Our research is in the field of nonlinear waves, lying at the nexus of applied mathematics and physics.  Mathematically, the nonlinear waves that we investigate are solutions to nonlinear, dispersive partial differential equations.  Physically, this field has been driven by applications ranging from classical (geophysical fluid dynamics) to modern physics (nonlinear optics and quantum/condensed matter) in which wave speed depends upon the countervailing effects of wave amplitude and wavelength.  In these physical environments, coherent structures such as solitons or solitary waves, periodic waves, and dispersive shock waves (DSWs) play a decisive role in the dynamics of large amplitude excitations.  The Dispersive Hydrodynamics Lab investigates the mathematics of coherent structures and their dynamics in nonlinear dispersive partial differential equations with particular application to fluid dynamics where we develop in-house experiments to test our mathematical predictions.  Research also focuses upon nonlinear wave applications in magnetic media where a variety of coherent spin wave, solitonic, and hydrodynamic-like structures are studied.  Methods employed include mathematical modeling, analysis, asymptotics, Whitham modulation theory, numerical analysis, and in-house experiment. Whenever possible, comparisons with experiment are carried out. 

The generation of DSWs represents a universal mechanism to resolve hydrodynamic singularities in dispersive media. Physical manifestations of DSWs include undular bores on shallow water and in the atmosphere (the Morning Glory), nonlinear diffraction patterns in optics, and matter waves in ultracold atoms. Any approximately conservative, nonlinear, hydrodynamic medium exhibiting weak dispersion can develop DSWs. The mathematical description of DSWs involves a synthesis of methods from hyperbolic quasi-linear systems, asymptotics, and soliton theory. This research is currently supported by the National Science Foundation through DMS-1816934. 

Ferromagnetic media provide a source of rich nonlinear, dispersive phenomena with practical import. Theoretical and technological developments have stimulated the field of nanomagnetism by the introduction of spin polarized currents as a means to excite magnetization dynamics at the nanometer scale in patterned environments. Strongly nonlinear magnetic solitons were recently observed in a nanomagnetic system. This solitary wave or "droplet" joins the domain wall and magnetic vortex as a fundamental and distinct object in nanomagnetism with similar potential for fruitful science. Marrying the Lab's research on fluid dynamics with the field of magnetic materials, we have also been developing the dispersive hydrodynamic description of magnetic materials.

External Listings:  Google Scholar, arXiv, ORCID

Manuscripts in Review

  1. Yifeng Mao and Mark A. Hoefer, Experimental investigations of linear and nonlinear periodic traveling waves in a viscous fluid conduit, arXiv:2202:05905 (2022).
  2. Mingyu Hu, Ezio Iacocca, and Mark Hoefer, The spin piston problem for a ferromagnetic thin film, arXiv:2112.15301 (2022).

​Journal Publications

  1. Samuel J. Ryskamp, Mark A. Hoefer, and Gino Biondini, Modulation theory for soliton resonance and Mach reflection, Proceedings of the Royal Society A, accepted.  (preprint)
  2. Mingyu Hu, Ezio Iacocca, Mark A. Hoefer, Spin-injection-generated shock waves and solitons in a ferromagnetic thin film​, IEEE Transactions on Magnetics 58 1–5 (2021). (preprint)
  3. Kiera van der Sande, Gennady A. El, and Mark A. Hoefer, Dynamic soliton–mean flow interaction with non-convex flux, Journal of Fluid Mechanics 928 A21 (2021). (reprint © Cambridge University Press)
  4. T. Congy, G. A. El, M. A. Hoefer, and M. Shearer, Dispersive Riemann problem for the Benjamin-Bona-Mahony equation, Studies in Applied Mathematics 147 1089–1145 (2021). (open access)
  5. Samuel J. Ryskamp, Mark A. Hoefer, and Gino Biondini, Oblique interactions between solitons and mean flows in the Kadomtsev-Petviashvili equation, Nonlinearity 34 3583–3617 (2021). (preprint)
  6. Adam L. Binswanger, Mark A. Hoefer, Boaz Ilan, and Patrick Sprenger, Whitham modulation theory for generalized Whitham equations and a general criterion for modulational instability, Studies in Applied Mathematics 147 724–751 (2021). (preprint).
  7. Samuel Ryskamp, Michelle D. Maiden, Gino Biondini, and Mark A. Hoefer, Evolution of truncated and bent gravity wave solitons: the Mach expansion problem, Journal of Fluid Mechanics 909, A24 (2021). (reprint © Cambridge University Press)
  8. Gino Biondini, Mark A. Hoefer, and A. Moro, Integrability, exact reductions and special solutions of the KP-Whitham equations, Nonlinearity 33 4114–4132 (2020). (preprint)
  9. P. Sprenger and M. A. Hoefer, Discontinuous shock solutions of the Whitham modulation equations as zero dispersion limits of traveling waves, Nonlinearity 33, 3268–3302 (2020). (preprint)
  10. M. D. Maiden, N. A. Franco, E. G. Webb, G. A. El, and M. A. Hoefer, Solitary wave fission of a large disturbance in a viscous fluid conduit, Journal of Fluid Mechanics 883, A10 (2019) (open access).
  11. Ezio Iacocca and Mark A. Hoefer, Perspectives on spin hydrodynamics in ferromagnetic materials, Physics Letters A, doi:10.1016/j.physleta.2019.125858 (2019).
  12. T. Congy, G. A. El, and M. A. Hoefer, Interaction of linear modulated waves with unsteady dispersive hydrodynamic states with application to shallow water waves, Journal of Fluid Mechanics 875, 1145-1174 (2019). (open access)
  13. Richard O. Moore and Mark A. Hoefer, Stochastic ejection of nanocontact droplet solitons via drift instability, Physical Review B 100, 014402 (2019). (reprint)
  14. Dalton V. Valentine, Michelle D. Maiden, and Mark A. Hoefer, Controlling Dispersive Hydrodynamic Wavebreaking in a Viscous Fluid Conduit, Physical Review Fluids 4, 074804 (2019). (reprint)
  15. Ezio Iacocca and Mark A. Hoefer, Hydrodynamic description of long-distance spin transport through noncollinear magnetization states:  the role of dispersion, nonlinearity, and damping, Physical Review B 99, 184402 (2019). (reprint)
  16. E. Iacocca, T-M. Liu, A. H. Reid, Z. Fu, S. Ruta, P. W. Granitzka, E. Jal, S. Bonetti, A. X. Gray, C. E. Graves, R. Kukreja, Z. Chen, D. J. Higley, T. Chase, L. Le Guyader, K. Hirsch, H. Ohldag, W. F. Schlotter, G. L. Dakovski, G. Coslovich, M. C. Hoffmann, S. Carron, A. Tsukamoto, M. Savoini, A. Kirilyuk, A. V. Kimel, Th. Rasing, J. Stöhr, R. F. L. Evans, T. Ostler, R. W. Chantrell, M. A. Hoefer, T. J. Silva, and H. A. Dürr, Spin-current-mediated rapid magnon localization and coalescence after ultrafast optical pumping of ferromagnetic alloys, Nature Communications 10, 1756 (2019). (open access) (news)
  17. Patrick Sprenger, Mark A. Hoefer, and Ezio Iacocca, Magnonic Band Structure Established by Chiral Spin-Density Waves in Thin Film Ferromagnets, IEEE Magnetics Letters 10, 1–5 (2019). (preprint)
  18. T. Congy, G. A. El, M. A. Hoefer, and M. Shearer, Nonlinear Schrödinger equations and the universal description of dispersive shock wave structure, Studies in Applied Mathematics 142, 241–268 (2019). (open access)
  19. Mark A. Hoefer, Noel F. Smyth, and Patrick Sprenger, Modulation theory solution for nonlinearly resonant, fifth order Korteweg-de Vries non-classical traveling dispersive shock waves, Studies in Applied Mathematics 142, 219–240 (2019).
  20. M. E. Mossman, M. A. Hoefer, K. Julien, P. Kevrekidis, P. Engels, Dissipative shock waves generated by a quantum-mechanical piston, Nature Communications 9, 4665 (2018). (open access)
  21. M. D. Maiden, D. V. Anderson, N. A. Franco, G. A. El, and M. A. Hoefer, Solitonic Dispersive Hydrodynamics: Theory and Observation, Physical Review Letters 120, 144101 (2018). (reprint)
  22. M. Ruth, E. Iacocca, P. G. Kevrekidis, and M. A. Hoefer, Transverse instabilities of stripe domains in magnetic thin films with perpendicular magnetic anisotropy. Physical Review B 97, 104428 (2018). (reprint)
  23. P. Sprenger, M. A. Hoefer, and G. A. El, Hydrodynamic Optical Soliton Tunneling, Physical Review E 97, 032218 (2018). (reprint)
  24. Denys Dutykh, Mark Hoefer, and Dimitrios Mitsotakis, Solitary wave solutions and their interactions for fully nonlinear water waves with surface tension in the generalized Serre equations, Theoretical and Computational Fluid Dynamics 32, 371–397 (2018).
  25. E. Iacocca, T. J. Silva, and M. A. Hoefer, Symmetry-broken dissipative exchange flows in thin-film ferromagnets with in-plane anisotropy, Phys. Rev. B 96, 134434 (2017) . (reprint)
  26. G. El, M. A. Hoefer, and M. Shearer, Stationary expansion shocks for a regularized Boussinesq system, Studies in Applied Mathematics 140, 27–47 (2017). (reprint) (1 of 4 2018 Highlights of the Year)
  27. P. A. P. Janantha, P. Sprenger, M. A. Hoefer, and M. Wu, Observation of self-cavitating envelope dispersive shock waves in Yttrium Iron Garnet thin films, Physical Review Letters 119, 024101 (2017). (reprint)  
  28. M. A. Hoefer, G. A. El, and A. M. Kamchatnov, Oblique spatial dispersive shock waves in nonlinear Schrödinger flows, SIAM Journal on Applied Mathematics 77, 1352-1374 (2017) (reprint)
  29. E. Iacocca and M. A. Hoefer, Vortex-Antivortex proliferation from an obstacle in thin film ferromagnets, Physical Review B 95, 134409 (2017). (reprint) 
  30. G. A. El, M. A. Hoefer, and M. Shearer, Dispersive and diffusive-dispersive shock waves for non-convex conservation laws, SIAM Review 59, 3-61 (2017).  (reprint)
  31. E. Iacocca, T. J. Silva, and M. A. Hoefer, Breaking of Galilean invariance in the hydrodynamic formulation of ferromagnetic thin films, Physical Review Letters 118, 017203 (2017). (reprint, news)
  32. P. Sprenger and M. A. Hoefer, Shock waves in dispersive hydrodynamics with non-convex dispersion, SIAM Journal on Applied Mathematics 77, 26-50 (2017). (reprint, news)
  33. M. D. Maiden and M. A. Hoefer, Modulations of viscous fluid conduit periodic waves, Proceedings of the Royal Society A 472, 20160533 (2016). (reprint, news)
  34. M. A. Hoefer and B. Ilan, Onset of transverse instabilities of confined dark solitons, Physical Review A 94, 013609 (2016). (reprint)
  35. M. D. Maiden, N. K. Lowman, D. V. Anderson, M. E. Schubert, and M. A. Hoefer, Observation of dispersive shock waves, solitons, and their interactions in viscous fluid conduits, Physical Review Letters 116, 174501 (2016). (reprint, movies, news)
  36. S. Chung, A. Eklund, E. Iacocca, S. M. Mohseni, S. R. Sani, L. Bookman, M. A. Hoefer, R. K. Dumas, and J. Åkerman, Magnetic droplet nucleation boundary in orthogonal spin-torque nano-oscillators, Nature Communications 7, 11209 (2016) (open access) (news)
  37. G. Biondini, G. A. El, M. A. Hoefer, and P. D. Miller, Dispersive Hydrodynamics:  Preface, Physica D 333, 1-5 (2016).  Special Issue on Dispersive Hydrodynamics.
  38. G. A. El and M. A. Hoefer, Dispersive shock waves and modulation theory, Physica D 333, 11-65 (2016).  Review article in Special Issue on Dispersive Hydrodynamics. (preprint).
  39. G. A. El, M. A. Hoefer, and M. Shearer, Expansion shock waves in regularized shallow-water theory, Proceedings of the Royal Society A 472, 20160141 (2016). (preprint)
  40. P. Wills, E. Iacocca, and M. A. Hoefer, Deterministic drift instability and stochastic thermal perturbations of magnetic dissipative droplet solitons, Phys. Rev. B 93, 144408 (2016). (reprint)
  41. L. D. Bookman and M. A. Hoefer, Perturbation theory for propagating magnetic droplet solitons, Proceedings of the Royal Society A 471, 2015 (2015). (author postprint)
  42. N. K. Lowman, M. A. Hoefer, and G. A. El, Interactions of large amplitude solitary waves in viscous fluid conduits, Journal of Fluid Mechanics 750, 372-384 (2014). (reprint © Cambridge University Press)
  43. M. D. Maiden, L. D. Bookman, and M. A. Hoefer, Attraction, merger, reflection, and annihilation in magnetic droplet soliton scattering, Physical Review B 89, 180409(R) (2014). (reprint)
  44. S. Chung, S. M. Mohseni, S. R. Sani, E. Iacocca, R. K. Dumas, T. N. Anh Nguyen, Ye. Pogoryelov, P. K. Muduli, A. Eklund, M. A. Hoefer, and J. Åkerman, Spin transfer torque generated magnetic droplet solitons (invited), Journal of Applied Physics 115, 172612 (2014). (reprint)
  45. M. A. Hoefer, Shock waves in dispersive Eulerian fluids, Journal of Nonlinear Science 24, 525-577 (2014). (reprint)
  46. E. Iacocca, R. K. Dumas, L. D. Bookman, M. S. Mohseni, S. Chung, M. A. Hoefer, and J. Åkerman, Confined dissipative droplet solitons in spin-valve nanowires with perpendicular magnetic anisotropy, Physical Review Letters 112, 047201 (2014). (reprint)
  47. L. D. Bookman, and M. A. Hoefer, Analytical theory of modulated magnetic solitons, Physical Review B 88, 184401 (2013). (reprint)
  48. N. K. Lowman and M. A. Hoefer, Dispersive hydrodynamics in viscous fluid conduits, Physical Review E 88, 023016 (2013). (reprint)
  49. N. K. Lowman and M. A. Hoefer, Fermionic shock waves:  distinguishing dissipative versus dispersive regularizations, Physical Review A88, 013605 (2013). (reprint)
  50. S. M. Mohseni, S. R. Sani, J. Persson,  T. N. Anh Nguyen, S. Chung, Ye. Pogoryelov, P. K. Muduli, E.  Iacocca, A. Eklund, R. K. Dumas, S.  Bonetti, A. Deac, M. A. Hoefer, and J. Åkerman, Spin Torque–Generated Magnetic Droplet Solitons, Science 339, 1295-1298 (2013). (reprint)
  51. N. K. Lowman and M. A. Hoefer, Dispersive shock waves in viscously deformable media, Journal of Fluid Mechanics 718, 524-557 (2013). (reprint © Cambridge University Press)
  52. M. A. Hoefer, M. Sommacal, and T. J. Silva, Propagation and control of nanoscale magnetic-droplet solitons, Physical Review B 85, 214433 (2012). (reprint)
  53. D. Yan, J. J. Chang, C. Hamner, M. Hoefer, P. G. Kevrekidis, P. Engels, V. Achilleos, D. J. Frantzeskakis, and J. Cuevas, Beating dark-dark solitons in Bose-Einstein condensates, Journal of Physics B 45, 115301 (2012).
  54. M. A. Hoefer and M. Sommacal, Propagating two-dimensional magnetic droplets, Physica D: Nonlinear Phenomena 241, 890-901 (2012). (preprint)
  55. M. A. Hoefer and B. Ilan, Dark solitons, dispersive shock waves, and transverse instabilities, SIAM Multiscale Modeling and Simulation 10, 306-341 (2012). (reprint)
  56. M. A. Hoefer, C. Hamner, J. J. Chang, and P. Engels, Dark-dark solitons and modulational instability in miscible, two-component Bose-Einstein condensates, Physical Review A 84, 041605(R) (2011). (reprint)
  57. C. Hamner, J. J. Chang, P. Engels, and M. A. Hoefer, Generation of dark-bright soliton trains in superfluid-superfluid counterflow, Physical Review Letters 106, 065302 (2011). (reprint)
  58. M. A. Hoefer and M. I. Weinstein, Defect modes and homogenization of periodic Schrödinger operators, SIAM Journal on Mathematical Analysis 43, 971-996 (2011). (reprint)
  59. A. Tovbis and M. A. Hoefer, Semiclassical dynamics of quasi-one-dimensional attractive Bose-Einstein condensates, Physics Letters A375, 726-632 (2010). (preprint)
  60. M. A. Hoefer, T. J. Silva, and M. W. Keller, Theory for a dissipative droplet soliton excited by a spin torque nanocontact, Physical Review B 82, 054432 (2010). (reprint)
  61. M. A. Hoefer and B. Ilan, Theory of two-dimensional oblique dispersive shock waves in supersonic flow of a superfluid, Physical Review A 80, 061601(R) (2009). (reprint)
  62. M. J. Ablowitz and M. A. Hoefer, Dispersive shock waves, Scholarpedia 4, 5562 (2009).
  63. M. J. Ablowitz, D. E. Baldwin, and M. A. Hoefer, Soliton generation and multiple phases in dispersive shock and rarefaction wave interaction, Physical Review E 80, 016603 (2009). (reprint)
  64. M. A. Hoefer, P. Engels, and J. J. Chang, Matter-wave interference in Bose-Einstein condensates:  a dispersive hydrodynamic approach,Physica D:  Nonlinear Phenomena 238, 1311-1320 (2009). (preprint)
  65. J. J. Chang, P. Engels, and M. A. Hoefer, Formation of dispersive shock waves by merging and splitting Bose-Einstein condensates,Physical Review Letters 101, 170404 (2008). (reprint)
  66. M. A. Hoefer, T. J. Silva, and M. D. Stiles, Model for a collimated spin-wave beam generated by a single-layer spin torque nanocontact,Physical Review B 77, 144401 (2008). (reprint)
  67. M. A. Hoefer, M. J. Ablowitz, and P. Engels, Piston dispersive shock wave problem, Physical Review Letters 100, 084504 (2008). (reprint)
  68. M. A. Hoefer and M. J. Ablowitz, Interactions of dispersive shock waves, Physica D:  Nonlinear Phenomena 236, 44-64 (2007). (preprint)
  69. P. Engels, C. Atherton, and M. A. Hoefer, Observation of Faraday waves in a Bose-Einstein condensate, Physical Review Letters 98, 095301 (2007). (reprint)
  70. M. A. Hoefer, M. J. Ablowitz, I. Coddington, E. A. Cornell, P. Engels, and V. Schweikhard, Dispersive and classical shock waves in Bose-Einstein condensates and gas dynamics, Physical Review A 74, 023623 (2006). (reprint)
  71. M. A. Hoefer, M. J. Ablowitz, B. Ilan, M. R. Pufall, and T. J. Silva, Theory of magnetodynamics induced by spin torque in perpendicularly magnetized thin films, Physical Review Letters 95, 267206 (2005). (reprint)