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Modulational instability of small amplitude periodic traveling waves in the $b$-family of Novikov equation

Xin Zhao, Lin Lu, Aiyong Chen

Abstract

We study the modulational instability of smooth, small-amplitude periodic traveling wave solutions to the $b$-family of Novikov equation with cubic nonlinearity with an arbitrary coefficient $b>0$. Our approach is based on applying spectral perturbation theory to the corresponding linearization process. We derive a modulation instability index dependent on the nonlinear parameter $b$ and the fundamental wave number, and prove that when this index is negative, sufficiently small periodic traveling waves in the Novikov equation $b$-family exhibit spectral instability to long-wavelength perturbations. This confirms the well-known Benjamin-Feir instability in the $b$-family of Novikov equation.

Modulational instability of small amplitude periodic traveling waves in the $b$-family of Novikov equation

Abstract

We study the modulational instability of smooth, small-amplitude periodic traveling wave solutions to the -family of Novikov equation with cubic nonlinearity with an arbitrary coefficient . Our approach is based on applying spectral perturbation theory to the corresponding linearization process. We derive a modulation instability index dependent on the nonlinear parameter and the fundamental wave number, and prove that when this index is negative, sufficiently small periodic traveling waves in the Novikov equation -family exhibit spectral instability to long-wavelength perturbations. This confirms the well-known Benjamin-Feir instability in the -family of Novikov equation.
Paper Structure (5 sections, 5 theorems, 146 equations, 3 figures)

This paper contains 5 sections, 5 theorems, 146 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Existence of small-amplitude periodic traveling wave solutions
  3. Spectral stability and spectral decomposition
  4. Unperturbed operators
  5. Analysis of modulational instability

Key Result

Lemma 2.1

For each $k > 0$, $d> 0$, there exists a family of small-amplitude $2\pi/k$-periodic traveling wave solutions to (y1.3) of the form for $|a|\ll1,$ where $w$ and $c$ depend analytically on $k$ and $a$. The function $w$ is a smooth, $2\pi$-periodic and even function with respect to $z$, and $c$ is an even function with respect to $a$. Moreover, as $a \to 0$, the following asymptotic expansion holds

Figures (3)

  • Figure 1: The periodic orbits determined by equation (\ref{['shs1']}).
  • Figure 2: A schematic diagram of the potential function $V (\varphi; d, c)$ for $d>0$ and $c>(b+1)b^{-\frac{b}{b+1}}d^{\frac{2}{b+1}}$. The horizontal coordinates of each black dot are marked in the graph.
  • Figure 3: The colored regions correspond to $g(k,b)<0$, the remaining blank regions correspond to $g(k,b)> 0$, the black curves are $g(k,b)=0$.

Theorems & Definitions (11)

  • Lemma 2.1
  • proof
  • Definition 3.1
  • Lemma 3.1
  • Definition 3.2
  • Lemma 4.1
  • proof
  • Theorem 5.1
  • proof
  • Lemma 5.1
  • ...and 1 more