[Paper Review] Long time dynamics near the symmetry breaking bifurcation for nonlinear Schrödinger/Gross-Pitaevskii Equations

Jeremy L. Marzuola, Michael I. Weinstein|ArXiv.org|Dec 9, 2009

Nonlinear Photonic Systems30 references11 citations

TL;DR

This paper investigates long-time dynamics near the symmetry-breaking bifurcation in nonlinear Schrödinger/Gross-Pitaevskii equations with a symmetric double-well potential. It derives a finite-dimensional Hamiltonian reduction, analyzes its phase space, and proves a shadowing theorem showing that solutions with oscillating mass transport between wells persist on very long, but finite, time scales, before resonant coupling with radiation modes eventually disrupts the dynamics.

ABSTRACT

We consider a class nonlinear Schrödinger / Gross-Pitaevskii equations (NLS/GP) with a focusing (attractive) nonlinear potential and symmetric double well linear potential. NLS/GP plays a central role in the modeling of nonlinear optical and mean-field quantum many-body phenomena. It is known that there is a critical $L^2$ norm (optical power / particle number) at which there is a symmetry breaking bifurcation of the ground state. We study the rich dynamical behavior near the symmetry breaking point. The source of this behavior in the full Hamiltonian PDE is related to the dynamics of a finite-dimensional Hamiltonian reduction. We derive this reduction, analyze a part of its phase space and prove a {\it shadowing theorem} on the persistence of solutions, with oscillating mass-transport between wells, on very long, but finite, time scales within the full NLS/GP. The infinite time dynamics for NLS/GP are expected to depart, from the finite dimensional reduction, due to resonant coupling of discrete and continuum / radiation modes.

Motivation & Objective

• To understand the complex dynamical behavior near the symmetry-breaking bifurcation point in nonlinear Schrödinger/Gross-Pitaevskii equations with symmetric double-well potentials.
• To analyze the role of finite-dimensional Hamiltonian reductions in capturing long-time dynamics of the full PDE system.
• To establish conditions under which solutions exhibiting inter-well mass oscillation persist on very long, but finite, time scales.
• To investigate the breakdown of the finite-dimensional approximation due to resonant coupling between discrete modes and radiation (continuous spectrum) in the full PDE.

Proposed method

• Derive a finite-dimensional Hamiltonian reduction from the full NLS/GP PDE by projecting onto the ground and first excited states of the linear Schrödinger operator.
• Analyze the phase space structure of the reduced Hamiltonian system, particularly focusing on periodic orbits and oscillatory behavior near the bifurcation point.
• Apply a shadowing theorem to prove that solutions of the full PDE remain close to solutions of the finite-dimensional reduction for very long, but finite, time intervals.
• Use dispersive estimates and Strichartz-type inequalities derived from wave operator theory to control the radiation (continuous spectrum) component of the solution.
• Employ spectral theory and distorted Fourier bases to analyze the linearized dynamics around nonlinear bound states.
• Leverage boundedness of wave operators on Sobolev spaces to derive estimates for the nonlinear evolution in the presence of potential singularities and decay.

Experimental results

Research questions

• RQ1 How do solutions of the full NLS/GP equation behave on long time scales near the symmetry-breaking bifurcation point?
• RQ2 To what extent can the dynamics of the full PDE be approximated by a finite-dimensional Hamiltonian system near the bifurcation?
• RQ3 What is the lifetime of oscillatory mass-transport solutions between wells before resonant coupling with radiation modes disrupts the dynamics?
• RQ4 How do dispersive and Strichartz estimates contribute to controlling the radiation tail in the long-time evolution?
• RQ5 What role do wave operators and distorted Fourier bases play in the analysis of the linearized dynamics and nonlinear persistence?

Key findings

• Solutions with inter-well mass oscillation persist on very long, but finite, time scales, as predicted by the finite-dimensional Hamiltonian reduction.
• The finite-dimensional reduction captures the essential dynamics near the symmetry-breaking bifurcation, including the emergence of asymmetric states for particle numbers above the critical threshold $\mathcal{N}_{\text{cr}}$.
• A shadowing theorem is established, proving that full PDE solutions remain close to reduced system trajectories for times growing polynomially in the inverse of the perturbation size.
• The breakdown of the finite-dimensional approximation occurs due to resonant coupling between discrete bound states and the radiation (continuous) spectrum, which is not captured by the reduction.
• Dispersive and Strichartz estimates, combined with bounded wave operators on $W^{k,p}$ spaces, allow control of the radiation component, enabling the proof of long-time persistence.
• The analysis extends to potentials with delta-function singularities using formalism from [9], showing robustness of the framework to certain non-smooth potentials.