Singular and Spatially Heterogeneous Perturbations of Solitary Waves

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Although quite different, waves on the surface of the ocean, vibrations in a long molecule, oscillations in a superheated plasma, the dynamics of electricity in power systems, and the motion of engineered materials have deep commonalities from a mathematical point of view. The state of the art for the modeling, analysis, and computer simulation of all these systems is very sophisticated. Nonetheless, there are seemingly simple, physically-relevant effects that profoundly complicate the way in which the systems are analyzed mathematically and that make current methods insufficient. This project aims to advance the mathematical treatment of waves to incorporate two broad classes of such physical effects. The first of these is spatial, that is, the effects related to the fact that a system is not uniform throughout its extent. For example, such heterogeneity arises from including bottom topography in water wave models, lamination in elastica, or structural variability in engineered materials. The second are so-called singular effects. These include accounting for surface tension in fluids, incorporating multiple species of ions in plasma models, and introduction of defects in molecular chains. The project aims to (a) devise models for these effects, (b) develop mathematically rigorous, broadly applicable, and highly accurate quantitative descriptions, and (c) implement novel algorithms for the simulation of such systems. The project will provide research training opportunities for undergraduate and graduate students. More technically, the goal is to understand how the above phenomena affect the existence, stability, and (especially) dynamics of coherent structures in nonlinear, hyperbolic and/or dispersive differential equations. Spatial heterogeneity ruins translation invariance, an essential ingredient for the existence of traveling waves. Singular effects are famously unpredictable. Nevertheless, incorporating these sorts of effects need not eliminate entirely the coherent structures. For instance, it may be that a traveling wave becomes a nanopteron, which is to say a traveling wave that is the superposition of a localized solitary core and a very small amplitude periodic wave. Nanopterons are already known to exist in the gravity-capillary wave problem and a variety of Hamiltonian lattice models. Another possibility is that a solitary wave propagates seemingly unchanged for an extremely long time but eventually deteriorates into dispersive waves. That is to say, the solitary wave transforms into a long-lived transient, or metastable, solution. This project’s principal aims are: identify systems that possess nanopteron solutions and establish their existence rigorously; advance the rigorous nanopteron theory to be more descriptive and to work in settings where the solitary core is large; devise and implement high-order symplectic integrators for Hamiltonian partial and lattice differential equations to simulate metastable solitary waves over very long time scales; prove rigorously the existence of metastable solitary waves; and identify new phenomenology related to spatially heterogeneous or singular perturbations of solitary waves through analysis and simulation.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.