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Several methods exist for finding ground (as well as excited) states ofnonlinear waves equations. In this paper we first introduce two modificationsof the so-called accelerated imaginary-time evolution method (AITEM). In ourfirst modification, time integration of the underlying gradient flow is doneusing exponential time differencing instead of using more standard methods. Inthe second modification, we present a generalization of the gradient flowmodel, motivated by the work of Nesterov, as well as that of Candes andcollaborators. Additionally, we consider combinations of these methods with theso-called spectral renormalization scheme. Finally, we apply these techniquesto the so-called Squared Operator Method, enabling convergence to excitedstates. Various examples are shown to illustrate the effectiveness of these newschemes, comparing them to standard ones established in the literature. In mostcases, we find significant reductions in the number of iterations needed toreach convergence.

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