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

TL;DR

This work analyzes the one-dimensional bi-harmonic NLS with mixed dispersion, focusing on ground states, their bifurcations, and the subsequent dynamics across subcritical, critical, and supercritical regimes. It combines exact solutions (where available), a Fourier-spectral numerical construction of ground states, and robust time-evolution simulations using exponential time-differencing to reveal rich dynamics beyond the standard scattering vs. blow-up dichotomy. The key findings include the existence of two ground-state branches (stable and unstable) for a range of powers, the emergence of a trichotomy in the critical/positively-dispersive regime, and self-similar blow-up profiles tied to the scale-invariant state $Q^{(0)}$, with detailed rate analyses. These results advance understanding of higher-order dispersion effects, soliton stability, and nontrivial threshold phenomena in PDEs with mixed dispersion, with implications for the long-time behavior of quartic solitons and related optical systems.

Abstract

We consider the one dimensional 4th order, or bi-harmonic, nonlinear Schrödinger (NLS) equation, namely, $i u_t - Δ^2 u - 2a Δu + |u|^α u = 0, ~ x,a \in \R$, $α>0$, and investigate the dynamics of its solutions for various powers of $α$, 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.

Figures (26)

• Figure 1: The difference between the numerically computed $Q$ from \ref{['Qhat']} and the exact solution from \ref{['E:a2']}.
• Figure 2: Profiles of ground state solutions to \ref{['E:1dGS']} with $b=2$. Left: $a=1$, $2 \leq \alpha \leq 10$. Right: cubic nonlinearity ($\alpha=2$), coefficient of lower dispersion $a= -1,0,1$.
• Figure 3: Profiles of ground state solutions to \ref{['E:1dGS']} with $b=2$ and cubic nonlinearity ($\alpha=2$). Left: $a=-\sqrt{2}$. Right: $a=1.4$.
• Figure 4: $\alpha=8$. Dependence of the ground state mass $M[Q^{(a)}]$ on the parameter $a$ for a fixed $b=2$. Mass (left), energy (middle), $L^{\infty}$ (right).
• Figure 5: $\alpha=2$. Dependence of the ground state mass $M[Q^{(a)}]$ on the parameter $a$ for a fixed $b=2$. Mass (left), energy (middle), $L^{\infty}$ (right).