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Continuum limit of the discrete nonlinear Klein-Gordon equation

Quentin Chauleur

Abstract

We study the convergence of solutions of the discrete nonlinear Klein-Gordon equation on an infinite lattice in the continuum limit, using recent tools developed in the context of nonlinear discrete dispersive equations. Our approach relies in particular on the use of bilinear estimates of the Shannon interpolation alongside controls on the growth of discrete Sobolev norms of the solution. We conclude by giving perspectives on uniform dispersive estimates for nonlinear waves on lattices.

Continuum limit of the discrete nonlinear Klein-Gordon equation

Abstract

We study the convergence of solutions of the discrete nonlinear Klein-Gordon equation on an infinite lattice in the continuum limit, using recent tools developed in the context of nonlinear discrete dispersive equations. Our approach relies in particular on the use of bilinear estimates of the Shannon interpolation alongside controls on the growth of discrete Sobolev norms of the solution. We conclude by giving perspectives on uniform dispersive estimates for nonlinear waves on lattices.
Paper Structure (9 sections, 5 theorems, 77 equations)

This paper contains 9 sections, 5 theorems, 77 equations.

Table of Contents

  1. Discrete framework and main result
  2. Growth of discrete Sobolev norms
  3. Strong convergence in the continuum limit
  4. The linear flow
  5. Linear flow on the nonlinearity
  6. Aliasing and Gronwall argument
  7. Decay estimates for discrete wave equations
  8. Dispersive estimates on lattices
  9. Uniform Strichartz estimates for the discrete Klein-Gordon equation

Key Result

Theorem 1

Let $s \in \mathbb{N}^*$ with $(p,d)$ satisfying param. Let $\phi \in \mathcal{C}(\mathbb{R};H^{s+2}(\mathbb{R}^d))$ be the unique solution of NLKG with initial condition $(\phi_0,\phi_1) \in H^{s+2}(\mathbb{R}^d) \times H^{s+1}(\mathbb{R}^d)$, and let $u$ be the unique solution of DNLKG with initia where $B$ and $C$ are constants depending on $d$, $p$, $s$ and $\|(\phi_0,\phi_1) \|_{H^{s+2}(\math

Theorems & Definitions (10)

  • Theorem 1
  • Proposition 1
  • proof
  • Proposition 2
  • proof
  • Lemma 1
  • proof
  • Proposition 3
  • proof
  • Conjecture 1