Dynamics of solutions in the 1d bi-harmonic nonlinear Schr\"odinger equation

We consider the one dimensional 4th order, or bi-harmonic, nonlinear Schr\"odinger (NLS) equation, namely, $i u_t - \Delta^2 u - 2a \Delta u + |u|^{\alpha} u = 0, ~ x,a \in \R$, $\alpha>0$, and investigate the dynamics of its solutions for various powers of $\alpha$, including the ground state solutions and their perturbations, leading to scattering or blow-up dichotomy when $a \leq 0$, or to a trichotomy when $a>0$. Ground state solutions are numerically constructed, and their stability is studied, finding that the ground state solutions may form two branches, stable and unstable, which dictates the long-term behavior of solutions. Perturbations of the ground states on the unstable branch either lead to dispersion or the jump to a stable ground state. In the critical and supercritical cases, blow-up in finite time is also investigated, and it is conjectured that the blow-up happens with a scale-invariant profile (when $a=0$) regardless of the value of $a$ of the lower dispersion. The blow-up rate is also explored.