2019/01/29 by Miguel Á. Alejo, Alejo, Miguel A., Eleomar Cardoso +1
Mathematics · Physics and Astronomy · #35Q53 #37K10 #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #Exactly Solvable and Integrable Systems (nlin.SI) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Nonlinear Photonic Systems #Nonlinear Waves and Solitons #Pattern Formation and Solitons (nlin.PS) #Primary 37K15 #Secondary 35Q51
paper · pdf · doi:10.48550/arxiv.1901.10409
openalex publication_date 2019/01/29 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We present a new variational characterization of breather solutions of any equation of the focusing Gardner hierarchy. This hierarchy is characterized by a nonnegative index n, and 2n+1 represents the order of the corresponding PDE member. In this paper, we first show the existence of such breathers, and that they are solutions of the (2n+1)th-order Gardner equation. Then we prove a variational universality property, in the sense that all these breather solutions satisfy the same fourth order stationary elliptic ODE, regardless the order of the hierarchy member. This fact also characterizes them as critical points of the same Lyapunov functional, that we also construct here. As by product of our approach, we find breather solutions of the hierarchy of (2n+1)th-order mKdV equations, as well as a respective characterization of them as solutions of a fourth order stationary elliptic ODE. We also extend part of these results to the periodic setting, presenting new breather solutions for the 5th and 7th mKdV members of the hierarchy. Finally, we prove ill-posedness results for the whole Gardner hierarchy, by using appropiately their breather solutions.