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Jumps and cusps: a new revival effect in local dispersive PDEs

Lyonell Boulton, George Farmakis, Beatrice Pelloni, David A. Smith

Abstract

We study the presence of a non-trivial revival effect in the solution of linear dispersive boundary value problems for two benchmark models which arise in applications: the Airy equation and the dislocated Laplacian Schr{ö}dinger equation. In both cases, we consider boundary conditions of Dirichlet-type. We prove that, at suitable times, jump discontinuities in the initial profile are revived in the solution not only as jump discontinuities but also as logarithmic cusp singularities. We explicitly describe these singularities and show that their formation is due to interactions between the symmetries of the underlying spatial operators with the periodic Hilbert transform.

Jumps and cusps: a new revival effect in local dispersive PDEs

Abstract

We study the presence of a non-trivial revival effect in the solution of linear dispersive boundary value problems for two benchmark models which arise in applications: the Airy equation and the dislocated Laplacian Schr{ö}dinger equation. In both cases, we consider boundary conditions of Dirichlet-type. We prove that, at suitable times, jump discontinuities in the initial profile are revived in the solution not only as jump discontinuities but also as logarithmic cusp singularities. We explicitly describe these singularities and show that their formation is due to interactions between the symmetries of the underlying spatial operators with the periodic Hilbert transform.
Paper Structure (7 sections, 9 theorems, 169 equations, 3 figures)

This paper contains 7 sections, 9 theorems, 169 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Statement of results
  3. Proof of Theorem \ref{['lkdvthm2']}
  4. Proof of Theorem \ref{['Revival in (0,b)']}
  5. Towards a general definition of revivals
  6. Notation
  7. The spatial operators

Key Result

Theorem 2.1

Let $u_0:[0,1]\longrightarrow {\mathbb R}$ be piecewise Lipschitz. Then, for all $t\in{\mathbb R}$, the solution of the boundary value problem pseD1 admits the representation where $U_{\mathcal{C}}({}\cdot{},t)\in\operatorname{C}(0,1)$ and with If $p,q\in{\mathbb N}$ are coprime, then where and Here $\mathcal{H}$ denotes the $1$-periodic Hilbert transform and $d_k^{p,q}$ are scalar coefficie

Figures (3)

  • Figure 1: A step function (dashed) and its periodic Hilbert transform (solid). When a given function has an isolated jump discontinuity, its Hilbert transform displays a cusp with a singularity of logarithmic order, see \ref{['logsing']}. (The cusps actually extend to infinity in either direction).
  • Figure 2: For $u_0$ a step function (blue) the graphs show the solution of \ref{['pseD1']} (purple), the function $U_{\mathcal{R}}$ (green) and the continuous function $U_{\mathcal{C}}$ (brick). See Theorem \ref{['lkdvthm2']}. The horizintal axis is $x\in[0,1]$. The time $t$ is fixed rational on the left and fixed irrational on the right. These graphs were constructed from an expansion in the first 600 eigenfunctions, approximated numerically.
  • Figure 3: For $u_0$ a step function (blue) the graphs show the solution of \ref{['disPDE']} (purple), the function $U_{\mathcal{R}}$ (green) and the continuous function $U_{\mathcal{C}}$ (brick). See Theorem \ref{['Revival in (0,b)']}. The horizontal axis is $x\in[0,1]$. The time $t$ is fixed rational with respect to $(0,b)$ on the left and fixed rational with respect to $(b,1)$ on the right. These graphs were constructed from an expansion in the first 250 eigenfunctions, approximated numerically.

Theorems & Definitions (23)

  • Theorem 2.1
  • Theorem 2.2
  • Proposition 3.1
  • proof
  • Proposition 3.2
  • proof
  • Lemma 3.1
  • proof
  • proof : Proof of Theorem \ref{['lkdvthm2']}
  • Proposition 4.1
  • ...and 13 more