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.

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The existence, uniqueness and spectral stability of periodic 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 also propose the Petviashvili's method to investigate the spectral stability of the periodic waves for the fractional Benjamin-Bona-Mahony equation, numerically.

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In this paper, we present results of existence and stability of odd periodic traveling wave solutions for the defocusing mass‐critical Korteweg‐de Vries equation. The existence of periodic wave trains is obtained by solving a constrained minimization problem. Concerning the stability, we use the Floquet theory to determine the behavior of the first three eigenvalues of the linearized operator around the wave, as well as the positiveness of the associated Hessian matrix.