$L^{\mathrm{\infty }}$-regularity thresholds for a generalized Gross-Pitaevskii equation: dispersive blow-up vs. nonlinear stability

We study a generalized Gross-Pitaevskii equation arising in the mean-field description of exciton-polariton Bose-Einstein condensates. The dynamics close to macroscopic wave functions $r_0{e}^{i{\omega }_{0}t+i{k}_{0}x}$ is governed by a stabilizing relaxation mechanism without suppressing dispersive blow-up. We show that this competition yields $L^{\mathrm{\infty }}$-regularity thresholds. More precisely, we construct solutions for which arbitrarily small perturbations from $C_b^4$ can develop finite time blow-up in $C_b^3$ whereas perturbations from $C_b^{4+\eta }$ for every $\eta >0$, satisfy a global-in-time $C_b^3$-stability estimate. The stability analysis is carried out in modulation spaces $M_{\mathrm{\infty },1}^k$.