Differential Equations and Nonlinear Analysis group

Seminar

Our seminar on Differential Equations and Nonlinear Analysis: we usually meet Tuesdays, 4:00 - 5:00pm in UH 2210, see details here
To speak at the seminar, contact Chunhua Shan at chunhua.shan@utoledo.edu or any of us.

Current students

graduate students:

Kursat Yilmaz (advisor: Arsie)
Yagmur Yilmaz (advisor: Shemyakova)
Meetra Nouri (advisor: Ou)
Xueliang Lu (advisor: Ou)
Chanaka Kottegoda (advisors: Arsie and Shan)

undergraduate research projects:

Jacob Imre (Fall 2019 -- Spring 2021) (advisor: Shemyakova)

Past graduate students

2019 Dr. Paduma Samarawickrama-Kuruppuge Eranga,
PhD thesis On the Reducibility of Systems of Quasi-Periodic Linear Functional Differential Equations , (advisor: Arsie),
now a senior lecturer at Department of Mathematics, University of Colombo, Sri Lanka

2015 Dr. Krishna Pokharel, PhD thesis: An Isospectral Flow for Complex Upper Hessenberg Matrices (advisor: Arsie),
now Assistant Professor at Young Harris College.

2014 Dr. Ramesh Karki, PhD thesis: Sobolev gradient semi-flows and applications to nonlinear problems (advisor: Arsie), now Assistant Professor at Indiana University East.

2012 Matthew Osborne, honors thesis (advisor: Arsie),
now a PhD student at Ohio State University.

2012 Dr. Douglas Oliver, M.Sc. thesis: Analysis of a Pseudo-Harmonic Tubular Bell (advisor: Arsie), now at Taylor University, Indiana.

Faculty

• Alessandro Arsie : Integrable systems, Differential Eqs (Ordinary, Partial and Functional), Mathematical Physics, Aspects of functional and real analysis related to DEs.
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  Recent publications
  • (preprint) A. Arsie, A. Buryak, P. Lorenzoni, P. Rossi, Flat F-manifolds, F-CohFTs, and integrable hierarchies
  • (preprint) A. Arsie, A. Buryak, P. Lorenzoni, P. Rossi, Semisimple flat F-manifolds in higher genus
  • R. Aceska, A. Arsie and R. Karki, On near-optimal time samplings for initial data best approximation, Matematiche, Vol. 74, No. 1, 173-190, 2019.
  • A. Arsie and Paduma Samarawickrama-Kuruppuge, On the reducibility of systems of analytic quasi-periodic linear functional differential equations, Journal of Mathematical Analysis and Applications, 2019.
  • D. Oliver and A. Arsie, Effects of Weighting the Ends of a Tubular Bell on Modular Frequencies, Journal of Mathematics and Music, 15 pages, 2019.
  • A. Arsie and P. Lorenzoni, F-manifolds, multi-flat structures and Painlevé transcendents, Asian Journal of Mathematics, Vol. 23, No. 5, 5877-904, 2019.
  • A. Arsie and P. Lorenzoni, Flat F-manifolds, Miura invariants and integrable systems of conservation laws , Journal of Integrable Systems, Volume 3, Issue 1, 2018
  • A. Arsie, P. Lorenzoni, Complex reflection groups, logarithmic connections and bi-flat F-manifolds, Lett.Math.Phys, 43 pages, 2017
  • A. Mansour, D. Offin, A. Arsie, Extensions to Chen's minimizing equal mass parallelogram solutions , 18 pages, Taiwanese J. Math. Volume 21, Number 6, 1437-1453, 2017
  • A. Arsie, P. Lorenzoni and A. Moro, Integrable viscous conservation laws , Nonlinearity, Volume 28, Number 6, 48 pages, 2015
  • A. Arsie, K. Pokharel, A normalizing isospectral flow on complex Hessenberg matrices , Journal of Mathematical Analysis and Applications, Volume 432, Issue 2, 787-805, 2015
  • A. Arsie, P. Lorenzoni, A. Moro, On integrable conservation laws, Proc. R. Soc. A, 2014
  • A. Arsie, P. Lorenzoni, Purely non-local Hamiltonian formalism, Kohno connections and $\vee$-systems, J. Math. Phys. 55, 113510, 2014
  • A. Arsie and Ch. Ebenbauer, A Hessenberg-Jacobi isospectral flow , Nonlinear Differential Equations and Applications (NoDEA), Birkaeuser, vol. 21, 2 , 2014

• Jeongoo Chen: Lie symmetries and differential invariants arising in problems of mathematical physics and differential geometry
  Recent publications
  • J. Cheh, On local congruence of immersions in homogeneous or nonhomogeneous spaces SIGMA (Symmetry, Integrability, and Geometry: Methods and Applications) 9, 2013
  • J. Cheh, P.J. Olver, J. Pohjanpelto, Algorithms for Differential Invariants of Symmetry Groups of Differential Equations, Foundations of Computational Mathematics, 2008
  • J. Cheh, P.J. Olver, and J. Pohjanpelto, Maurer–Cartan equations for Lie symmetry pseudogroups of differential equations, Journal of Mathematical Physics, 46, 2005

• Biao Ou: Partial differential equations, applied analysis
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  Recent publications
  • W. Chen, C. Li, B. Ou, Positive regular solutions to a singular integral equation, Int. J. Pure Appl. Math. 52, 4, 2009
  • W. Chen, C. Li, B. Ou, Classification of solutions for a system of integral equations, Communications in Partial Differential Equations, 2005

• Chunhua Shan : Differential equations, dynamical systems, and mathematical biology
  Recent publications
  • C. Shan, G. Fan and H. Zhu, Periodic Phenomena and Driving Mechanisms in Transmission of West Nile Virus with Maturation Time, J. Dynam. Differential Equations, 2019.
  • C. Shan and Q. Huang, Direct and Indirect Effects of Contaminants on Two Competing Species in a Polluted Aquatic Environment, J. Math. Biol. 78 (2019), 739-766.
  • X. Zhang, C. Shan, Z. Jin and H. Zhu, Complex Dynamics of Epidemic Models on Adaptive Networks, J. Differential Equations 266 (2019), 803-832.
  • M. Li, W. Liu, C. Shan, Y. Yi, Turning points and relaxation oscillation cycles in epidemic models, SIAM J. Appl. Math., 2016
  • C. Shan, Y. Yi, H. Zhu, Nilpotent singularities and dynamics in an SIR type of compartmental model with hospital resources, J. Differential Equations (2016)
  • C. Rousseau, C. Shan and H. Zhu, Finite cyclicity of some graphics through a nilpotent point of saddle type inside quadratic systems, Qual. Theor. Dyn. Syst. 15, 237-256, 2016

• Ekaterina Shemyakova : Differential Geometry, Mathematical Physics, Computational Mathematics. Algebraic theory of differential operators. Supergeometry.
  Recent publications
  • E. Shemyakova, On a Batalin-Vilkovisky operator generating higher Koszul brackets on differential forms , conditionally accepted to Lett.Math.Phys.
  • (preprint) E. Shemyakova, Th. Voronov, On super Plücker embedding and possible application to cluster algebras .
  • D. Hobby, E. Shemyakova, Laplace invariants for differential operators, accepted to Illinois J.Math., 2020.
  • E. Shemyakova, Th. Voronov, On differential operators over a map, thick morphisms of supermanifolds, and symplectic micromorphisms, Differential Geometry and its Applications, accepted, 2020.
  • E. Shemyakova, Factorization of Darboux Transformations of Arbitrary Order for Two-dimensional Schroedinger operator, Illinois J. of Math., 2019
  • E. Shemyakova, Th. Voronov, Differential operators on the algebra of densities and factorization of the generalized Sturm-Liouville operator , Lett.Math.Phys, 2019
  • S. Li (undergraduate student), E. Shemyakova, Th. Voronov, Differential operators on the superline, Berezinians, and Darboux transformations, Lett.Math.Phys, 2017
    
  • D. Hobby, E. Shemyakova, Classification of Multidimensional Darboux Transformations: First Order and Continued Type, Symmetry, Integrability and Geometry: Methods and Applications (SIGMA), 2017
  • E. Shemyakova, An algorithm for construction of orbits of Darboux transformations of type I for third order hyperbolic operators of two variables, Programming and Computer Software, issue devoted to Computer Algebra, 2016
  • E. Shemyakova, Invertible Darboux Transformations of Type I , Programming and Computer Software, issue devoted to Computer Algebra, 2015
  • S. Hill (undergraduate student), E. Shemyakova, Th. Voronov, Darboux transformations for differential operators on the superline , Russian Mathematical Surveys, (6), pp. 207-208, 70, 2015

Events that we organize

March, 2022 FIELDS workshop "Supergeometry and bracket structures in mathematics and physics", the Fields Institute, Canada. Organizers: Ekaterina Shemyakova, Alexander Odesski (Brock, Canada), Theodore Voronov (Manchester, UK).

June 5 - June 9, 2020 Special session on "Applications of dynamical systems in mathematical biology'', The 13th AIMS Conference on Dynamical Systems, Differential Equations and Applications, Atlanta, GA, USA. Organizers: Guihing Fan (Columbus State University), and Chunhua Shan.

September 14-15, 2019 Special Session " Supergeometry, Poisson Brackets, and Homotopy Structures" at the AMS meeting at U. of Wisconsin-Madison, Madison, WI. Organizers: Ekaterina Shemyakova, Theodore Voronov (Manchester, UK).

July 21-25, 2019 Special Session on "Recent Advances on Vector-borne Diseases'', Annual Society on Mathematical Biology Meeting, University of Montreal, Montreal, Canada. Organizers: Guihing Fan (Columbus State University), Rongsong Liu (University of Wyoming), and Chunhua Shan.

October 20-21, 2018 Special session "Advances on Analytical and Geometric Aspects of Differential Equations" at the AMS meeting at the University of Michigan, Ann Arbor. Organizers: Alessandro Arsie, Chunhua Shan, Ekaterina Shemyakova.

April 21-22, 2018 Special session "Dynamical systems, Geometric Structures and Special Functions, I" at the AMS meeting at Northeastern University, Boston, MA. Organizers: Alessandro Arsie and Oksana Bihun (University of Colorado, Colorado Springs)

January 10-13, 2018 Special Session "Dynamical Systems with Applications to Mathematical Biology" at Joint Mathematics Meetings, San Diego, CA. Organizers: Guihong Fan (Columbus State University), Jing Li (California State University Northridge) and Chunhua Shan.

Funding

2017 - 2022 Chunhua Shan (PI) Bifurcation Theory and Applications in Mathematical Epidemiology, Simons Foundation, Collaboration Grants for Mathematicians. $42,000.

2017 - 2020 Ekaterina Shemyakova (PI) (amended to David Hobby as PI, and Shemyakova as co-PI, because of the change of the affiliation by Shemyakova) RUI: Algebraic, Differential-Geometric, and Computational Aspects of Darboux Transformations in Classical and Super Settings, NSF, DMS, Geometric Analysis program. $118,374.00

summer 2017 Alessandro Arsie. INDAM Stipend grants (sponsored by the GNFM - Italian National Group for Mathematical Physics) for collaborative work with Prof. P. Lorenzoni. INDAM is the National Institute for Higher Mathematics in Rome.

summer 2015 Alessandro Arsie. INDAM Stipend grants (sponsored by the GNFM - Italian National Group for Mathematical Physics) for collaborative work with Prof. P. Lorenzoni. INDAM is the National Institute for Higher Mathematics in Rome.

2015-2016 Alessandro Arsie (co-PI), Judy Lambert (PI), Doug Oliver, Sekhar Pindiprolu, Tod Shockey, William B. Weber, Jr. (co-PIs). Ohio Board of Regents, Improving Teaching Quality State Grants Program, New Learning Standards in Math and Science through a Technology Lens in NW Ohio.

2011-2012 Alessandro Arsie (PI), Tod Shockey (co-PI), Willam B. Weber (co-PI), Mark Templin (co-PI). The Woodward High School Mathematics and Science Partnership Project, Phase III.

2010-2011 Alessandro Arsie (PI), Tod Shockey (co-PI), Willam B. Weber (co-PI), Mark Templin (co-PI). The Woodward High School Mathematics and Science Partnership Project, Phase II.

2010-2011 Alessandro Arsie (PI), Dynamical Systems of Lax type, Integrability and Applications, Summer URAF, internal grant.

2010-2011 Alessandro Arsie (PI), Integrable PDEs of hydrodynamic type and bi-Hamiltonian deformations, Visiting scholar grant, internal grant to invite at UT Prof. Paolo Lorenzoni.