Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Scattering for the Quartic Generalized Benjamin-Bona-Mahony Equation

A. George Morgan

Abstract

The generalized Benjamin-Bona-Mahony equation (gBBM) is a model for nonlinear dispersive waves which, in the long-wave limit, is approximately equivalent to the generalized Korteweg-de Vries equation (gKdV). While the long-time behaviour of small solutions to gKdV is well-understood, the corresponding theory for gBBM has progressed little since the 1990s. Using a space-time resonance approach, I establish linear dispersive decay and scattering for small solutions to the quartic-nonlinear gBBM. To my knowledge, this result provides the first global-in-time pointwise estimates on small solutions to gBBM with a nonlinear power less than or equal to five. Owing to nonzero inflection points in the linearized gBBM dispersion relation, there exist isolated space-time resonances without null structure, but in the course of the proof I show these resonances do not obstruct scattering.

Scattering for the Quartic Generalized Benjamin-Bona-Mahony Equation

Abstract

The generalized Benjamin-Bona-Mahony equation (gBBM) is a model for nonlinear dispersive waves which, in the long-wave limit, is approximately equivalent to the generalized Korteweg-de Vries equation (gKdV). While the long-time behaviour of small solutions to gKdV is well-understood, the corresponding theory for gBBM has progressed little since the 1990s. Using a space-time resonance approach, I establish linear dispersive decay and scattering for small solutions to the quartic-nonlinear gBBM. To my knowledge, this result provides the first global-in-time pointwise estimates on small solutions to gBBM with a nonlinear power less than or equal to five. Owing to nonzero inflection points in the linearized gBBM dispersion relation, there exist isolated space-time resonances without null structure, but in the course of the proof I show these resonances do not obstruct scattering.
Paper Structure (34 sections, 16 theorems, 298 equations, 4 figures)

This paper contains 34 sections, 16 theorems, 298 equations, 4 figures.

Table of Contents

  1. Introduction
  2. The Scattering Problem. Existing Results.
  3. Main Result
  4. Strategy of the Proof: the Method of Space-Time Resonances
  5. Space-Time Resonance Framework for gBBM
  6. Steps of the Proof
  7. Notation
  8. Linear Estimates
  9. Proof of Theorem \ref{['thm:main_thm']}
  10. Proof of Proposition \ref{['prop:bootstrap_close']}
  11. Part 0: Setup
  12. Part I: $H^s_{x}$ Bounds
  13. Part II: Determination of Space-time Resonances
  14. Part III: Weighted $L^2_x$ and $L^{\infty}_{\xi}$ Bounds Near Resonances
  15. Anomalous Resonance
  16. ...and 19 more sections

Key Result

Theorem 1.1

If $u_{0}\in H^{1}_{x}\left(\mathbb{R}\right)\cap C^{2}_{x}\left(\mathbb{R}\right),$ then there exists a unique $u(t,x)$ satisfying eqn:gbbm_INTRO. Further, the initial-data-to-solution map is continuous.

Figures (4)

  • Figure 1: Plot of the group velocity $\omega'(\xi)$ for the linearized BBM and the linearized KdV.
  • Figure 3: A sketch of $\Phi(\xi)$. The $+--$ curve has roots at $-\eta_0$ and $0$. CAUTION : While the thickness of the lines makes it look like $+++$ has only two second-order roots, zooming in shows that there are in fact four first-order roots at $\approx 0.9, 1.1, 14.1$, and $14.3$.
  • Figure 4: The red solid line denotes $\Xi(\eta)=3\omega(\eta) - \omega(r(\eta))-\omega(3\eta-r(\eta))$ near $\eta_{0}\approx 5.0762$. The blue dashed line emphasizes that this function does not vanish at $3\sqrt{3}$.
  • Figure 5: Plot of $r\left(\xi\right)$. There are vertical asymptotes at $|\xi|=1$. The purple dashed line intersects the graph twice, at $\pm\sqrt{3}$.

Theorems & Definitions (30)

  • Theorem 1.1
  • Remark 1.3
  • Theorem 1.4: Main Theorem
  • Definition 1.5
  • Definition 1.6
  • Definition 1.7: LP Projections
  • Lemma 1.8: LP Decomposition
  • Theorem 2.1: Dispersive Estimates on LP Pieces
  • proof
  • Corollary 2.2: Dispersive Estimate for linBBM Flow
  • ...and 20 more