On ill-posedness for the Gabitov--Turitsyn equation

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 $s_m = \tfrac{d}{2} - \tfrac{2}{p},$ coinciding with the monomial NLS. For $s \geq \max(s_m,0)$, local well-posedness is known to hold in $H^s$, while for $s < s_m$, we show that the data-to-solution map fails to be $C^{p+1}$ in $H^s$. Second, we identify $s_i = \tfrac{d}{2} - \tfrac{4}{p},$ below which we conjecture that norm inflation occurs. We resolve this conjecture in $H^s$ in the case $s < \min (s_i, 0)$ -- specifically for the one-dimensional cubic model -- and in the case $1 \leq s < s_i$. In the case $s_i \geq 1$, we establish norm inflation by showing that suitable solutions undergo {\em energy equipartition}: a rapid renormalization of kinetic and potential energy.