Mathematical Sciences: Nonlinear Wave Motion

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9404265 Ablowitz This project involves the study of solutions to a class of physically significant nonlinear wave equations. New reductions of the self dual Yang-Mills system will be considered. The self dual equations have been shown to be a master integrable system from which virtually all known soliton equations are contained as special cases. Novel classes of solutions will be studied including those recently found which can be expressed in terms of classical automorphic functions whose solution structure possess complex singularities which are natural boundaries in the complex plane. Nonlinear integrable systems will be used in computational simulations in order to establish the existence of and to analytically understand the phenomena of numerical chaos. New solutions to physically important forced nonlinear wave equations will be developed. Many of the nonlinear equations we are studying are universally important asymptotic approximations in physical phenomena. Applications include nonlinear optical wave propagation, internal and surface waves in fluid dynamics, lattice dynamics, and relativity. The use of solitons in the technological application of long distance communication has been widely noted. In this research project, multisoliton solutions and their interaction properties will be analyzed in order to significantly improve the effectiveness of soliton based communication systems. This is usually referred to as wave length division multiplexing (WDM) in optical communications. One of the forced nonlinear wave systems which will be studied involves moving pressure distributions in surface waves, such as ships moving at critical speeds.