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Instability of the peaked traveling wave in a local model for shallow water waves

Fábio Natali, Dmitry E. Pelinovsky, Shuoyang Wang

TL;DR

The study proves the linear and nonlinear instability of the peaked traveling wave in a local shallow-water model related to the Hunter–Saxton equation, within the energy space $H^1_{ m per}(\mathbb{T}) \cap W^{1,\infty}(\mathbb{T})$. It derives a proper linearization around the peaked profile, showing spectrum confined to the vertical strip $-\pi/4 \le \Re(\lambda) \le \pi/4$ and establishing instability in $L^2(\mathbb{T})$, while a nonlinear instability result follows from gradient growth at the peak via characteristic methods. A truncated linearized operator reduces the problem to a 1D real-line operator $D_0$ and proves the strip location; the full linearized spectrum matches the truncated one, with zero arising from domain considerations at the peak. Numerical experiments demonstrate that the instability of the peaked wave is not captured by the limit of spectrally stable smooth traveling waves, evidenced by the Hessian's lowest eigenvalue diverging as $c \to c_*$, underscoring a fundamental discontinuity between smooth and peaked stability analyses.

Abstract

The traveling wave with the peaked profile arises in the limit of the family of traveling waves with the smooth profiles. We study the linear and nonlinear stability of the peaked traveling wave by using a local model for shallow water waves, which is related to the Hunter--Saxton equation. The evolution problem is well-defined in the function space $H^1_{\rm per} \cap W^{1,\infty}$, where we derive the linearized equations of motion and study the nonlinear evolution of co-periodic perturbations to the peaked periodic wave by using methods of characteristics. Within the linearized equations, we prove the spectral instability of the peaked traveling wave from the spectrum of the linearized operator in a Hilbert space, which completely covers the closed vertical strip with a specific half-width. Within the nonlinear equations, we prove the nonlinear instability of the peaked traveling wave by showing that the gradient of perturbations grow at the wave peak. By using numerical approximations of the smooth traveling waves and the spectrum of their associated linearized operator, we show that the spectral instability of the peaked traveling wave cannot be obtained in the limit along the family of the spectrally stable smooth traveling waves.

Instability of the peaked traveling wave in a local model for shallow water waves

TL;DR

The study proves the linear and nonlinear instability of the peaked traveling wave in a local shallow-water model related to the Hunter–Saxton equation, within the energy space . It derives a proper linearization around the peaked profile, showing spectrum confined to the vertical strip and establishing instability in , while a nonlinear instability result follows from gradient growth at the peak via characteristic methods. A truncated linearized operator reduces the problem to a 1D real-line operator and proves the strip location; the full linearized spectrum matches the truncated one, with zero arising from domain considerations at the peak. Numerical experiments demonstrate that the instability of the peaked wave is not captured by the limit of spectrally stable smooth traveling waves, evidenced by the Hessian's lowest eigenvalue diverging as , underscoring a fundamental discontinuity between smooth and peaked stability analyses.

Abstract

The traveling wave with the peaked profile arises in the limit of the family of traveling waves with the smooth profiles. We study the linear and nonlinear stability of the peaked traveling wave by using a local model for shallow water waves, which is related to the Hunter--Saxton equation. The evolution problem is well-defined in the function space , where we derive the linearized equations of motion and study the nonlinear evolution of co-periodic perturbations to the peaked periodic wave by using methods of characteristics. Within the linearized equations, we prove the spectral instability of the peaked traveling wave from the spectrum of the linearized operator in a Hilbert space, which completely covers the closed vertical strip with a specific half-width. Within the nonlinear equations, we prove the nonlinear instability of the peaked traveling wave by showing that the gradient of perturbations grow at the wave peak. By using numerical approximations of the smooth traveling waves and the spectrum of their associated linearized operator, we show that the spectral instability of the peaked traveling wave cannot be obtained in the limit along the family of the spectrally stable smooth traveling waves.

Paper Structure

This paper contains 8 sections, 9 theorems, 118 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Evolution and conserved quantities
  3. Traveling wave with the smooth and peaked profiles
  4. Linearization at the smooth and peaked traveling waves
  5. Truncated linearized equation
  6. Full linearized evolution
  7. Nonlinear evolution
  8. Numerical approximations

Key Result

Theorem 1

For every $\eta_0 \in H^1_{\rm per}(\mathbb{T}) \cap W^{1,\infty}(\mathbb{T})$, there exist $\tau_0 > 0$ and a unique solution $\eta \in C^0((-\tau_0,\tau_0),H^1_{\rm per}(\mathbb{T}) \cap W^{1,\infty}(\mathbb{T})) \cap C^1((-\tau_0,\tau_0),L^2(\mathbb{T}) \cap L^{\infty}(\mathbb{T}))$ of the evolut

Figures (6)

  • Figure 1: (a) The solid lines represent the smooth profiles $\eta$ for $c = 1.03, 1.07$. The dashed line represents the peaked profile $\eta_*$ for $c = c_*$. (b) The wave amplitude versus the wave speed $c$ for smooth profiles in $(1,c_*)$ and cusped profiles in $(c_*,c_{\infty})$, where $c_*\approx 1.1107$ (dashed line) and $c_\infty\approx 1.1850$ (dashed-dotted line).
  • Figure 2: The solution profiles $\hat{\eta}$ in Fourier space (\ref{['eta_n']}) in log-log coordinates for $c = 1.03,1.07$ with $N=300$ grid points and $\epsilon = 10^{-14}$ tolerance. The black dashed line represents the peaked profile $\eta_*$ for $c = c_*$.
  • Figure 3: Eigenfunctions corresponding to the first four eigenvalues for five values of $c$ in $(1,c_*)$. The grid in physical space is chosen to be $N=300$. The solution profiles obtained from equation (\ref{['newton']}) are used for diagonalization, and all eigenfunctions are plotted on $[-\pi,\pi]$ with positive slope near $-\pi$.
  • Figure 4: The absolute value of eigenfunctions corresponding to the first four eigenvalues in Fourier space is plotted versus $m\in \{1,\dots, N\}$ for five values of $c$ in $(1,c_*)$. The grid in physical space is chosen to be $N=300$, and the solution profiles $\hat{\eta}$ are obtained from equations (\ref{['newton']}) and (\ref{['eta_n']}).
  • Figure 5: The dependence of the first four eigenvalues of the spectral problem (\ref{['eigvalue-prob']}) is plotted versus $c$ for $c \in (1,c_*)$ obtained with the finite-difference method (left) and with the Fourier collocation method (right). Eigenvalues computed for the peaked profile with $c = c_*$ are marked as circles & crosses.
  • ...and 1 more figures

Theorems & Definitions (24)

  • Theorem 1
  • proof
  • Remark 1
  • Remark 2
  • Remark 3
  • Lemma 1
  • Lemma 2
  • proof
  • Remark 4
  • Remark 5
  • ...and 14 more