Nonlinear Wave Motion

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Abstract: 0303756, PI: Mark Ablowitz, University of CaloradoTitle: Nonlinear Wave MotionThe solutions and properties of a class of nonlinear wave equations and related nonlinear systems which arise frequently in application will be studied by analytical, asymptotic and computational methods. New solutions of multi-dimensional equations and related linear scattering problems will be investigated. A prototypical system is the Kadomtsev-Petviashvili (KP) equation, which is a two-space one-time dimensional extension of the Korteweg-deVries equation. Associated with the linearization of the KP equation is the nonstationary Schrodinger equation which itself is a prominent equation in mathematics and physics. Important recent discoveries by the PI include finding new real, localized, multi-lump solutions to the KP equation and new classes of eigenfuctions to the nonstationary Schrodinger equation. These solutions are related to a positive integer, referred to as the charge, which is a type of winding number or index. The characterization of these solutions in terms of the charge and other indices will continue. New classes of KP solutions will be sought. Reductions of the four dimensional self-dual Yang Mills (SDYM) system, which is viewed as a "master" integrable system, leads to the study of novel nonlinear ordinary differential equations whose solutions possess unusual features. Special cases are the classical Darboux-Halphen system and Chazy equation, in general position. The solutions of these systems are related to modular/automorphic functions; and in the case of Chazy, it is related to the well known Ramanujan functions. Research involving new reductions of SDYM will continue. The investigation of differential-difference nonlinear Schrodinger (NLS) equations has shown that new vector extensions of a previously derived scalar difference NLS equation has soltion solutions and is integrable by the inverse scattering transform. The scalar and vector difference NLS systems reduce in the continuous limit to the physically important NLS equations. New solutions and properties of this vector difference NLS equation will be studied. Recent experimental and theoretical studies of water waves has shown that modulation of periodic waves exhibit nonrepeatible, chaotic dynamics whereas localized soltion soltuions do not possess these properties. This work was motivated by earlier research by the PI on computational chaos. Current research indicates that this phenomena also occurs in nonlinear optics and appears to be universal in character. This infinite dimensional and possibly universal chaotic dynamics will be studied in detail.The dynamics of wave systems with large amplitude is often referred to as nonlinear wave motion. Unlike small amplitude phenomena where substantial and wide ranging theory is available, the mathematical investigation of nonlinear wave motion is still at an early stage of development. Nonlinear wave equations, such as the ones described in this proposal, are centrally important in many physical applications. Two examples are water waves and nonlinear optics, including fiber optic communications. Extremely stable, localized nonlinear waves called solitons, is a subject which is closely related to the research investigations in this project. The study of nonlinear optics has focused in recent years on the study of localized large amplitude pulses such as solitons. Such pulses, are used in a variety of ways such as the shaping and controlling of light beams. In fiber optic communications, understanding the properties of large amplitude optical pulses are important for the next generation of communication systems. The mathematical discoveries made in the field of nonlinear fiber optic waves only a few years years ago are now at the cusp of commercial application. It is expected that publication of all new results will be published in prominent journals.