This paper studies the propagation of plasmons on graphene when the Drude weight is varied in time. The phenomenon of plasmon propagation is modeled by considering the graphene as a conductive sheet. Under the assumption that the field is oscillatory in the direction parallel to the sheet, it can be shown that the coupled electromagnetic field can be reduced to a single time-dependent equation describing the current density on the sheet. The current density depends on the wave number \xi and is shown to satisfy an integro-differential equation in time. Well-posedness for this equation is established. A numerical scheme to solve the current equations based on convolution quadrature is developed. An approximate equation, based on large \xi with the physical interpretation of a quasi-static approximation, is derived and its accuracy assessed. The phenomena of wave reversal and parametric amplification are studied. Numerical calculations are conducted to address several theoretical issues as well as to demonstrate the main ideas.
Bibliographical noteFunding Information:
\ast Received by the editors November 14, 2018; accepted for publication (in revised form) March 27, 2019; published electronically June 20, 2019. http://www.siam.org/journals/siap/79-3/M122688.html Funding: This work was partially supported by the National Science Foundation through awards DMS 1211884 and DMS 1440471. \dagger School of Mathematics, University of Minnesota, Minneapolis, MN 55455-0134 (wil00108@umn. edu, firstname.lastname@example.org). \ddagger Department of Applied Mathematics and Statistics, Colorado School of Mines, Golden, CO 80401-1887 (email@example.com).
- Maxwell's equations
- Time reversal
- Wave propagation