[Paper Review] Stable Directions for Degenerate Excited States of Nonlinear Schr

Stephen J. Gustafson, Tuoc Phan|arXiv (Cornell University)|Apr 12, 2010

Advanced Mathematical Physics Problems4 citations

TL;DR

This paper studies nonlinear Schrödinger equations with radial, decaying potentials and constructs finite-codimension stable manifolds in phase space that govern solutions converging to degenerate excited states at infinity. For both attractive ($\lambda = 1$) and repulsive ($\lambda = -1$) nonlinearities, it establishes the existence of stable directions in the dynamics near these states, even when linear stability analysis fails due to degeneracy.

ABSTRACT

We consider nonlinear Schr\{o}dinger equations, $i\partial_t \psi = H_0 \psi + \lambda |\psi|^2\psi$ in $\mathbb{R}^3 imes [0,\infty)$, where $H_0 = -\Delta + V$, $\lambda=\pm 1$, the potential $V$ is radial and spatially decaying, and the linear Hamiltonian $H_0$ has only two eigenvalues $e_0 2e_1$) cases, we construct certain finite-codimension regions of the phase space consisting of solutions converging to these excited states at time infinity (stable directions).

Motivation & Objective

• To analyze the long-time dynamics of nonlinear Schrödinger equations with degenerate excited states in three dimensions.
• To address the challenge of constructing stable directions when linearized dynamics around excited states are non-hyperbolic due to degeneracy.
• To extend the theory of asymptotic stability beyond non-degenerate states, particularly in the presence of radial, decaying potentials.
• To provide a rigorous framework for understanding convergence to excited states under nonlinear evolution, even when standard spectral methods fail.

Proposed method

• Formulates the nonlinear Schrödinger equation as $i\partial_t \psi = H_0 \psi + \lambda |\psi|^2\psi$ with $H_0 = -\Delta + V$, $V$ radial and decaying.
• Identifies the excited states as bound states corresponding to the second eigenvalue $e_1$ of $H_0$, assuming only two eigenvalues exist.
• Applies center manifold reduction and Lyapunov-Perron method to construct finite-codimension stable manifolds in phase space.
• Uses spectral theory and radial symmetry to handle the degeneracy in the linearized operator around the excited state.
• Employs a Lyapunov-type functional and energy estimates to control the nonlinear evolution near the excited state.
• Establishes convergence to the excited state in the $L^2$-based topology by showing the existence of stable directions in the phase space.

Experimental results

Research questions

• RQ1 Can stable manifolds be constructed for degenerate excited states in the nonlinear Schrödinger equation with radial, decaying potentials?
• RQ2 How does the dynamics behave near excited states when the linearized operator has zero or negative eigenvalues, indicating degeneracy?
• RQ3 What role does the sign of the nonlinearity ($\lambda = \pm 1$) play in the existence and structure of stable directions?
• RQ4 Can finite-codimension stable manifolds be rigorously established for both attractive and repulsive nonlinearities in three dimensions?
• RQ5 How does radial symmetry and decay of the potential influence the construction of stable directions in the phase space?

Key findings

• For both $\lambda = 1$ and $\lambda = -1$, the paper constructs finite-codimension stable manifolds in the phase space that govern solutions converging to the excited state as $t \to \infty$.
• The stable directions are shown to exist despite the degeneracy of the excited state, which invalidates standard linear stability analysis.
• The construction relies on a refined spectral analysis and the use of center manifold techniques to handle the non-hyperbolic nature of the linearized dynamics.
• The existence of stable directions is established in a neighborhood of the excited state in the phase space, ensuring asymptotic convergence for a codimension-finite set of initial data.
• The method applies uniformly to both attractive and repulsive nonlinearities, demonstrating robustness in the presence of degeneracy.
• The results extend the theory of asymptotic stability to degenerate excited states in three-dimensional nonlinear Schrödinger systems with radial potentials.