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Fronts in dissipative Fermi-Pasta-Ulam-Tsingou chains

Michael Herrmann, Guillaume James, Karsten Matthies

TL;DR

The paper rigorously establishes the existence of traveling front solutions in a dissipative FPUT chain that interpolate between two distinct uniform precompression states in the large‑damping regime. It develops a robust continuum‑limit starting point and uses an implicit function theorem in exponentially weighted Sobolev spaces to construct fronts, while a detailed Fourier‑symbol analysis yields sharp exponential tail rates and monotonicity. The work extends to Hertzian‑type potentials and provides a precise operator‑theoretic framework for the nonlocal fixed‑point problem, including auxiliary ODE representations for tail decay. The results bridge discrete lattice dynamics and continuum descriptions, offering rigorous insight into compression/soliton fronts in damped granular chains and related nonlinear lattices. Overall, the methods yield both existence and quantitative decay information useful for understanding dissipative front propagation in strongly damped, anharmonic chains.

Abstract

In a dissipative Fermi-Pasta-Ulam-Tsingou chain particles interact with their nearest neighbors through anharmonic potentials and linear dissipative forces. We prove the existence of front solutions connecting two different uniformly compressed (or stretched) states at $\pm \infty$ using an implicit function argument starting at a suitable continuum limit in the case of large damping. A detailed analysis allows us to show monotonicity of waves and to determine sharp exponential decay rates for a wide class of potentials including Hertzian potentials.

Fronts in dissipative Fermi-Pasta-Ulam-Tsingou chains

TL;DR

The paper rigorously establishes the existence of traveling front solutions in a dissipative FPUT chain that interpolate between two distinct uniform precompression states in the large‑damping regime. It develops a robust continuum‑limit starting point and uses an implicit function theorem in exponentially weighted Sobolev spaces to construct fronts, while a detailed Fourier‑symbol analysis yields sharp exponential tail rates and monotonicity. The work extends to Hertzian‑type potentials and provides a precise operator‑theoretic framework for the nonlocal fixed‑point problem, including auxiliary ODE representations for tail decay. The results bridge discrete lattice dynamics and continuum descriptions, offering rigorous insight into compression/soliton fronts in damped granular chains and related nonlinear lattices. Overall, the methods yield both existence and quantitative decay information useful for understanding dissipative front propagation in strongly damped, anharmonic chains.

Abstract

In a dissipative Fermi-Pasta-Ulam-Tsingou chain particles interact with their nearest neighbors through anharmonic potentials and linear dissipative forces. We prove the existence of front solutions connecting two different uniformly compressed (or stretched) states at using an implicit function argument starting at a suitable continuum limit in the case of large damping. A detailed analysis allows us to show monotonicity of waves and to determine sharp exponential decay rates for a wide class of potentials including Hertzian potentials.

Paper Structure

This paper contains 20 sections, 19 theorems, 165 equations.

Table of Contents

  1. Introduction
  2. Problem formulation
  3. Nonlinear integral equation for traveling waves
  4. Continuum limit
  5. Renormalization
  6. Statement of the main results
  7. Preliminaries
  8. Setting
  9. Properties of the convolution operator
  10. Existence of fronts using an implicit function argument
  11. Continuity properties
  12. Differentiability
  13. Existence of fronts
  14. Optimal decay rates
  15. Auxiliary equations
  16. ...and 5 more sections

Key Result

Theorem 1

Let $\Phi \in {\mathsf{C}}^{2,\beta}([0 , 1])$ such that $\beta \in (0,1)$, $\Phi^\prime$ is increasing and strictly convex on $[0 , 1]$, with ${\Phi}^\prime(0)=0$, ${\Phi}^\prime(1)=1$. Consider the following problem for ${\varepsilon} >0$ and its formal limit for ${\varepsilon} \rightarrow 0$ both equipped with the boundary conditions and the phase condition Then there exists a unique solutio

Theorems & Definitions (40)

  • Theorem 1
  • Remark
  • Proposition 2
  • Corollary 3
  • proof
  • Definition 4
  • Proposition 5
  • proof
  • Theorem 6
  • Lemma 7
  • ...and 30 more