In this paper, 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\"odinger 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 with 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, our method offers a promising alternative for solving a wide range of evolutionary partial differential equations.

翻译：本文评估了用于求解一维非线性分数阶色散与耗散演化方程的新型数值方法的性能。这些方法基于时间分裂积分器与使用埃尔米特和傅里叶展开的拟谱离散化的仿射组合。通过数值计算标准及分数阶非线性薛定谔方程与复金兹堡-朗道方程的孤子解动力学，并将结果与标准分裂积分器所得结果进行对比，我们验证了所提方法的有效性。详尽的数值研究表明，在哈密顿问题情形下，新方法在精度和计算成本方面均能与传统组合分裂方案相媲美。此外，该方法可直接应用于不可逆模型，其性能优于高阶辛积分器——后者因需使用负时间步长而可能变得不稳定。最后，我们讨论了旨在提升效率的数值方法潜在改进方向，以及该方法在当代非线性光学系统中耗散孤子研究中的可能应用。总体而言，我们的方法为求解广泛类型的演化偏微分方程提供了一种有前景的替代方案。