Nonlinear Wave Motion

University Of Colorado At Boulder, Boulder CO

Investigators

Abstract

Wave motion is fundamental in nature: Light waves impact the way people see, sound waves allow people to hear, and water waves are critical to the motion of ships and tidal energy. This project is focused on nonlinear wave propagation, that is the study of large amplitude or high-power signals, and aims to enhance the ability of scientists to develop insight and understanding of large amplitude waves and of the associated dynamics of wave energy transfer. The results of the investigation will benefit applications of nonlinear wave motion, which include tsunamis, rogue or freak waves, high power lasers, and the propagation of large changes in pressure waves such as those that occur in shock waves and blast waves. For linear waves or small amplitude waves a substantial theory is available and, in many situations, exact solutions can be found. For nonlinear waves the available theory is less developed and there are relatively few nonlinear wave equations for which exact solutions are known. However, the so-called Inverse Scattering Transform method has been used to find solutions and understand important properties of the underlying equations and has led to a wide class of solutions of physically significant nonlinear wave equations. This method can be used to find so-called solitons solutions, stable localized waves which have special interaction properties and have been found to apply widely, for example in water waves, electromagnetic waves, lattice dynamics, magnetic waves, and elasticity. The investigator will consider extensions of this method to new classes of nonlinear equations, such as fractional nonlinear wave equations and their soliton solutions. Fractional equations themselves have numerous applications, including to the study of amorphous materials. Additionally, the method will be applied to study the properties of a class of multidimensional solitons, that is lump-type waves that vanish in all directions, which are relevant also for equations that arise in quantum mechanics and optics. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.