Broadly speaking, I am interested in nonlinear wave phenomena and the mathematical models that describe them. The central theme in my research is the characterization of coherent structures, including dispersive shock waves, solitary waves, and their generalization. I use a combination of asymptotic analysis, numerical computation, and dynamical systems theory in my research.
My PhD thesis was titled "Generalized Riemann problems in Dispersive Hydrodynamics." You can check it out here.
Publications
PS, M. Hoefer, B. Ilan. Whitham modulation theory and two-phase instabilities for generalized nonlinear Schrödinger equations with full dispersion. Accepted to SIAM Journal on Applied Mathematics (2024)
PS, T. Bridges, M. Shearer. Traveling Wave Solutions of the Kawahara Equation Joining Distinct Periodic Waves. Journal of Nonlinear Science 33(5), 79 (2023)
A Binswanger, M Hoefer, B Ilan, PS, Whitham modulation theory for generalized Whitham equations and a general criterion for modulational instability. Studies in Applied Math 142, 724-751 (2021)
PS, MA Hoefer, Discontinuous shock solutions of the Whitham modulation equations as zero dispersion limits of traveling waves. Nonlinearity 33 (7), 2020
PS, M Hoefer, E Iacocca, Magnonic Band Structure Established by Chiral Spin-Density Waves in Thin Film Ferromagnets , IEEE Magnetics Letters 10 (2019)
PS, MA Hoefer, GA El, Hydrodynamic optical soliton tunnelingPhysical Review E 97 (3), 032218 (2018)
MA Hoefer, NF Smyth, PS, Modulation theory solution for nonlinearly resonant, fifth‐order Korteweg–de Vries, nonclassical, traveling dispersive shock waves, Studies in Applied Mathematics (2018)
PAP Janantha, PS, MA Hoefer, M Wu, Observation of self-cavitating envelope dispersive shock waves in yttrium iron garnet thin films, Physical Review Letters 119 (2), 024101 (2017)
PS, MA Hoefer, Shock Waves in Dispersive Hydrodynamics with Nonconvex Dispersion SIAM Journal on Applied Mathematics 77 (1), 26-50 (2017)
For an up-to-date list of publications, please refer to my Google Scholar Page
Current projects
Traveling wave solutions of abcd-Boussinesq systems (with T. Bridges and D. Ratliff)
Modulation theory and the computation of traveling waves in systems with nonlocal, non-polynomial dispersion
Characterization of long-time solutions of Riemann problem for discrete conservation laws (with C. Chong, M. Herrmann, M. Hoefer and P. Kevrekidis)
Transcriticial flow in nonlinear Schrödinger models with complex Wadati potentials (with S. Chandramouli)
Modeling dispersive shock waves in Bose-Einstein Consdensates with spin-orbit coupling (with B. Ilan and S. Sharan)
Collaborators
Mark Hoefer (CU Boulder)
Michael Shearer (NC State)
Gennady El (Northumbria)
Tom Bridges (Surrey)
Ezio Iacocca (CU Colorado Springs)
Boaz Ilan (UC Merced)
Adam Binswanger (currently PhD student at UC Merced)
Noel Smyth (A tribute paper in memeoriam of Noel can be found here)