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NLS approximation for a scalar FPUT system on a 2D square lattice with a cubic nonlinearity

Ioannis Giannoulis, Bernd Schmidt, Guido Schneider

Abstract

We consider a scalar Fermi-Pasta-Ulam-Tsingou (FPUT) system on a square 2D lattice with a cubic nonlinearity. For such systems the NLS equation can be derived to describe the evolution of an oscillating moving wave packet of small amplitude which is slowly modulated in time and space. We show that this NLS approximation makes correct predictions about the dynamics of the original scalar FPUT system for the strain and the displacement variables.

NLS approximation for a scalar FPUT system on a 2D square lattice with a cubic nonlinearity

Abstract

We consider a scalar Fermi-Pasta-Ulam-Tsingou (FPUT) system on a square 2D lattice with a cubic nonlinearity. For such systems the NLS equation can be derived to describe the evolution of an oscillating moving wave packet of small amplitude which is slowly modulated in time and space. We show that this NLS approximation makes correct predictions about the dynamics of the original scalar FPUT system for the strain and the displacement variables.
Paper Structure (7 sections, 5 theorems, 93 equations)

This paper contains 7 sections, 5 theorems, 93 equations.

Table of Contents

  1. Introduction
  2. The Fourier transformed FPUT system
  3. Derivation of the NLS equation
  4. The error estimates
  5. Discussion
  6. The approximation result for the displacement variables $q_{m,n}$
  7. Small variations of the interaction force $W'$

Key Result

Theorem 1.1

Let $s_A > 4$ and $(k_0,l_0) \neq (0,0)$ chosen in such a way that the subsequent non-resonance condition nonres is satisfied. For all $T_0 > 0$, $C_1 > 0$, $C_2 > 0$ there exist $\varepsilon_0 > 0$ and $C_3 > 0$ such that for all $\varepsilon \in (0,\varepsilon_0)$ the following holds. In case $k_0 and let $\psi_{u,m,n}$, $\psi_{v,m,n}$ be defined by uapprox1-vapprox1 with $B$ given by ABrelation

Theorems & Definitions (10)

  • Theorem 1.1
  • Remark 1.2
  • Remark 1.3
  • Remark 3.1
  • Lemma 4.1
  • Lemma 5.1
  • Remark 5.2
  • Theorem 5.3
  • Remark 5.4
  • Corollary 5.5