A Toolkit For Steady States of Nonlinear Wave Equations: Continuous Time Nesterov and Exponential Time Differencing Schemes

Several methods exist for finding ground (as well as excited) states of nonlinear waves equations. In this paper we first introduce two modifications of the so-called accelerated imaginary-time evolution method (AITEM). In our first modification, time integration of the underlying gradient flow is done using exponential time differencing instead of using more standard methods. In the second modification, we present a generalization of the gradient flow model, motivated by the work of Nesterov, as well as that of Candes and collaborators. Additionally, we consider combinations of these methods with the so-called spectral renormalization scheme. Finally, we apply these techniques to the so-called Squared Operator Method, enabling convergence to excited states. Various examples are shown to illustrate the effectiveness of these new schemes, comparing them to standard ones established in the literature. In most cases, we find significant reductions in the number of iterations needed to reach convergence.