Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Orbital Stability of Smooth Solitary Waves for the Novikov Equation

Brett Ehrman, Mathew A. Johnson, Stéphane Lafortune

Abstract

We study the orbital stability of smooth solitary wave solutions of the Novikov equation, which is a Camassa-Holm type equation with cubic nonlinearities. These solitary waves are shown to exist as a one-parameter family (up to spatial translations) parameterized by their asymptotic endstate, and are encoded as critical points of a particular action functional. As an important step in our analysis we must study the spectrum the Hessian of this action functional, which turns out to be a nonlocal integro-differential operator acting on $L^2(\mathbb{R})$. We provide a combination of analytical and numerical evidence that the necessary spectral hypotheses always holds for the Novikov equation. Together with a detailed study of the associated Vakhitov-Kolokolov condition, our analysis indicates that all smooth solitary wave solutions of the Novikov equation are nonlinearly orbitally stable.

Orbital Stability of Smooth Solitary Waves for the Novikov Equation

Abstract

We study the orbital stability of smooth solitary wave solutions of the Novikov equation, which is a Camassa-Holm type equation with cubic nonlinearities. These solitary waves are shown to exist as a one-parameter family (up to spatial translations) parameterized by their asymptotic endstate, and are encoded as critical points of a particular action functional. As an important step in our analysis we must study the spectrum the Hessian of this action functional, which turns out to be a nonlocal integro-differential operator acting on . We provide a combination of analytical and numerical evidence that the necessary spectral hypotheses always holds for the Novikov equation. Together with a detailed study of the associated Vakhitov-Kolokolov condition, our analysis indicates that all smooth solitary wave solutions of the Novikov equation are nonlinearly orbitally stable.
Paper Structure (14 sections, 15 theorems, 162 equations, 4 figures)

This paper contains 14 sections, 15 theorems, 162 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Smooth Solitary Wave Solutions
  3. Existence of Smooth Solitary Waves
  4. Smooth Solitary Waves as Critical Points
  5. Spectral Properties of $\mathcal{L}$
  6. Analytical Results on the Spectrum of ${\mathcal{L}}$
  7. Evans Function Computations
  8. Computation of Smooth Solitary Wave Solutions
  9. Construction of the Evans Function
  10. Numerical Evans Function Calculations
  11. Orbital Stability Analysis
  12. Coercivity of $\Lambda$
  13. Orbital Stability Result
  14. Analysis of the Vakhitov-Kolokolov Condition

Key Result

Theorem 2.1

For each fixed $c>0$ and $0<a<\frac{3\sqrt{3}c^2}{16}$ there exists a unique, smooth solitary wave solution $u(x,t)=\phi(x-ct)$ of e:nov2 which satisfies the conditions along with $\phi'(0)=0$. Further, we note that $\phi$ is even and monotonically decreasing on $(0,\infty)$.

Figures (4)

  • Figure 1: A schematic drawing of the effective potential $V$ for a given $c>0$ and an admissible value of $a\in\left(0,\frac{3\sqrt{3}c^2}{16}\right)$. Note that the parameter $a$ is defined as a function of $k$ in \ref{['kcond']} below.
  • Figure 2: The solid curve shows the plot of $\phi$ as a function of $x$ obtained though our shooting method for $c=1$ and $a=3\sqrt{3}/32$. The dashed line shows the graph of $\mu$ obtained from $\phi$ through the profile equation \ref{['e:profile1']} as $\mu=a/(c-\phi)^{3/2}$.
  • Figure 3: Contours for our various numerical computations. (a) A depiction of the contour $B$ used to numerically find a lower bound for $\lambda_-(\widetilde{\mathcal{L}})$. (b) A depiction of the contours $\Gamma_1$ and $\Gamma_2$ used to verify the simplicity of the eigenvalue at $\lambda=0$ for $\mathcal{L}$ as well as that $\mathcal{L}$ has only one negative eigenvalue.
  • Figure 4: A plot of $\mathcal{F}(\mu(\cdot;k))$ vs. $k\in(0,\sqrt{c}/2)$ for $c=1$. Note, in particular, that it follows that \ref{['e:F_dec']} holds for all $k\in(0,\sqrt{c}/2)$.

Theorems & Definitions (39)

  • Theorem 2.1
  • Remark 2.2
  • Remark 2.3
  • Corollary 2.4
  • Remark 2.5
  • Remark 2.6
  • Proposition 2.7
  • proof
  • Corollary 2.8
  • Lemma 3.2
  • ...and 29 more