[Paper Review] On weak interaction between a ground state and a non-trapping potential

Scipio Cuccagna, Masaya Maeda|arXiv (Cornell University)|Sep 19, 2013

Advanced Mathematical Physics Problems26 references3 citations

TL;DR

This paper establishes asymptotic stability of moving ground states for the nonlinear Schrödinger equation with a non-trapping potential by combining Birkhoff normal forms and charge transfer model theory. It proves that ground states with non-zero speed are asymptotically stable under small potentials, large speeds, or large spatial separation, with radiation decaying to zero via a Fermi Golden Rule mechanism.

ABSTRACT

We show that ground states of the NLS moving at nonzero speed are asymptotically stable if they either stay far from the potential, or the potential is small, or the ground state has large speed.

Motivation & Objective

• To analyze the long-time dynamics of ground state solutions to the nonlinear Schrödinger equation with a non-trapping potential.
• To establish asymptotic stability of moving ground states when the potential is weak, the state has high speed, or is spatially separated from the potential.
• To extend previous results on orbital stability by incorporating a non-trapping potential as a perturbation.
• To rigorously justify the decay of internal modes into radiation using a Fermi Golden Rule condition.

Proposed method

• Apply Birkhoff normal forms to treat the non-trapping potential as a small perturbation of the unperturbed NLS equation.
• Use charge transfer model theory from [40] to handle continuous mode dispersion and radiation damping.
• Employ a Fermi Golden Rule condition to describe the decay of discrete internal modes into the radiation continuum.
• Construct an effective Hamiltonian framework to track the evolution of the ground state parameters (frequency, velocity, phase).
• Use a time-dependent gauge transformation to reduce the problem to a moving frame where the ground state appears stationary.
• Establish convergence of the state parameters (ω(t), v(t)) to limiting values ω₊, v₊ via energy and dispersive estimates.

Experimental results

Research questions

• RQ1 Under what conditions does a moving ground state of the NLS equation remain asymptotically stable when subjected to a non-trapping potential?
• RQ2 How does the presence of a non-trapping potential affect the decay of internal modes into radiation?
• RQ3 What role does the Fermi Golden Rule play in ensuring long-time stability of the ground state in the presence of a potential?
• RQ4 Can the dynamics of the ground state be described by a limiting state with conserved parameters ω₊ and v₊, even under weak perturbations?
• RQ5 How does the spatial separation between the ground state and the potential influence the stability and convergence of the solution?

Key findings

• Ground states with non-zero speed are asymptotically stable if they are far from the potential, the potential is small, or the speed is large.
• The solution converges to a limiting state of the form $ e^{i\theta(t) + \frac{i}{2}v_+ \cdot x} \phi_{\omega_+}(x - y(t)) + e^{it\Delta}h_+ $ in $ H^1 $ as $ t \to \infty $, with $ \|h_+\|_{H^1} \leq C\epsilon $.
• The remainder $ r(t,x) $ decomposes into a decaying part $ A(t,x) $ with $ \|A(t,\cdot)\|_{L^\infty} \to 0 $ and a dispersive part $ \widetilde{r} $ satisfying $ \|\widetilde{r}\|_{L^p_t W^{1,q}_x} \leq C\epsilon $ for any admissible pair $ (p,q) $.
• The parameters $ \omega(t) $ and $ v(t) $ converge to limiting values $ \omega_+ $ and $ v_+ $, with $ |\omega_+ - \omega_1| + |v_+| \leq C\epsilon $.
• The Fermi Golden Rule condition ensures the decay of internal modes into radiation, with the decay rate governed by the non-degeneracy of the resonance condition.
• The existence of the limit $ \lim_{t\to\infty} \mathcal{U}^{-1}(t,0)h(t) = h_+ $ in $ H^1 $ confirms the radiation term vanishes asymptotically.