully discrete and ultra-discrete versions

We present a class of solutions of the two-dimensional Toda lattice equation, its fully discrete analogue and its ultra-discrete limit. These solutions demonstrate the existence of soliton resonance and web-like structure in discrete integrable systems such as differential-difference equations, difference equations and cellular automata (ultra-discrete equations).

2005-04-09math-phmath.MPnlin.CG

Wave phenomena of the Toda lattice with steplike initial data

We give a survey of the long-time asymptotics for the Toda lattice with steplike constant initial data using the nonlinear steepest descent analysis and its extension based on a suitably chosen $g$-function. Analytic formulas for the leading term of the asymptotic solutions of the Toda shock and rarefaction problems (including the case of overlapping background spectra) are given and complemented by numerical simulations. We provide an explicit formula for the modulated solution in terms of Abelian integrals on the underlying hyperelliptic Riemann surface.

2016-02-10math-phmath.MP

Stability of the Periodic Toda Lattice: Higher Order Asymptotics

In a recent paper we have considered the long time asymptotics of the periodic Toda lattice under a short range perturbation and we have proved that the perturbed lattice asymptotically approaches a modulated lattice. In the present paper we capture the higher order asymptotics, at least away from some resonance regions. In particular we prove that the decay rate is $O(t^{-1/2})$. Our proof relies on the asymptotic analysis of the associated Riemann-Hilbert factorization problem, which is here set on a hyperelliptic curve. As in previous studies of the free Toda lattice, the higher order asymp…

2010-05-25math-phmath.MPnlin.SI

更多相关论文收起更多论文

Friedmann-Robertson-Walker度规的冲击波细化

Shock-Wave Refinement of the Friedmann-Robertson-Walker Metric

本文通过在冲击波面上匹配FRW与TOV度规构建宇宙学冲击波模型，并提出FRW的自相似微扰可在无宇宙学常数或暗能量的情况下解释宇宙加速膨胀。

2023-09-25gr-qcmath-phmath.MP

Nanoptera in a Period-2 Toda Chain

We study asymptotic solutions to a singularly-perturbed, period-2 Toda lattice and use exponential asymptotics to examine `nanoptera', which are nonlocal solitary waves with constant-amplitude, exponentially small wave trains. With this approach, we isolate the exponentially small, constant-amplitude waves, and we elucidate the dynamics of these waves in terms of the Stokes phenomenon. We find a simple asymptotic expression for the waves, and we study configurations in which these waves vanish, producing localized solitary-wave solutions. In the limit of small mass ratio, we derive a simple an…

2018-09-18math-phmath.DSmath.MP

表现出加速膨胀的广义相对论激波

General Relativistic Shock Waves that Exhibit an Accelerated Expansion

Rutgers University（罗格斯大学）

本文构造并分析了一类新的精确广义相对论激波，通过匹配自相似渐近Friedmann时空与Tolman-Oppenheimer-Volkoff时空实现纯辐射情形下的存在性证明，其加速膨胀无需宇宙常数，或为暗能量提供替代解释。

2026-08-03gr-qcmath-phmath.MP

广义 Fokker-Planck 方程的可积性质与多扭结解

Integrability properties and multi-kink solutions of a generalised Fokker-Planck equation

本文通过 Cole-Hopf 变换分析广义 Fokker-Planck 方程，在小粘性区域利用几何构造求得行波与多扭结渐近解，揭示了激波轨迹及多扭结的融合裂变动力学，并探讨了 Bäcklund 变换的作用。

2024-10-29math-phmath.MPnlin.SI

Classical and Quantum Strings in plane waves, shock waves and spacetime singularities: synthesis and new results

Key issues of classical and quantum strings in gravitational plane waves, shock waves and spacetime singularities are synthetically understood. This includes the string mass and mode number excitations, energy-momentum tensor, scattering amplitudes, vaccum polarization and wave-string polarization effect. The role of the real pole singularities characteristic of the tree level string spectrum (real mass resonances) and that of spacetime singularities is clearly exhibited. This throws light on the issue of singularities in string theory which can be thus classified and fully physically characte…

2003-02-26astro-phgr-qchep-ph