F. Natali, D.E. Pelinovsky, and S. Wang
Instability of the peaked traveling wave in a local model for shallow water waves
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 H1 intersecting W1,inf 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 halfwidth.
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.
Keywords:
Camassa-Holm equation; Hunter-Saxton equation; traveling waves with peaked profile;
spectral stability; nonlinear stability.