Nonlinear Wave Motion

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NSF Award Abstract - DMS-0070792Mathematical Sciences: Nonlinear Wave MotionAbstract0070792 AblowitzInvestigation of a new class of solutions to certain physically significant multidimensional nonlinear wave equations and associated linear scattering problems will continue. This work follows the discovery by the principal investigator that these solutions can be associated with an integer, referred to as the charge. Solutions and properties of a class of nonlinear differential-difference equations which arise in the study of fiber optic arrays will be obtained by employing analytical, computational and approximation/asymptotic techniques. Recent joint experimental-theoretical investigations of water waves by the principal investigator and collaborators has demonstrated that infinite dimensional chaotic dynamics can eventually result from the well known modulational instability of the classical Stokes water wave. The underlying mechanism governing this phenomenon has recently been uncovered. Further experiments and analysis are planned. Preliminary analysis has indicated that this dynamical scenario can also be observed in fiber optics.The dynamics of wave systems with large amplitude is often referred to as nonlinear wave motion. In contrast to the substantial theory available to understand small amplitude phenomena, the mathematical description of nonlinear wave motion is still in an early stage of development. Applications range from the study of large amplitude ocean waves to the propagation of pulses of light in fiber optics. Extremely stable, localized nonlinear waves, called solitons, are closely related to the investigations in this project. Solitons are being investigated worldwide due their potential use in high data rate communications over fiber-optic cables. The mathematical discoveries made in the field of nonlinear waves only a few years ago are now at the cusp of commercial