Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Scattering of Rational Solutions to the Half-Wave Maps Equation

Gaspard Ohlmann

TL;DR

This work studies rational solutions of the one-dimensional Half-Wave Maps equation in the non-singular spectrum regime. It develops a precise scattering framework, showing that scattering in a scattering-situation sense implies Sobolev scattering to an explicit rational profile, and proves a local condition that guarantees such scattering. The authors construct global N-soliton-like solutions with spectra arbitrarily close to prescribed targets, and demonstrate that the scattering map is the identity on non-singular data. They also establish a diagonal traveling-wave characterization showing that scattering to traveling waves forces the original solution to be traveling, thereby ruling out nontrivial scattering to traveling waves except in the traveling case. These results provide a robust mechanism to realize and classify global dynamics of rational HWM solutions and connect integrable structure to asymptotic stability in Sobolev spaces.

Abstract

This article studies the rational solutions of the Half-Wave Maps equation (HWM) in the non-singular spectrum case. We first provide characterizations to what we call \emph{scattering behavior}, and show that they imply scattering in Sobolev norm. We then provide a local condition implying \emph{scattering behavior}. Building on this, we show that any solution with non-singular spectrum scatters and give an explicit formula for the function to which the solution is scattering. This allows us to show that the scattering map is the identity. Additionally, we create, for any given number of spins and any target non-singular spectrum, global solutions of (HWM) with a spectrum arbitrarily close to the target. Finally, using a diagonal characterization of traveling waves, we show that if a wave scatters to a traveling wave, it is a scattering wave.

Scattering of Rational Solutions to the Half-Wave Maps Equation

TL;DR

This work studies rational solutions of the one-dimensional Half-Wave Maps equation in the non-singular spectrum regime. It develops a precise scattering framework, showing that scattering in a scattering-situation sense implies Sobolev scattering to an explicit rational profile, and proves a local condition that guarantees such scattering. The authors construct global N-soliton-like solutions with spectra arbitrarily close to prescribed targets, and demonstrate that the scattering map is the identity on non-singular data. They also establish a diagonal traveling-wave characterization showing that scattering to traveling waves forces the original solution to be traveling, thereby ruling out nontrivial scattering to traveling waves except in the traveling case. These results provide a robust mechanism to realize and classify global dynamics of rational HWM solutions and connect integrable structure to asymptotic stability in Sobolev spaces.

Abstract

This article studies the rational solutions of the Half-Wave Maps equation (HWM) in the non-singular spectrum case. We first provide characterizations to what we call \emph{scattering behavior}, and show that they imply scattering in Sobolev norm. We then provide a local condition implying \emph{scattering behavior}. Building on this, we show that any solution with non-singular spectrum scatters and give an explicit formula for the function to which the solution is scattering. This allows us to show that the scattering map is the identity. Additionally, we create, for any given number of spins and any target non-singular spectrum, global solutions of (HWM) with a spectrum arbitrarily close to the target. Finally, using a diagonal characterization of traveling waves, we show that if a wave scatters to a traveling wave, it is a scattering wave.

Paper Structure

This paper contains 26 sections, 28 theorems, 435 equations.

Table of Contents

  1. Introduction
  2. The Half-Wave maps equation
  3. Results
  4. Scattering behavior and Local condition
  5. Behavior assuming a scattering situation
  6. Scattering behavior implies scattering
  7. Sufficient conditions for a scattering behavior starting at $t=0$
  8. Creation of $N$-solitons solutions with non-degenerate spectrum
  9. Defining the approximate initial condition
  10. Inductive definition of the sequence
  11. Growth estimates for the soundness of the fixpoint
  12. Asymptotic spectrum of the created solution
  13. Systematic scattering, trivial scattering map
  14. Guaranteed scattering behavior for large times
  15. The scattering map is the identity
  16. ...and 11 more sections

Key Result

Theorem 1

Let $\mathbf{m}$ be a rational solution of HWMS and define $x_j$, $\mathbf{s}_j$ as Further define $S$, $\nu$, $D$ as Then, if there exists $t_0$ such that then $\mathbf{m}$ scatters to some function Moreover, $x_j$ and $\mathbf{s}_j$ satisfy

Theorems & Definitions (55)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Theorem 5
  • Definition 2.1
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • ...and 45 more