My research interests lie generally in applied nonlinear mathematics and physical applied mathematics. A central theme is the study of nonlinear differential equations via analytical and numerical means. My research can be roughly divided into the following three areas.

Nonlinear waves/Nonlinear optics

My research on nonlinear waves focuses on waves in higher spatial dimensions, including in particular an emerging field of nonlinear optics known as nonlinear topological photonics. My prior work has centered on traveling edge waves in photonic graphene. I have also worked on 2D dispersive shock waves, and waves in 2D granular lattices.

Research highlights:

  • Developed asymptotic theories for traveling edge waves in photonic graphene;
  • Demonstrated the existence of classical 1D NLS solitons in nonlinear photonic graphene;
  • Discovered strong transmission and reflection of edge modes in bounded photonic graphene;

Relevant publications:

  • [DOI] M. J. Ablowitz, A. Demirci, and Y.-P. Ma. Dispersive shock waves in the Kadomtsev-Petviashvili and Two Dimensional Benjamin-Ono equations. Physica D 333, 84-98 (2016).
  • [DOI] C. Chong, P. G. Kevrekidis, M. J. Ablowitz, and Y.-P. Ma. Conical wave propagation and diffraction in two-dimensional hexagonally packed granular lattices. Phys. Rev. E 93, 012909 (2016).
  • [DOI] M. J. Ablowitz and Y.-P. Ma. Strong transmission and reflection of edge modes in bounded photonic graphene. Opt. Lett. 40, 4635-4638 (2015).
  • [DOI] M. J. Ablowitz, C. W. Curtis, and Y.-P. Ma. Adiabatic dynamics of edge waves in photonic graphene. 2D Materials 2 (2), 024003 (2015).
  • [DOI] M. J. Ablowitz, C. W. Curtis, and Y.-P. Ma. Linear and nonlinear traveling edge waves in optical honeycomb lattices. Phys. Rev. A 90, 023813 (2014).

Topologically protected dispersion relation and edge soliton in photonic graphene.

Transmission of linear edge mode in bounded photonic graphene.

Pattern formation/Applied dynamical systems

My research on pattern formation focuses on spatially localized states in dissipative systems, drawing upon dynamical systems techniques such as bifurcation theory and numerical continuation. My prior work has centered on localized states in forced oscillatory systems. I have also worked on localized states in the 1D Brusselator model.

Research highlights:

  • Discovered new bifurcation structures of 1D localized patterns mediated by a central defect;
  • Analyzed new depinning dynamics of 1D localized patterns featuring successive phase slips;
  • Derived new scaling laws for the bifurcation structures of 1D localized patterns;
  • Extended the above to radially symmetric localized patterns in higher dimensions;
  • Discovered new expansion and contraction dynamics of fully 2D supercritical localized hexagonal patterns;

Relevant publications:

  • [DOI] Y.-P. Ma and E. Knobloch. Two-dimensional localized structures in harmonically forced oscillatory systems. Physica D 337, 1-17 (2016).
  • [DOI] J. C. Tzou, Y.-P. Ma, A. Bayliss, B. J. Matkowsky, and V. A. Volpert. Homoclinic snaking near a codimension-two Turing-Hopf bifurcation point in the Brusselator model. Phys. Rev. E 87, 022908 (2013).
  • [DOI] A. R. Champneys, E. Knobloch, Y.-P. Ma and T. Wagenknecht. Homoclinic snakes bounded by a saddle-center periodic orbit. SIAM J. Appl. Dyn. Syst. 11(4), 1583-1613 (2012).
  • [DOI] Y.-P. Ma and E. Knobloch. Depinning, front motion and phase slips. Chaos 22, 033101 (2012).
  • [DOI] Y.-P. Ma, J. Burke and E. Knobloch. Defect-mediated snaking: A new growth mechanism for localized structures. Physica D 239, 1867-1883 (2010).

Bifurcation diagram of defect-mediated snaking and the associated solution profiles.

Competition between expansion and contraction of supercritical planar localized hexagonal patterns.

Climate dynamics/Fluid mechanics

My research on climate dynamics focuses on minimal models of climate components poorly represented in general circulation models. I have also worked on some fluid problems including convection and self-similar flows.

Research highlights:

  • Proposed a new Ising model to explain the geometric characteristics of Arctic melt ponds.

Relevant publications:

  • [PDF] Y.-P. Ma, I. Sudakov, and K. M. Golden. Ising model for melt ponds on Arctic sea ice. Preprint arXiv:1408.2487 (2014).
  • [DOI] P. Weidman and Y.-P. Ma. The competing effects of wall transpiration and stretching on Homann stagnation-point flow. Eur. J. Mech. B-Fluid 60, 237-241 (2016).
  • [DOI] Y.-P. Ma and E. A. Spiegel. A diagrammatic derivation of (convective) pattern equations. Physica D 240, 150-165 (2011).

Isolated and clustered phases in an Ising model for Arctic melt ponds.

`Ever tried. Ever failed. No matter. Try Again. Fail again. Fail better.' --- Samuel Beckett