  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.
