  We evaluate the performance of novel numerical methods for solving
one-dimensional nonlinear fractional dispersive and dissipative evolution
equations. The methods are based on affine combinations of time-splitting
integrators and pseudo-spectral discretizations using Hermite and Fourier
expansions. We show the effectiveness of the proposed methods by numerically
computing the dynamics of soliton solutions of the the standard and fractional
variants of the nonlinear Schr{\"o}dinger equation (NLSE) and the complex
Ginzburg-Landau equation (CGLE), and by comparing the results with those
obtained by standard splitting integrators. An exhaustive numerical
investigation shows that the new technique is competitive when compared to
traditional composition-splitting schemes for the case of Hamiltonian problems
both in terms accuracy and computational cost. Moreover, it is applicable
straightforwardly to irreversible models, outperforming high-order symplectic
integrators which could become unstable due to their need of negative time
steps. Finally, we discuss potential improvements of the numerical methods
aimed to increase their efficiency, and possible applications to the
investigation of dissipative solitons that arise in nonlinear optical systems
of contemporary interest. Overall, the method offers a promising alternative
for solving a wide range of evolutionary partial differential equations.
