2025/10/13 by Matthew Kowalski, Kowalski, Matthew
Mathematics · #Advanced Mathematical Physics Problems
paper · pdf · doi:10.48550/arxiv.2510.11887
We investigate the well- and ill-posedness theory for the Gabitov--Turitsyn equation, which models the long-time dynamics of pulses in dispersion-managed optical fibers. We identify two critical regularities, corresponding to two scaling pseudo-symmetries, that demarcate regimes of ill-posedness. First, we identify sm = \tfracd2 - \tfrac2p, coinciding with the monomial NLS. For s ≥ max(sm,0), local well-posedness is known to hold in Hs, while for s < sm, we show that the data-to-solution map fails to be Cp+1 in Hs. Second, we identify si = \tfracd2 - \tfrac4p, below which we conjecture that norm inflation occurs. We resolve this conjecture in Hs in the case s < min (si, 0) -- specifically for the one-dimensional cubic model -- and in the case 1 ≤ s < si. In the case si ≥ 1, we establish norm inflation by showing that suitable solutions undergo \em energy equipartition: a rapid renormalization of kinetic and potential energy.