Spectral Theory, Analysis of Eigenfunctions, Nodal Domains, Index Theory:

• 12.  “Uniform upper bounds on Courant sharp Neumann eigenvalues of chain domains” (with Thomas Beck and Yaiza Canzani).  Submitted (2023).
• 11.  “Nodal Set Openings on Perturbed Rectangular Domains” (with Thomas Beck and Marichi Gupta).  Accepted to Annales Henri Poincaré (2023).
• 10. “Eigenvalue spacing for 1d Schrödinger operators” (with Luc Hillairet).  Published in Asymptotic Analysis, Vol. 133, No. 1-2 (2023), 267–289.
• 9.  “Spectral minimal partitions, nodal deficiency and the Dirichlet-to-Neumann map:  The generic case” (with Greg Berkolaiko, Yaiza Canzani, and Graham Cox).  Published in Calculus of Variations and PDE, Vol. 61, No. 6 (2022), Paper No. 203.
• 8. “Limiting Eigenfunctions of Sturm-Liouville operators Subject to a Spectral Flow” (with Tom Beck, Isabel Bors, Grace Conte, and Graham Cox). Published in Annales mathématiques du Québec, Vol. 45, No. 2 (2021), 249-269.
• 7. “Nodal line estimates for the second Dirichlet eigenfunction” (with Thomas Beck and Yaiza Canzani).  Published in Journal of Spectral Theory, Vol. 11, No. 1 (2021), 323-353.
• 6. “Nodal deficiency, spectral flow, and the Dirichlet-to-Neumann map” (with Greg Berkolaiko and Graham Cox).  Published in Letters in Mathematical Physics, Vol. 109, No. 7 (2019), 1611–1623.
• 5. “Manifold decompositions and indices of Schrödinger operators” (with Graham Cox and Chris Jones). Published in Indiana University Mathematics Journal, Vol. 66, No. 5 (2017), 1573-1602.
• 4. “A Morse index theorem for elliptic operators on bounded domains” (with Graham Cox and Chris Jones). Published in Communications in Partial Differential Equations, Vol. 40, No. 8 (2015), 1467-1497.
• 3. “Nonconcentration for partially rectangular billiards” (with Luc Hillairet). Published in Analysis & PDE, Vol. 5, Issue 4 (2012), 831-854.
• 2. “An eigenfunction concentration result for polygonal billiards” (with Andrew Hassell and Luc Hillairet). Published in Communications in Partial Differential Equations, Vol. 34, Issue 5 (2009), 475-485.
• 1. “Eigenfunctions For Partially Rectangular Billiards” (written with the help of Maciej Zworski). Published in Communications in Partial Differential Equations, Vol. 31 (2006), 775-790.

  Scattering Theory, Conic Diffraction, Dispersive/Strichartz Estimates & Microlocal Analysis in PDEs:

• 13.  “Newton polygons and resonances of multiple delta-potentials” (with Kiril Datchev and Jared Wunsch).  Accepted to Transactions of the AMS (2023).
• 12.  “Price’s Law for the massless Dirac-Coulomb system” (with Dean Baskin and Jesse Gell-Redman).  Submitted (2022).
• 11.  “Pointwise dispersive estimates for Schrödinger operators on product cones” (with Blake Keeler).  Published in Journal of Differential Equations, Vol. 320 (2022), 419-468.
• 10.  “The radiation field on product cones” (with Dean Baskin).  Published in Advances in Mathematics, Vol. 408 (2022), Paper No. 108589.
• 9. “Locating resonances on hyperbolic cones” (with Dean Baskin). Published in Mathematical Research Letters, Vol. 26, No. 2 (2019), 365-381.
• 8. “L^p-bounds on spectral clusters associated to polygonal domains” (with Matt Blair and G. Austin Ford).  Published in Revista Matematica Iberoamericana, Vol. 34, No. 3 (2018).
• 7. “Strichartz estimates on exterior polygonal domains” (with Dean Baskin and Jared Wunsch). Published in AMS Contemporary Mathematics, Volume 630 – Centre de Recherches Mathematiques Proceedings on Spectral Invariants on Non-compact and Singular Spaces (2014), 291-306.
• 6. “Strichartz estimates for the wave equation on flat cones”  (with Matt Blair and G. Austin Ford).  Published in International Mathematics Research Notices,  Vol. 2013 (2013), 562-591.
• 5.“Wave operator bounds for singular perturbations of the Laplacian in 1d” (with Vincent Duchêne and Michael Weinstein). Published in Journal of Mathematical Physics, Vol. 52, Issue 1 (2011), 013505.
•  4. “Strichartz estimates for the Schrödinger equation on polygonal domains” (with Matt Blair, G. Austin Ford and Sebastian Herr).  Published in Journal of Geometric Analysis, Vol. 22, No. 2 (2012), 339-351.
• 3. “Strichartz estimates on Schwarzschild black hole backgrounds” (with Jason Metcalfe, Daniel Tataru, and Mihai Tohaneanu). Published in Communications in Mathematical Physics, Vol. 293, No. 1 (2010), 37-83.
  
•  2. “Strichartz estimates and local smoothing estimates for asymptotically flat Schrödinger equations” (with Jason Metcalfe and Daniel Tataru). Published in Journal of Functional Analysis, Vol. 255, Issue 6 (2008), 1497-1553.
• 1. “Wave packet parametrices for evolutions governed by PDO’s with rough symbols” (with Jason Metcalfe and Daniel Tataru). Published in Proceedings of the American Mathematical Society, Vol. 136, No. 2 (2008), 597-604.

 

      Fluid Mechanics – Theory & Experiment:

• 9.  “Damping for fractional wave equations and applications to water waves” (with Thomas Alazard and Jian Wang).  Submitted (2023).
• 8.  “Numerical Algorithms for Water Waves with Background Flow over Obstacles and Topography” (with David Ambrose, Roberto Camassa, Rich McLaughlin, Quentin Robinson and Jon Wilkening).  Published in Advances in Computational Mathematics, Vol. 48, no. 4 (2022), Paper No. 46.
• 7.  “Dispersion control on diffusive microchannels” (with Garam Lee, Alan Luner and Dan Harris).  Published in Microfluidics and Nanofluidics, Vol. 25 (2021), 34.
• 6.  “On the stability of traveling wave solutions to thin-film and long-wave models for thin film flows inside a tube” (with Roberto Camassa, H. Reed Ogrosky and Sterling Swygert).  Published in Physica D, Vol. 415 (2021), 132750.
• 5.  “Nonnegative Weak Solutions of Thin Film Equations Related to Viscous Flows in Cylindrical Geometries” (with Sterling Swygert and Roman Taranets).  Published in Journal of Evolution Equations, Vol. 20, No. 4 (2020), 1227–1249.
• 4. “Small data global solutions for the Camassa-Choi equations” (with Ben Harrop-Griffiths). Published in Nonlinearity, Vol. 31, No. 5  (2018), 1868-1904.
• 3. “A numerical study of traveling waves for a model of thin films in cylindrical domains” (with Roberto Camassa, Reed Ogrosky and Nathan Vaughn). Published in Physica D, Vol. 333 (2016), 254-265. The numerical component uses these codes by N. Vaughn implemented in AUTO to track various traveling wave solutions for equations governing the flow of thin films along the interior of a cylinder.
• 2. Appendix to “Euler Equation on a Rotating Surface” (with Michael E. Taylor). Published in Journal of Functional Analysis, Vol. 270 (2016), 3884-3945. The appendix reports findings on stability of striated patterns for fluids constrained to the surface of spheres. The numerical component uses this code solving a number of truncated linear algebraic systems discussed in the paper.
• 1. “On the regularity of the flow map for the gravity-capillary equations” (with Ming Chen, Dan Spirn and J. Doug Wright).  Published in Journal of Functional Analysis, Vol. 264, No. 3 (2013), 752-782.

 

    Lattice Problems, Quantum Graphs, Topological Physics & Optics:

• 9. “QGLAB: A MATLAB Package for Computations on Quantum Graphs” (with Roy Goodman and Grace Conte).  Submitted (2023).
• 8.  “Defect resonances of truncated crystal structures” (with Jianfeng Lu and Alex Watson).  Published in SIAM Journal on Applied Mathematics, Vol. 82, No. 1 (2022), 49-74.
• 7.  “A local test for global extrema in the dispersion relation of a periodic graph” (with Greg Berkolaiko, Graham Cox and Yaiza Canzani).  Published in Pure & Applied Analysis, Vol. 4, No. 2 (2022), 257-286.
• 6.  “Edge-localized states on quantum graphs in the limit of large mass” (with Greg Berkolaiko and Dmitry Pelinovsky).  Published in Annales de l’Institut Henri Poincare: Analyse nonlineaire , Vol. 38, No. 5 (2021), 1295–1335.
• 5.  “Bulk soliton dynamics in bosonic topological insulators,”  (with Miguel Bandres, Braxton Osting and Mikael Rechtsman).  Submitted (2020).
• 4. “Spectral band degeneracies of π/2-rotationally invariant periodic Schrödinger operators” (with Rachael Keller, Braxton Osting and Michael Weinstein).  Published in Multiscale, Modeling and Simulation, Vol. 16, No 4 (2018), 1684-1731.  This is a corrected version where we have made important adjustments to Corollary 4.2 detailing the structure of the dispersion bands in reflection symmetric cases.  We have also made a few other small minor changes.  
• 3. “Ground state for the dumbbell graph” (with Dmitry Pelinovsky).  Published in Applied Mathematics Research eXpress, Vol. 2016 (2016), 98-145. The code used to study the bound states can be found here and requires the code constructing the Laplacian on the graph as well as the publicly available Newton solver nsoli from Tim Kelley. This is an updated version from the published that corrects an error at the very end of the paper. The technical asymptotics for large mass are correct, but the translation of those asymptotics to the energy had a small mistake that was caught by Riccardo Adami and Enrico Serra. The journal is now out of print, so we have corrected this mistake in 6 above.
•  2. “Trapping of bound states by diffracting beams in photorefractive crystals” (with Hanan Sheinfux, Braxton Osting, Mikael Rechtsman and Moti Segev).  Short version accepted to CLEO Proceedings 2013. This reports non-Hamiltonian dynamics at moderate intensity.
• 1. “Bose-Einstein condensation transition studies for atoms confined in Laguerre-Gaussian laser modes” (with T.G. Akin, S. Kennedy, B. Dribus, L. Johnson, J. Alexander and E.R.I. Abraham). This is a physics paper written by Akin and Abraham regarding work I collaborated on as an undergraduate at the University of Oklahoma. Published in Optics Communications, Vol. 285 (2012), 84-89.

 

   Nonlinear dispersive PDE in Mathematical Physics – Analysis, Existence & Stability Theory of Stationary Solutions, Dynamics (applications in Density Functional Theory, Nonlinear Optics, General Relativity,Fluids,…):

• 21.  “The essential spectrum of periodically-stationary solutions of the complex Ginzburg-Landau equation” (with John Zweck, Yuri Latushkin and Christopher K.R.T. Jones).  Published in Journal of Evolution Equations, Vol. 21, No. 3 (2021), 3313-3329.
• 20. “Symmetry Breaking in Density Functional Theory due to Dirac Exchange for a Hydrogen Molecule” (with Mike Holst, Houdong Hu, Jianfeng Lu, Duo Song and John Weare).  Published in Journal of Nonlinear Science, Vol. 32, No. 6 (2022), Paper No. 89.
• 19. “A symplectic perspective on constrained eigenvalue problems” (with Graham Cox).  Published in Journal of Differential Equations, Vol. 266, Issue 6 (2019), 2924-2952.
• 18. “The profile decomposition for the hyperbolic Schrödinger equation” (with Benjamin Dodson, Benoit Pausader, Dan Spirn).  Published in Illinois Journal of Mathematics, Vol. 62, Nos. 1-4 (2018), 293-320. This version correctly points to a similar estimate in work of Rogers-Vargas from 2006. Our proof is different though and there are several applications including the full profile decomposition included, so we hope the community will find the contribution useful.  This is a corrected version thanks to work of Carneiro-Oliveira-Sousa that contains an updated appendix noting that Strichartz Extremizers ARE NOT GAUSSIANS!!  But, they do exist and we show some interesting numerical simulations of what they should look like.  Further work is needed in this direction!
• 17. “Dark Soliton Scattering in Symmetric and Asymmetric Double Potential Barriers” (with Fotini Tsitoura, Zacharias Anastassi, Panos Kevrekidis, and Dmitri Frantzeskakis).  Published in Physics Letters A, Vol. 381, No. 31 (2017), 2514-2520.
• 16. “Nonlinear Bound States in a Schrödinger-Poisson System with External Potential” (with Sarah Raynor and Gideon Simpson). Published in SIAM Journal on Applied Dynamical Systems, Vol. 16, No. 1 (2017), 226-251.
• 15. “Gross-Pitaevskii vortex motion with critically-scaled inhomogeneities” (with Matthias Kurzke and Dan Spirn).  Published in SIAM Journal on Mathematical Analysis, Vol. 49, No. 1 (2017), 471-500.
• 14. “Dark solitons near potential and nonlinearity steps” (with Fotini Tsitoura, Zacharias Anastassi, Panos Kevrekidis, and Dmitri Frantzeskakis). Published in Physical Review A, Vol. 94 (2016), 063612.
• 13. “The long-time existence of self-trapping and Josephson tunneling solutions to the nonlinear Schrödinger Equation” (with Roy Goodman and Michael Weinstein). Published in Discrete and Continuous Dynamical Systems – A, Vol. 35, No. 1 (2015), 225-246.
• 12. “The structure of global attractors for dissipative Zakharov systems with forcing on the torus” (with Burak Erdogan, Katie Newhall and Nikos Tzirakis). Published in SIAM Journal on Applied Dynamical Systems, Vol. 14, No. 4 (2015), 1978-1990. The article reports findings that follow directly from running the forward solving attached code solving a modified Zakharov system using the Kassam-Trefethen exponential time-differencing methods and with initial data generated by that code plugged into AUTO using the attached Fortran 90 code and constants file.
• 11. “Dynamics near the minimal mass soliton for the Korteweg-de Vries equation” (with Sarah Raynor and Gideon Simpson). Published in Dynamical Systems, Vol. 29, No. 2 (2014), 285-299.
•  10. “On instability for the quintic nonlinear Schrödinger equation of some approximate periodic solutions” (with Scipio Cuccagna). Published in Indiana University Mathematics Journal, Vol. 61, Issue 6 (2013), 2053-2083.
• 9. “Spectral analysis for matrix Hamiltonian operators” (with Gideon Simpson). Published in Nonlinearity, Vol. 24 (2011), 389-429.
• 8. “Small data scattering and soliton stability in H^{-1/6} for the quartic KdV equation.” (with Herbert Koch).  Published in Analysis & PDE, Vol. 5, No. 1 (2012), 145-198.
• 7. “Long time dynamics near the symmetry breaking bifurcation for nonlinear Schrödinger/Gross-Pitaevskii equations” (with Michael Weinstein). Published in Discrete and Continuous Dynamical Systems A, Vol. 28, No. 4 (2010), 1505-1554.
• 6. “A system of ODE’s for a perturbation of a minimal mass soliton” (with Sarah Raynor and Gideon Simpson). Published in Journal of Nonlinear Science, Vol. 20, No. 4 (2010), 425-461.
•  5. “A class of stable perturbations for a minimal mass soliton in three dimensional saturated nonlinear Schrödinger equations”. Published in SIAM Journal of Mathematical Analysis, Vol. 42, Issue 3 (2010), 1382-1403.
• 4. “Dispersive estimates using scattering theory for matrix Hamiltonian equations”.  Published in Discrete and Continuous Dynamical Systems – Series A, Vol. 30, No. 4  (2011), 995-1036.
•  3. My UC Berkeley Ph.D. Thesis supervised by Daniel Tataru. Also, here is a copy of my talk regarding the research. There will be some further details added for the resulting publications, but the main ideas are all here. Filed (2007).
•  2. “Soliton Splitting by External Delta Potentials” (with Maciej Zworski and Justin Holmer). This is a numerical companion to the result above. If you are interested in the numerical methods, feel free to e-mail me. Published in Journal of Nonlinear Science, Vol. 17, No. 4 (2007), 349-367.
• 1. “Fast Soliton Scattering by Delta Impurities” (with Maciej Zworski and Justin Holmer). Published in Communications in Mathematical Physics, Vol. 274, No. 1 (2007), 187-216.

 

     Material Science & Stochastic Spin Systems:

• 7.  “Non-local SPDE limits of spatially-correlated-noise driven spin systems derived to sample a canonical distribution” (with Yuan Gao, Jonathan Mattingly and Katie Newhall).  Published in Physical Review E, Vol. 102, No. 5 (2020), 052112.
• 6. “A Proximal-Gradient algorithm for crystal surface evolution” (with Katy Craig, Jian-Guo Liu, Jianfeng Lu and Li Wang).  Published in Numerische Mathematik, Vol. 152, No. 3 (2022), 631-662.
• 5. “Analysis of a fourth order exponential PDE arising from a crystal surface jump process with Metropolis-type transition rates” (with Yuan Gao, Anya Katsevich, Jian-Guo Liu, and Jianfeng Lu).  Published in Pure and Applied Analysis, Vol. 3, No. 4 (2021), 595-612.
• 4.  “Analysis of a continuum theory for broken bond crystal surface models with evaporation and deposition effects” (with Yuan Gao, Jian-Guo Liu and Jianfeng Lu).  Published in Nonlinearity, Vol. 33, No. 8 (2020), 3816-3846.
• 3. “Limiting Behaviors of High Dimensional Stochastic Spin Ensembles” (with Yuan Gao, Kay Kirkpatrick, Jonathan Mattingly and Katherine A. Newhall). Published in Communications in Mathematical Sciences, Vol. 19, No. 2 (2021), pp. 453-494.
• 2. “Asymmetry in crystal facet dynamics of homoepitaxy in a continuum model” (with Jian-Guo Liu, Jianfeng Lu and Dio Margetis).  Published in Physica D, Vol. 393 (2019), 54-67.
• 1. “The relaxation of a general family of broken bond crystal surface models” (with Jon Weare). Published in Physical Review E, Vol. 88 (2013), 032403.

 

    Quasilinear and Degenerate Dispersion Equations, Weak Turbulence Models:

• 8.  “Local well-posedness for a quasilinear Schrödinger equation with degenerate dispersion” (with Ben Harrop-Griffiths). Published in Indiana University Mathematics Journal, Vol. 71, no. 4 (2022), 1585-1626.
• 7.  “Quasilinear Schrödinger equations III: Large Data and Short Time” (with Jason Metcalfe and Daniel Tataru).  Published in Archive for Rational Mechanics and Analysis, 242(2), (2021) 1119-1175.
• 6. “Existence and uniqueness of solutions for a quasilinear Kdv equation with degenerate dispersion” (with Pierre Germain and Ben Harrop-Griffiths). Published in Communications in Pure and Applied Mathematics, Vol. 72 (2019), 2449-2484.
• 5. “Compactons and their variational properties for degenerate KdV and NLS in dimension 1” (with Pierre Germain and Ben Harrop-Griffiths).  Published in Quarterly of Applied Mathematics, Vol. 78, No. 1 (2020), 1-32.
• 4. “On discrete rarefaction waves in an NLS toy model for weak turbulence” (with Sebastian Herr). Published in Indiana University Mathematics Journal, Vol. 65, No. 3 (2016), 753-777.
•  3. “Quasilinear Schrödinger equations II: Small data and cubic nonlinearities” (with Jason Metcalfe and Daniel Tataru).  Published in Kyoto Journal of Mathematics, Vol. 54, No. 3 (2014), 529-546.
• 2. “Behavior of a model dynamical system with applications to weak turbulence” (with Jim Colliander, Tadahiro Oh and Gideon Simpson).  Published in Experimental Mathematics, Vol. 22, Issue 3 (2013), 250-264.
•  1. “Quasilinear Schrödinger equations I: Small data and quadratic interactions” (with Jason Metcalfe and Daniel Tataru). Published in Advances in Mathematics, Vol. 231, Issue 2 (2012), 1151-1172.

 

  Numerical Analysis:

• 4.  “Quantitative bounds on Impedance-to-Impedance operators with applications to fast direct solvers for PDEs” (with Thomas Beck and Yaiza Canzani).  Published in Pure and Applied Analysis, Vol. 4, No. 2 (2022), 225-256.
• 3.  “Quadrature by Two Expansions: Evaluating Laplace Layer Potentials using Complex Polynomial and Plane Wave Expansions”  (with Lingyun Ding, Jingfang Huang, and Zhuochao Tang).   Published in Journal of Computational Physics, Vol. 428 (2021), 109963.
• 2. “Trigonometric integrators for quasilinear wave equations” (with Ludwig Gauckler, Jianfeng Lu, Frederic Rousset and Katharina Schratz). Published in Mathematics of Computation, Vol. 88, No. 316 (2018), 717-749.
• 1. “Strang splitting methods for quasilinear Schrödinger equations” (with Jianfeng Lu).  Published in Communications in Mathematical Sciences, Vol. 13, No. 5 (2015), 1051-1074.

 

  Markov Chains, Spectral Optimization and Graph Partitioning:

• 5.  “An escape time formulation for subgraph detection and partitioning of directed graphs” (with Zach Boyd, Nico Fraiman, Peter Mucha, and Braxton Osting).  Accepted to SIAM Journal of Matrix Analysis and Applications (2023).
• 4.  “A metric on directed graphs and Markov chains based on hitting probabilities” (with Zach Boyd, Nico Fraiman, Peter Mucha, Braxton Osting and Jon Weare).  Published in SIAM Journal on Mathematics of Data Science, Vol. 3, No. 2 (2021), 467-493.
• 3.  “On the behavior of 1-Laplacian Ratio Cuts on nearly rectangular domains” (with Wesley Hamilton and Hau-tieng Wu).  Published in Information and Inference, Vol. 10, No. 4 (2021), 1563-1610.
• 2. “Spectrally optimized pointset configurations” (with Braxton Osting).  Published in Constructive Approximation, Vol. 46, No. 1 (2017), 1-35.
• 1. “An isoperimetric inequality for an integral operator on flat tori” (with Elena Cherkaev and Braxton Osting). Published in Illinois Journal of Mathematics, Vol. 59, No. 3 (2015), 773-793.

Nonlinear PDEs on Geometric Backgrounds:

• 6.  ”Existence and stability of Schrödinger solitons on non-compact manifolds” (with David Borthwick, Roland Donninger and Enno Lenzmann). Published in SIAM J. Math. Anal. 51-5 (2019), pp. 3854-3901.
• 5. “Dispersive estimates for scalar and matrix Schrödinger operators on H^{n+1}” (with David Borthwick). Published in Mathematical Physics, Analysis and Geometry, Vol. 1 (2015), 18:22.
• 4. “Higher dimensional vortex standing waves for nonlinear Schrödinger equations” (with Michael E. Taylor).  Published in Communications in Partial Differential Equations, Vol. 41, No. 3 (2015), 398-446.
• 3. “Nonlinear bound states on weakly homogeneous spaces” (with Hans Christianson, Jason Metcalfe and Michael Taylor).  Published in Communications in Partial Differential Equations, Vol. 39 (2014), 34-97.
• 2. “Nonlinear quasimodes near elliptic periodic geodesics” (with Pierre Albin, Hans Christianson and Laurent Thomann). Published in Physica D: Nonlinear Phenomena, Vol. 241 (2012), 409-425.
• 1. “Existence and stability of solitons for the nonlinear Schrödinger equation on hyperbolic space” (with Hans Christianson).  Published in Nonlinearity, Vol. 23 (2010), 89-106. Original version contained a mistake (corrected in the revised version here) regarding the sub-additivity of energy minimizers and their subsequent orbital stability. The claim should read that for sufficiently large mass, sub-critical problems can be solved via energy minimization and as a result then one can apply the theory of Cazenave-Lions to get orbital stability. The results here are generally corrected and subsumed by the larger work with Metcalfe and Taylor here (#3 above).

Miscellaneous:

• 1. “Counting numerical sets with no small atoms” (with Andy Miller). This article reports findings in Table 1 that follow directly from running code that computes the cardinality of numerical sets with no small atom.  Similarly, in Table 2 data is recorded from running code that computes the cardinality of symmetric numerical sets with no small atom. Both codes are written in Fortran 90.  Published in Journal of Combinatorial Theory A, Vol. 117, Issue 6 (2010), 650-667.