A constrained gentlest ascent dynamics and its applications to finding excited states of Bose-Einstein condensates

TL;DR

This paper extends the gentlest ascent dynamics to a constrained version for finding saddle points with specific Morse indices, and applies it to compute excited states of Bose-Einstein condensates, demonstrating effectiveness through numerical results.

Contribution

The paper introduces a constrained gentlest ascent dynamics (CGAD) for locating saddle points with specified Morse indices and applies it to excited states of BECs.

Findings

CGAD accurately finds constrained saddle points of the energy functional.

The method converges exponentially near nondegenerate saddle points.

Numerical results confirm the robustness and physical relevance of the approach.

Abstract

In this paper, the gentlest ascent dynamics (GAD) developed in [W. E and X. Zhou, Nonlinearity, 24 (2011), pp. 1831--1842] is extended to a constrained gentlest ascent dynamics (CGAD) to find constrained saddle points with any specified Morse indices. It is proved that the linearly stable steady state of the proposed CGAD is exactly a nondegenerate constrained saddle point with a corresponding Morse index. Meanwhile, the locally exponential convergence of an idealized CGAD near nondegenerate constrained saddle points with corresponding indices is also verified. The CGAD is then applied to find excited states of single-component Bose--Einstein condensates (BECs) in the order of their Morse indices via computing constrained saddle points of the corresponding Gross--Pitaevskii energy functional under the normalization constraint. In addition, properties of the excited states of BECs in the linear/nonlinear cases are mathematically/numerically studied. Extensive numerical results are reported to show the effectiveness and robustness of our method and demonstrate some interesting physics.