ABSTRACT
This chapter introduces the mathematical and computational methods that are important in the analysis of optics and several motivating examples illustrating regular and singular perturbations. It includes several formal definitions of the big and little order relations, asymptotic sequences and series. Of particular importance to us is the notion of a uniform asymptotic expansion. The chapter discusses the evolution of several types of solutions to linear wave equations, and shows how the notions of group velocity and dispersion emerge from an asymptotic analysis of these solutions using the method of steepest descent. The chapter also introduces the boundary value problem associated with electromagnetic fields that are confined by regions of elevated refractive index. It also discusses the numerical and graphical algorithms most commonly encountered in dealing with problems in nonlinear optics. The chapter also includes specifically the coding of the nls equation and strongly encourage the reader to do some computations.
