Universality in Nonlinear Waves and Related Topics

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This project studies the phenomenon of universality in the context of models for the motion of large waves in several physical settings such as surface water waves and electromagnetic waves in optical fibers. Universality refers to situations in which the same or very similar wave patterns appear despite the waves being set into motion by quite different mechanisms, or even in quite different physical systems. For instance, rogue waves on the sea surface are frequently characterized as consisting of a large central peak with a distinctive dip on either side. It does not matter much the conditions under which the rogue wave appears — the pattern is nearly always the same. The aim of this research is to determine what patterns should be expected, as this knowledge can be used in applications ranging from device design to disaster mitigation, and why they occur. Furthermore, because modeling is a process that involves numerous ad-hoc assumptions, it is important to understand which features predicted by a model are independent of those assumptions, and universality gets to the heart of this question. Graduate students and early-career researchers will join the investigator in this study, which enhances its impact beyond scientific inquiry and into education and training of the next generation of scientists. Mathematically, the study of universality is related to asymptotic analysis, specifically involving double-scaling limits to localize the coordinates near a point of interest, while a parameter in the model or solution becomes large. The investigator will study such double-scaling limits in various asymptotic models for nonlinear waves given by integrable evolution equations. Broadening the scope slightly, several specific questions involving asymptotic analysis of mathematical models for nonlinear waves will be addressed, including (i) determining the small-dispersion asymptotics of solutions of the defocusing nonlinear Schrödinger equation and the intermediate long-wave and Benjamin-Ono equations; (ii) analyzing the features of a new family of transcendental solutions of the focusing nonlinear Schrödinger equation termed "rogue waves of infinite order"; (iii) studying the degeneration of specific solution families of Painlevé equations. The investigator will combine and further develop techniques from the fields of integrable systems and asymptotic analysis to address these questions. Anticipated outcomes include a first proof of universal wave breaking in the defocusing nonlinear Schrödinger equation, new results on the small-dispersion asymptotic behavior of solutions of the intermediate long-wave equation (a nonlocal model for internal waves in stratified fluids interpolating between the shallow-water Korteweg-de Vries limit and the deep-water Benjamin-Ono limit), development of a new analytical technique for asymptotic analysis of nonlocal Riemann-Hilbert problems, and the discovery of new information about the solution space of Painlevé equations and the focusing nonlinear Schrödinger equation.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.