NSF Award
2405535
Many naturally occurring and artificially induced phenomena can be studied via models of discrete, heterogeneous media. Examples include the oscillations of DNA strands and muscle proteins, the resonances within granular crystals, and the flow of growth hormones among the cells of plant leaves. These examples are discrete in the sense that they can be separated into multiple components that remain distinct from each other but also interact together. They are heterogeneous because the material properties of their components are not all identical but instead vary in some identifiable manner, perhaps via intentional structural choices or due to flaws and defects. Both the discrete structure of a medium and the heterogeneous selection of its material data profoundly influence the kinds of dynamics that it can, or cannot, experience. This project will develop and refine the mathematical theory of waves in such media by studying increasingly more complicated dimensions of heterogeneity, by developing more precise and versatile quantifications of existing wave structures, and by identifying new kinds of dynamics that can propagate in the presence of physical discretization and material heterogeneity. The project will also provide topics coursework in nonlinear waves for advanced undergraduates and foster community engagement in department seminars and colloquia.<br/><br/>The project will study the effects of material heterogeneity in lattice differential equations. These arise after a traveling wave ansatz from the equations of motion for infinite chains of nonlinearly coupled discrete oscillators. Varying the oscillator masses and/or the coupling potentials yields material heterogeneities. Particularly meaningful structures result from allowing the material data to repeat periodically throughout the lattice. Recent work has shown that the classical solitary wave in monatomic lattices of identical particles and springs perturbs into a nanopteron or generalized solitary wave in dimer lattices of alternating particles or springs. The nanopteron traveling wave profile consists of small amplitude, highly oscillatory far field periodic ripples superimposed on an exponentially localized core. The project will adapt the construction of nanopterons to lattices whose material data repeats with a larger period than the dimer and whose structure destroys the useful physical symmetries of dimers, two significant and unresolved complications. One likely change is that the periodic ripple will need to be replaced by a quasiperiodic structure. Additionally, the project will revisit dimer nanopterons and extract more precise estimates on the amplitudes and phase shifts of the periodic ripples, as well as develop simpler model equations that transparently capture the functional-analytic complexities of these problems. Throughout, the main technical challenges will be the nonlocal structure of the traveling wave problems, a consequence of the lattice's discrete structure; singular perturbations arising from various physical limiting regimes of interest; multiple solvability conditions induced by the higher-order heterogeneities; and exponentially small, or possibly large, factors, that arise in capturing the precise ripple amplitudes.<br/><br/>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.
