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

Abstract The travelling wave with the peaked profile is usually considered as a limit in the family of travelling waves with the smooth profiles. We study the linear and nonlinear stability of the peaked travelling wave by using a local model for shallow water waves, which is an extended version of 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 linearised equations of motion and study the nonlinear evolution of co-periodic perturbations to the peaked periodic wave by using the method of characteristics. Within the linearised equations, we prove the spectral instability of the peaked travelling wave from the spectrum of the linearised 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 travelling wave by showing that the gradient of perturbations grows at the wave peak. By using numerical approximations of the smooth travelling waves and the spectrum of their associated linearised operator, we show that the spectral instability of the peaked travelling wave cannot be obtained as a limit in the family of the spectrally stable smooth travelling waves.