Periodic traveling-wave solutions for regularized dispersive equations: sufficient conditions for orbital stability with applications

Nonlinear stability of periodic-wave solutions for systems of dispersive equations

“…) is a solution of (3.26) in the sense of distributions, then ϕ ∈ H n per ([0, L]), for all n ∈ N. This result can be proved by an iteration method and is a consequence of the fact that ϕ ∈ 1 (see [19,Proposition 3.1]). In particular, using the previous theorem, we have that ϕ (ω,W ) → ϕ (ω0,0) , as (ω, W ) → (ω 0 , 0), in L ∞ per ([0, L]).…”

“…Since ϕ (ω0,0) does not belong to H 1 per,e ([0, L]) (because it is odd), we conclude that G (ω0,0) is oneto-one. A simple analysis also shows that G (ω0,0) is also surjective (see, for instance, [19,Theorem 3.2]). As consequence of the Implicit Function Theorem we establish the desired result.…”

“…To prove Theorem 1.2, we follow the strategy in [7,19,30,33]. Note that solutions of (1.5) are critical points of the functional (1.10).…”

“…Proof of Theorem 1.2. In this section, we basically rewrite the arguments in [7,19,30,33] to establish the theory of orbital stability for coupled dispersive systems. The next proposition is a classical result inserted in [24] adapted to our case.…”

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

“…5, the existence of minimizers for the energy functional 𝑉 ∶= 𝑃 − 𝐸 with fixed momentum 𝑃 and mass 𝑀 has been established. The periodic wave obtained by this minimization problem is smooth in terms of the independent parameters 𝐴 and 𝑐 in (6) and it has been determined that they are spectrally stable in the sense of Definition 1. The orbital stability of the periodic minimizers is then established by assuming that the 2-by-2 determinant {𝑃, 𝑀} 𝐴,𝑐 is nonzero (see the precise definition of orbital stability in the last section of this paper).…”