Dynamics of solutions in the 1d bi-harmonic nonlinear Schrödinger equation

Abstract

We consider the one dimensional 4th order, or bi-harmonic, nonlinear Schrödinger (NLS) equation, namely, $i{u}_t-\hat{I}{”}^{2}u-2a\hat{I}”u+|u{|}^{\hat{I}}\pm u=0,x,a\in \mathbb{R}$, $\hat{I}\pm >0$, and investigate the dynamics of its solutions for various powers of $\hat{I}\pm$, including the ground state solutions and their perturbations, leading to scattering or blow-up dichotomy when $a\le 0$, 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.

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