Linear Instability and Uniqueness of the Peaked Periodic Wave in the Reduced Ostrovsky Equation

Abstract

We show that the peaked periodic traveling wave of the reduced Ostrovsky equations with quadratic and cubic nonlinearity is spectrally unstable in the space of square integrable periodic functions with zero mean and the same period. The main novelty of our result is that the spectrum of a linearized operator at the peaked periodic wave completely covers a closed vertical strip of the complex plane. In order to obtain this instability, we prove an abstract result on spectra of operators under compact perturbations. This justifies the truncation of the linearized operator at the peaked periodic wave to its differential part for which the spectrum is then computed explicitly. π −π U (z)dz = 0, Date: November 1, 2019.

Spectral instability of the peaked periodic wave in the reduced Ostrovsky equations

“…In [8] we proved that the unique peaked solution (1.5) of the reduced Ostrovsky equation (1.1) is linearly unstable with respect to square integrable perturbations with zero mean and the same period. This was done by obtaining sharp bounds on the exponential growth of the L 2 norm of the perturbations in the linearized time-evolution problem v t = @ z Lv.…”

“…At c = c ⇤ solutions to the boundary-value problem (1.3) are peaked at the points z = ±⇡ where U (±⇡) = c ⇤ . Uniqueness and Lipschitz continuity of the peaked solutions to the boundary-value problem (1.3) were proven in [8] for p = 1 (see [1,4] for a generalization). We denote this unique (up to translation) peaked solution by U ⇤ (z).…”

“…This was done by obtaining sharp bounds on the exponential growth of the L 2 norm of the perturbations in the linearized time-evolution problem v t = @ z Lv. No claims regarding the spectral instability of the peaked periodic wave were made in [8]. In [14], explicit solutions of the spectral stability problem M v = @ ⇠ v were constructed, but since this construction violated the periodic boundary conditions on the perturbation term, it did not provide an answer to the spectral stability question.…”

Instability of H1-stable peakons in the Camassa–Holm equation

“…The linear instability was found from the solution of the truncated linearized equation obtained by the method of characteristics and from the estimates on the solution of the full linearized equation. Nonlinear instability was not studied in [15] due to the lack of global well-posedness results on solutions of the reduced Ostrovsky equation (1.17) in H 1 . Compared to [15], we show here that the full linearized equation (1.11) can be solved by method of characteristics without truncation and that the nonlinear instability of peakons in the CH equation (1.2) can be concluded from the linear instability of perturbations in H 1 ∩ C 1 0 .…”

“…Nonlinear instability was not studied in [15] due to the lack of global well-posedness results on solutions of the reduced Ostrovsky equation (1.17) in H 1 . Compared to [15], we show here that the full linearized equation (1.11) can be solved by method of characteristics without truncation and that the nonlinear instability of peakons in the CH equation (1.2) can be concluded from the linear instability of perturbations in H 1 ∩ C 1 0 . The remainder of the article is organized as follows.…”