Modelling high intensity laser pulse propagation in air using the modified Korteweg-de Vries equation

Abstract

Ultrafast laser pulse experiments and applications are entering a phase that challenges the validity of mathematical models utilised to model longer pulses in nonlinear optics. This thesis aims to propose a possible mathematical model for high intensity laser pulse propagation in air through a multiple scales expansion of Maxwell’s equations and discuss a method on how to solve the corresponding differential equation, known as the modified Korteweg-de Vries equation in to the small dispersion regime. This equation is solvable using a technique named the scattering transform and due to weak dispersion the equation can be solved asymptotically. The method is based on using the asymptotic WKB approximation for the forward scattering problem and reformulating the inverse scattering as a Riemann-Hilbert problem. Both analytical steps and numerical procedures needed to use the method is discussed and implemented. A full example calculation using a particular initial condition is performed and some challenges using the method for more general initial conditions are discussed.