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Multiple-Scale Analyses of Forced, Nonlinear Waves: Graphene Hydrodynamics and Surface Water Waves


Nonlinear interactions in physical systems make analyzing the dynamics challenging. Multiple-scale analyses are asymptotic perturbation techniques useful for analyzing nonlinear systems that possess scale-separation between the relevant physical scales. Graphene is a two-dimensional lattice of carbon atoms which exhibits many unique electrical properties, such as a viscous, hydrodynamic regime where the electrons become strongly interacting. We consider both the Dirac fluid and Fermi liquid regimes in gated graphene, and we investigate the one-dimensional propagation of electronic solitons. By leveraging the scale-separation between the wave-propagation time scale and nonlinear-interaction time scale, we utilize a multiple-scale analysis to derive a Korteweg-de Vries (KdV)-Burgers equation governing the wave's evolution. We numerically solve the KdV-Burgers equation and analyze the viscous decay and entropy production. Finally, we propose experimental realizations of these effects to measure the shear viscosity of graphene.

Surface water waves also possess nonlinear interactions and permit nonlinear, periodic waves of permanent form known as Stokes waves in intermediate- and deep-water. Wind forcing causes wave growth and decay, but it can also influence wave shape. To study the effect of wind on the shape of these nonlinear Stokes waves, we again utilize scale-separation between the wave-propagation time scale and the nonlinear-interaction time scale. We then analytically solve the resulting system for three different wind-induced surface pressure profiles (Jeffreys, Miles, and generalized Miles) to calculate the wind-induced changes to the wave's shape statistics, growth rate, and phase speed. These results are constrained by existing large eddy simulations and are consistent with prior laboratory experiments.

Shallow water supports surface waves known as solitary waves that balance nonlinearity and dispersion. By applying the same Jeffreys-type surface pressure forcing, we analyze the effect of wind on shallow-water wave shape. Another multiple-scale analysis yields a KdV-Burgers equation for the wave's profile, and we solve this numerically to investigate the wave's skewness and asymmetry resulting from the solitary waves's wind-induced, bound, dispersive tail. Extending this analysis to a planar, sloping bathymetry instead yields a variable-coefficient KdV-Burgers equation. Numerically solving this equation reveals the interaction of wind-forcing and shoaling on wave growth, width change, and rear-shelf generation.

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