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The Benjamin-Feir instability in KdV-like equations with general dispersion and monomial nonlinearity

Bhavna Kaushik, Bernard Deconinck

Abstract

Nonlinear waves in dispersive media can be succeptible to modulational instabilities. We examine a category of scalar equations, with general dispersion and monomial nonlinearity, including a large variety of KdV-like equations. For small-amplitude traveling wave solutions, we provide a complete characterization of the spectrum near the origin of the linear operator obtained from linearizing about periodic traveling waves. We prove rigorously that, when the modulational instability is present, the spectrum connected to the origin consists of curves that invariably form a closed figure-eight pattern.

The Benjamin-Feir instability in KdV-like equations with general dispersion and monomial nonlinearity

Abstract

Nonlinear waves in dispersive media can be succeptible to modulational instabilities. We examine a category of scalar equations, with general dispersion and monomial nonlinearity, including a large variety of KdV-like equations. For small-amplitude traveling wave solutions, we provide a complete characterization of the spectrum near the origin of the linear operator obtained from linearizing about periodic traveling waves. We prove rigorously that, when the modulational instability is present, the spectrum connected to the origin consists of curves that invariably form a closed figure-eight pattern.
Paper Structure (12 sections, 22 theorems, 276 equations, 3 figures, 2 tables)

This paper contains 12 sections, 22 theorems, 276 equations, 3 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Small amplitude periodic traveling waves
  3. The complete Benjamin-Feir spectrum
  4. Main Results
  5. Results for $N$ even
  6. Results for $N$ odd.
  7. Properties of the operator
  8. Proof of the expansion of the basis $\Sigma$
  9. Proof of Proposition \ref{['p:1']}
  10. Even values of $\mathbf{N}$
  11. Odd values of $N$
  12. Proof of the Block-Diagonalization

Key Result

Lemma 2.1

If $\eta\in H^s_{\mathrm{even}}(\mathbb{T})$, with $s=\max(1,1+\sigma)$ and $\sigma$ from Hypothesis (H2) satisfies e:q for some $c\in\mathbb{R}$, $\rho>0$, and if $c-N|\alpha|\|\eta^{N-1}\|_{L^\infty}\geqslant\epsilon>0$ for some $\epsilon$, then $\eta\in H^\infty_{\rm even}(\mathbb{T})$.

Figures (3)

  • Figure 1: Comparison of the analytical result (blue) for the unstable spectrum near the origin for the modified KdV equation with $\alpha=-1$ (thus $N=3$, $\jmath(k)=1+k^2$), with numerical results (red) using a 9-th order Stokes expansion, with $a=0.02$, $\rho=1.5$. Hill's method DK uses $5$ Fourier modes, $11\times 11$ matrices, and $201$ equally spaced Floquet exponents $\mu\in [-0.01, 0.01]$.
  • Figure 2: Comparison of the analytical result (blue) for the unstable spectrum near the origin for 4 different Whitham-like equations with increasing degree of nonlinearity $N$, with $\alpha=1$, $\jmath(k)=\sqrt{\tanh k/k}$, with numerical results (red) using a 9-th order Stokes expansion, with $a=0.02$, $\rho=1.5$. Hill's method DK is used with $5$ Fourier modes, $11\times 11$ matrices, $210$ equally spaced Floquet exponents. For $N=2$, $\mu\in [-1/2, 1/2]$, for $N=3$, $\mu\in [-0.06, 0.06]$, for $N=4$, $\mu\in [-0.0001,0.0001]$, for $N=5$, $\mu\in [-0.002, 0.002]$. The relative sizes of these Whitham figure-eight curves is noticeable, see also Fig. \ref{['fig:combo']}.
  • Figure 3: The unstable spectrum near the origin, for the Whitham equation ($N=2$) and its generalizations with higher-degree ($N=3,4,5$) monomial nonlinearity, with $a=0.02$ and $\rho=1.5$.

Theorems & Definitions (43)

  • Lemma 2.1: Regularity
  • proof
  • Theorem 2.2: Existence
  • proof
  • Lemma 3.1
  • proof
  • Lemma 3.2: Expansion of the basis $\Sigma$
  • proof
  • Definition 3.3
  • Definition 3.4
  • ...and 33 more