Today, engineers face an increasing challenge in advanced engineering applications that are based on efficient mathematical models for propagation and transition phenomena. Propagation aspects implying commutative and/or additive consequences of quantum physics are used extensively in the design of long-range transmission systems. Differential geometry is adapted for solving nonlinear partial differential equations with a very great number of variables for transitions in complex optoelectronics systems. Special mathematical functions are used in modeling very small-scale material properties (energy levels and induced transitions) in quantum physics for the design of nanostructures in microelectronics. Time series with extremely high transmission rates are used for multiplexed transmission systems for large communities, such as traffic in computer networks or transportations, financial time series, and time series of fractional order in general. All these advanced engineering subjects require efficient mathematical models in the development of classical tools for complex systems such as differential geometry, vector algebra, partial differential equations, and time series dynamics. The objective in such applications is to take into consideration efficiency aspects of mathematical and physical models required by basic phenomena of propagation and transitions in complex systems, such as in situations implying physical limits as long distances propagation phenomena (solitons), quantum transitions in nanostructures, complex systems with great number of variables, and infinite spatiotemporal extension of material media. This special issue seeks high-quality research and review papers in developments and methods for efficient mathematical approaches for propagation phenomena and transitions in complex systems with applications in experimental physics and engineering.

Potential topics include, but are not limited to:

• Accurate and efficient mathematical models for long-distances propagation phenomena
• Specific methods for solving nonlinear partial differential equations describing wave propagation and transitions in nonlinear optics and optoelectronics
• Mathematical tools for analyzing the dynamics of complex systems with applications in nanostructures, microelectronics, and image processing
• Dynamical models for infinite spatiotemporal extension of material media or for highly repetitive phenomena