A normal form for Hamiltonian-Hopf bifurcations in generalized nonlinear Schrodinger equations

TL;DR

This paper derives a simplified normal form for Hamiltonian-Hopf bifurcations in generalized nonlinear Schrödinger equations, accurately predicting solution dynamics and oscillations near bifurcation points, with implications for solution stability.

Contribution

It introduces a second-order nonlinear ODE normal form for Hamiltonian-Hopf bifurcations, capturing complex dynamics and blow-up phenomena in generalized nonlinear Schrödinger equations.

Findings

Normal form accurately describes solution dynamics near bifurcations

Complex nonlinear coefficient leads to finite-time blow-up

Real coefficient admits long-lasting oscillations

Abstract

A normal form is derived for Hamiltonian-Hopf bifurcations of solitary waves in generalized nonlinear Schr\"odinger equations. This normal form is a simple second-order nonlinear ordinary differential equation that is asymptotically accurate in describing solution dynamics near Hamiltonian-Hopf bifurcations. When the nonlinear coefficient in this normal form is complex, which occurs if the second harmonic of the Hopf bifurcation frequency falls inside the continuous spectrum of the system, the solution of this normal form will blow up to infinity in finite time, meaning that solution oscillations near Hamiltonian-Hopf bifurcations will strongly amplify and eventually get destroyed. When the nonlinear coefficient of the normal form is real, the normal form can admit periodic solutions, which correspond to long-lasting solution oscillations in the original PDE system. Quantitative comparison between the normal form's predictions and true PDE solutions is also made in several numerical examples, and good agreement is obtained.