Analytical Methods for Nonlinear Oscillators and Solitary Waves

The most well-known analytical method is the perturbation method, which has led to the great discovery of Neptune in 1846, and since then mathematical prediction and empirical observation became two sides of a coin in physics. However, the perturbation method is based on the small parameter assumption, and the obtained solutions are valid only for weakly nonlinear equations, which have greatly limited their applications to modern physical problems. To overcome the shortcomings, many mathematicians and physicists have been extensively developing various technologies for several centuries, however, there is no universal method for all nonlinear problems, and mathematical prediction with remarkably high accuracy is still much needed for modern physics, for example, the solitary waves traveling along an unsmooth boundary, the low-frequency property of a harvesting energy device, the pull-in voltage in a micro-electromechanical system. Now various effective analytical methods have appeared in the open literature, e.g., the homotopy perturbation method and the variational iteration method. An analytical solution provides a fast insight into its physical properties of a practical problem, e.g., frequency-amplitude relation of a nonlinear oscillator, solitary wave in an optical fiber, pull-in instability of a microelectromechanical system, making mathematical prediction even more attractive in modern physics.

Nonlinear physics has been developing into a new stage, where the fractal-fractional differential equations have to be adopted to describe more accurately discontinuous problems, and it becomes ever more difficult to find an analytical solution for such nonlinear problems, and the analytical methods for fractal-fractional differential equations have laid the foundations for nonlinear physics.

The Research Topic is a review of the state of the art of fields of fractional calculus and analytical methods for nonlinear physics, it aims at developing new concepts, new mathematical frameworks, and new analytical methods for nonlinear problems to trigger new research frontiers in future.
This Research Topic welcomes the following articles:
• Fractal-fractional models for solitary waves traveling along the unsmooth boundary or in a fractal medium;
• Pull-in stability of MEMS system;
• Nonlinear control of the bioprinting and 3D-printing processes;
• Low-frequency technology for energy harvesting and green buildings;
• Nonlinear phenomena arising in nanoscale fluids and nanomaterials.