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Soliton Amplification in the Korteweg-de Vries Equation by Multiplicative Forcing

Rik W. S. Westdorp, Hermen Jan Hupkes

TL;DR

This work analyzes soliton stability for the forced Korteweg-de Vries equation with small multiplicative forcing $u_t = -\partial_x^3 u - 2u \partial_x u + \epsilon f(\epsilon t/E) u$. The authors develop a modulation framework in a co-moving, dynamically rescaled frame, decomposing the solution as a modulated soliton $\phi_{c(t)}(x-\xi(t))$ plus a remainder and deriving precise modulation equations for the amplitude and phase. They extend linear stability to asymmetric weighted spaces to handle dilation, establish short-time control via a Duhamel argument, and obtain long-time control with time-varying weights, together with energy-based estimates that yield global-in-time bounds. The main result shows that the soliton remains orbitally stable under forcing, with the amplitude and phase following leading-order drift laws and the perturbation decaying in a moving, exponentially weighted norm; this provides a rigorous foundation for soliton stability under nonconservative perturbations and paves the way toward stochastic forcing analyses.}

Abstract

We study the stability and dynamics of solitons in the Korteweg-de Vries (KdV) equation with small multiplicative forcing. Forcing breaks the conservative structure of the KdV equation, leading to substantial changes in energy over long times. We show that, for small forcing, the inserted energy is almost fully absorbed by the soliton, resulting in a drastically changed amplitude and velocity. We decompose the solution to the forced equation into a modulated soliton and an infinite dimensional perturbation. Assuming slow exponential decay of the forcing, we show that the perturbation decays at the same exponential rate in a weighted Sobolev norm centered around the soliton.

Soliton Amplification in the Korteweg-de Vries Equation by Multiplicative Forcing

TL;DR

This work analyzes soliton stability for the forced Korteweg-de Vries equation with small multiplicative forcing . The authors develop a modulation framework in a co-moving, dynamically rescaled frame, decomposing the solution as a modulated soliton plus a remainder and deriving precise modulation equations for the amplitude and phase. They extend linear stability to asymmetric weighted spaces to handle dilation, establish short-time control via a Duhamel argument, and obtain long-time control with time-varying weights, together with energy-based estimates that yield global-in-time bounds. The main result shows that the soliton remains orbitally stable under forcing, with the amplitude and phase following leading-order drift laws and the perturbation decaying in a moving, exponentially weighted norm; this provides a rigorous foundation for soliton stability under nonconservative perturbations and paves the way toward stochastic forcing analyses.}

Abstract

We study the stability and dynamics of solitons in the Korteweg-de Vries (KdV) equation with small multiplicative forcing. Forcing breaks the conservative structure of the KdV equation, leading to substantial changes in energy over long times. We show that, for small forcing, the inserted energy is almost fully absorbed by the soliton, resulting in a drastically changed amplitude and velocity. We decompose the solution to the forced equation into a modulated soliton and an infinite dimensional perturbation. Assuming slow exponential decay of the forcing, we show that the perturbation decays at the same exponential rate in a weighted Sobolev norm centered around the soliton.
Paper Structure (9 sections, 22 theorems, 229 equations, 1 figure)

This paper contains 9 sections, 22 theorems, 229 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Modulation system
  3. Linear stability on weighted spaces
  4. Short-time control
  5. Long-time control
  6. Energy evolution
  7. Proof of main result
  8. Decompositions
  9. Time-varying weights

Key Result

Theorem 1.1

Pick $c_*,E_{\mathrm{max}}>0$, $w\in(0,\sqrt{c_*}/3)$, and $p\in[0,\tfrac{1}{4})$. There exist a weight $w_{\infty}\in(0,w)$ and constants $\delta_1,C_1,c_{\mathrm{min}},c_{\mathrm{max}}>0$ with $c_{\mathrm{min}}<c_*<c_{\mathrm{max}}$ such that the following holds true. For each $E\in (0, E_{\mathrm each $\overline{v}_*\in H^2\cap H^1_w$ for which $\|\overline{v}_*\|^2_{H^1}+\|\overline{v}_*\|^2_{

Figures (1)

  • Figure 1: Graph of the weight-functions $W_-$ and $W_+$ defined in \ref{['eqn:weightdef']}. We remark that $w_{\mathrm{min}}$ and $w_\infty$ defined in \ref{['eqn:wmin']} approach $w$ upon letting $E_{\mathrm{max}}\downarrow 0$.

Theorems & Definitions (42)

  • Theorem 1.1: See \ref{['sec:proofmain']}
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Proposition 3.1
  • Lemma 3.2
  • proof
  • proof : Proof of \ref{['prop:smoothing']}
  • Proposition 4.1: Short-time control
  • ...and 32 more