[Paper Review] An example of stable excited state on nonlinear Schrödinger equation with nonlocal nonlinearity

Masaya Maeda, Satoshi Masaki|arXiv (Cornell University)|Sep 13, 2011

Advanced Mathematical Physics Problems4 citations

TL;DR

This paper establishes global well-posedness for a nonlinear Schrödinger equation with nonlocal nonlinearity in a broader function space ($\Sigma^{1/2}$) than previous results, and demonstrates the existence of stable excited states with arbitrarily high Morse index—challenging the conventional belief that high Morse index implies instability. The key contribution is constructing explicit examples of such stable excited states via spectral analysis of the action functional's Hessian.

ABSTRACT

In this article, we consider nonlinear Schrödinger equation with nonlocal nonlinearity which is a generalized model of the Schrödinger-Poisson system (Schrödinger-Newton equations) in low dimensions. We first prove the global well-posedness in wider space than in previous result and show the stability of standing waves including excites states. It turns out that an example of stable excite states with high Morse index is contained. Several examples of traveling-wave-type solutions are also given.

Motivation & Objective

• To extend the global well-posedness of the nonlinear Schrödinger equation with nonlocal nonlinearity to a wider function space ($\Sigma^{1/2}$) than previously known.
• To investigate the stability of standing wave solutions, particularly excited states, in the context of nonlocal nonlinearity.
• To construct explicit examples of stable excited states with high Morse index, challenging the conventional instability expectation for such states.
• To analyze the spectral properties of the Hessian of the action functional to determine stability conditions beyond the standard Grillakis-Shatah-Strauss framework.

Proposed method

• The authors introduce a modified nonlinear Schrödinger equation (H') with a nonlocal nonlinearity that allows solutions in $\Sigma^{1/2}$, a space weaker than $\Sigma$, by removing the unbounded third term in the nonlinearity expansion.
• They exploit conservation laws to control the time evolution of the unbounded terms in the nonlinearity, enabling well-posedness in $\Sigma^{1/2}$.
• The stability of standing waves is analyzed using the Grillakis-Shatah-Strauss theory, focusing on the Morse index $n(S_{\omega}^{\prime\prime}(\phi_{\omega}))$ and the second derivative of the frequency-to-mass map $d^{\prime\prime}(\omega)$.
• Explicit standing wave solutions are constructed in one dimension using Hermite functions, and their stability is determined by analyzing the spectrum of the Hessian $S_{\omega}^{\prime\prime}(\phi_{\omega})$ in a finite-dimensional subspace.
• The spectral analysis is reduced to a $3 \times 3$ matrix problem on the span of $\{\Omega_{n-2}, \Omega_n, \Omega_{n+2}\}$, where $\Omega_n$ are Hermite functions.
• The characteristic polynomials of the Hessian blocks are derived and analyzed to determine the number of negative eigenvalues, which determines the Morse index and stability.

Experimental results

Research questions

• RQ1 Can global well-posedness be established in a function space weaker than $\Sigma$ for the nonlinear Schrödinger equation with nonlocal nonlinearity?
• RQ2 Are there stable excited states with Morse index greater than one in this nonlocal model, contrary to classical expectations?
• RQ3 How does the nonlocal nonlinearity, which behaves like a linear potential at infinity, affect the stability of standing waves?
• RQ4 Can explicit examples of such stable high-index excited states be constructed and verified analytically?
• RQ5 What is the role of the frequency-to-mass map's second derivative $d^{\prime\prime}(\omega)$ in determining stability when the Morse index is high?

Key findings

• The equation (H') is globally well-posed in the space $\Sigma^{1/2}$, which is a strictly larger space than $\Sigma$, by removing the unbounded third term in the nonlinearity expansion.
• The paper constructs explicit examples of standing wave solutions using Hermite functions $\Omega_n$ in one dimension.
• For the case (II) solutions, the Hessian $S_{\omega}^{\prime\prime}(\Omega_n)$ has two negative eigenvalues and one positive eigenvalue, implying a Morse index of two.
• The frequency-to-mass map satisfies $d^{\prime\prime}(\omega) > 0$ at $M=1$, and since the Morse index is two and the condition $n(S_{\omega}^{\prime\prime}) - \max(d^{\prime\prime}/|d^{\prime\prime}|, 0)$ is even, the standing wave is stable.
• The paper proves the existence of stable excited states with arbitrarily high Morse index, contradicting the common belief that high Morse index implies instability.
• The stability of the excited state is confirmed via spectral analysis of the Hessian, showing that the number of negative eigenvalues and the behavior of $d^{\prime\prime}(\omega)$ satisfy the stability criterion despite high Morse index.