On the existence, uniqueness, and stability of periodic waves for the fractional Benjamin–Bona–Mahony equation

Abstract

The existence, uniqueness, and stability of periodic traveling waves for the fractional Benjamin-Bona-Mahony equation is considered. In our approach, we give sufficient conditions to prove a uniqueness result for the single-lobe solution obtained by a constrained minimization problem. The spectral stability is then shown by determining that the associated linearized operator around the wave restricted to the orthogonal of the tangent space related to the momentum and mass at the periodic wave has no negative eigenvalues. We propose the Petviashvili's method to investigate the spectral stability of the periodic waves for the fractional Benjamin-Bona-Mahony equation, numerically. Some remarks concerning the orbital stability of periodic traveling waves are also presented.

Traveling waves and transverse instability for the fractional Kadomtsev-Petviashvili equation

“…Since we do not have the exact solutions for the general values of α we construct the solitary wave solutions of the fKdV equation numerically by using the Petviashvili iteration method. This method has been widely used for fractional equations [3,8,34,36]. We refer to [8] and [36] for details of the method while generating solitary wave solutions of the fKdV equation.…”

Periodic waves of the modified KdV equation as minimizers of a new variational problem

“…Let ψ ∈ H 1 per be a solution of the stationary equation ( 2) satisfying (11). The periodic wave with the profile ψ is stable in the time evolution of the mKdV equation (1) if…”

“…Stability of periodic waves in the defocusing version of the mKdV equation has been studied in all details [9]. We consider the focusing version of the mKdV equation (1).…”

“…Stability of a particular family of positive periodic waves with b = 0 has been proven in [2], but no general results on stability of these periodic waves are available in the literature to the best of our knowledge. A new variational formulation of the periodic waves with b = 0 was developed in our previous works with F. Natali [17,18] (see also the follow-up work [1]).…”

“…The purpose of this paper is to explore the new variational formulation of travelling periodic waves and to detect numerically which periodic waves with b = 0 are stable and which are unstable in the time evolution of the mKdV equation (1).…”