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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., Comment: 37 pages, 14 figures

Inspired by some recent experiments and numerical works related to nanoresonators, we perform classical molecular dynamics simulations to investigate the thermal expansion and the ability of the device to act as a strain sensor assisted by thermally-induced vibrations. The proposed model consists in a chain of atoms interacting anharmonically with both ends clamped to thermal reservoirs. We analyze the thermal expansion and resonant frequency shifts as a function of temperature and the applied strain. For the transversal modes the shift is approximately linear with strain. We also present analytical results from canonical calculations in the harmonic approximation showing that thermal expansion is uniform along the device. This prediction also works when the system operates in a nonlinear oscillation regime at moderate and high temperatures., Comment: 12 pages, 6 figures