Accelerated Imaginary-time Evolution Methods for the Computation of Solitary Waves

TL;DR

This paper introduces two accelerated imaginary-time evolution methods for computing solitary waves in multiple dimensions, demonstrating improved convergence and performance over existing methods like Petviashvili's.

Contribution

The paper proposes two novel accelerated methods with different normalization techniques, providing convergence analysis and showing superior performance in computing solitary waves.

Findings

The first method converges if and only if the solitary wave is linearly stable.

The second method significantly outperforms the first and Petviashvili methods.

Both methods are effective across various examples and dimensions.

Abstract

Two accelerated imaginary-time evolution methods are proposed for the computation of solitary waves in arbitrary spatial dimensions. For the first method (with traditional power normalization), the convergence conditions as well as conditions for optimal accelerations are derived. In addition, it is shown that for nodeless solitary waves, this method converges if and only if the solitary wave is linearly stable. The second method is similar to the first method except that it uses a novel amplitude normalization. The performance of these methods is illustrated on various examples. It is found that while the first method is competitive with the Petviashvili method, the second method delivers much better performance than the first method and the Petviashvili method.