In this paper we establish the orbital stability of periodic traveling waves for a general class of dispersive equations. We use the Implicit Function Theorem to guarantee the existence of smooth solutions depending of the corresponding wave speed. Essentially, our method establishes that if the linearized operator has only one negative eigenvalue which is simple and zero is a simple eigenvalue the orbital stability is determined provided that a convenient condition about the average of the wave is satisfied. We use our approach to prove the orbital stability of periodic dnoidal waves associated with the Kawahara equation.
Periodic Traveling-wave solutions for regularized dispersive equations: Sufficient conditions for orbital stability with applications
Orbital Stability of Standing Waves for BNLS]{Orbital Stability of Standing Waves for a fourth-order nonlinear Schrödinger equation with the mixed dispersions
On spectral and orbital stability for the Klein--Gordon equation coupled to an anharmonic oscillator
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