Linghai Zhang
Associate Professor of Mathematics
Department of Mathematics, Lehigh University, Bethlehem, Pennsylvania 18015
Office Telephone: 610-758-4116. Fax: 610-758-3767. Email: liz5@lehigh.edu
EDUCATION
- BA: Department of Mathematics, Beijing Normal University. 1982-1986
- MA: Division of Applied Mathematics, Beijing Institute of Applied Physics and Computational Mathematics. The People's Republic of China. 1986-1989
- Ph.D: Department of Mathematics, The Ohio State University. 1994-1999
AWARDS
- Franz Summer Research Award: Lehigh University. 2003
- Research Grant from Howard Hughs Medical Institute (Biosystems Dynamics Summer Institute at Lehigh University):
(with Professor R. Michael Burger, Department of Biological Sciences, Lehigh University, and Professor Ping-Shi Wu, Department of Mathematics, Lehigh University).
Modeling local synaptic control in neural coincidence detectors. 2007-2008
- Research Grant from Reidler Research Foundation. Modeling propagation of nerve impulses arising from synaptically coupled neuronal networks. 2003-2005
- Faculty Research Grants from Lehigh University. 2005 and 2008
RESEARCH INTERESTS
- Synaptically Coupled Neuronal Networks, Mathematical Biology
- Differential Equations and Dynamical Systems
- Decay Estimates and Exact Limits with Sharp Rates of Global Solutions of Dissipative Partial Differential Equations
SELECTED PUBLICATIONS
- Linghai Zhang,
On stability of traveling wave solutions in synaptically coupled neuronal networks. Differential and Integral Equations, 16(2003), number 5, 513-536.
- Linghai Zhang,
Existence, uniqueness and exponential stability of traveling wave solutions of some integral differential equations arising from neuronal networks.
Journal of Differential Equations, 197(2004), number 1, 162-196.
- Linghai Zhang,
Traveling waves of a singularly perturbed system of integral-differential equations arising from neuronal networks.
Journal of Dynamics and Differential Equations, 17(2005), number 3, 489-522.
- Linghai Zhang, Dynamics of neuronal waves.
Mathematische Zeitschrift, 255(2007), number 2, 283-321.
- Linghai Zhang,
How do synaptic coupling and spatial temporal delay influence traveling waves in nonlinear nonlocal neuronal networks?
SIAM Journal on Applied Dynamical Systems, 6(2007), number 3, 597-644.
- Linghai Zhang, New Principles of Evans Functions.
- Linghai Zhang, Dynamics of nonlinear traveling waves in synaptically coupled neuronal networks.
- Linghai Zhang, Speeds of traveling waves of some integral differential equations.
- Linghai Zhang, Speeds of traveling waves arising from synaptically coupled neuronal networks.
- Linghai Zhang, Traveling wave of an integral differential equation with two kinds of delays.
- Linghai Zhang, Solutions to some open problems in n-dimensional fluid dynamics. Published in
``Progress in Nonlinear Analysis Research''. 2008. Pages 73-139.
- Linghai Zhang, The modified Fourier splitting method and exact limits in n-dimensional fluid dynamics.
- Linghai Zhang, A one-dimensional dissipative evolution equation arising from fluid dynamics.
- Linghai Zhang, Exact limits of solutions of a partial differential equation arising from n-dimensional fluid dynamics.
- Linghai Zhang, New results of a general n-dimensional incompressible Navier-Stokes equations. Journal of Differential Equations, (2008).
Here are several lectures given at research institutions or at international conferences in applied mathematics.
Integral differential equations:
Talk-One, Talk-Two, Talk-Three, Talk-Four, Talk-Five,
Partial differential equations:
Talk-Six, Talk-Seven, Talk-Eight, Talk-Nine, Talk-Ten.
Here are the latest version of my
Curriculum Vitae,
Research Statement,
Teaching Statement
and
Service Statement.
For a complete list of my research publications, please go to the website of the American Mathematical Society (AMS):
www.ams.org/mathscinet
SELECTED RESEARCH TOPICS
- Mathematical neuroscience: traveling waves modeled with nonlinear singularly perturbed systems of integral differential equations.
I investigate the influence of biological mechanisms on existence and stability of traveling wave solutions and wave speeds of model equations arising from
synaptically coupled neuronal networks. I have introduced two very important concepts: speed index functions and stability index functions, to study the existence, uniqueness and stability of traveling waves in mathematical neuroscience.
- Global solutions of Cauchy problems for nonlinear dissipative partial differential equations. In particular, I investigate decay estimates and exact limits with sharp rates of the global solutions. I invented the modified Fourier splitting method to establish exact limits and decay estimates with sharp rates of global solutions to the Cauchy problems for many dissipative partial differential equations.
My research activities involve the following model equations:
- scalar integral differential equations with spatial temporal delay
- nonlinear singularly perturbed systems of integral differential equations
- nonlinear Benjamin-Ono-Korteweg-de Vries-Burgers equations
- nonlinear Korteweg-de Vries-Burgers equations
- nonlinear Benjamin-Bona-Mahony-Burgers equations
- n-dimensional incompressible Navier-Stokes equations
- n-dimensional incompressible Magnetohydrodynamics equations
POSITIONS HELD
- Associate Professor: Department of Mathematics, Lehigh University. 14 East Packer Avenue, Bethlehem, Pennsylvania 18015 USA. August 2008-present.
- Visiting Assistant Professor: Mathematical Biosciences Institute, The Ohio State University. 231 West 18th Avenue, Columbus, Ohio 43210 USA.
September-December, 2006
- Assistant Professor: Department of Mathematics, Lehigh University. 14 East Packer Avenue, Bethlehem, Pennsylvania 18015 USA. August 2002-August 2008
- Dunham Jackson Assistant Professor: School of Mathematics, University of Minnesota. 206 Church Street Southeast, Minneapolis, Minnesota 55455 USA. 1999-2002
- Graduate Teaching Associate: Department of Mathematics, The Ohio State University. 231 West 18th Avenue, Columbus, Ohio 43210 USA. 1994-1999
- Research Assistant: Division of Applied Mathematics, Beijing Institute of Applied Physics and Computational Mathematics.
P. O. Box: 8009, Beijing 100088 People's Republic of China. 1989-1994
TEACHING
To return to the Department of Mathematics, please click
here
To return to Lehigh University, please click
here
SOME OTHER WEBSITES:
- Department of Applied Mathematics, The University of Washington
- Department of Mathematics, The University of Texas at Austin
- School of Mathematics, Georgia Institute of Technology
- Department of Mathematics, The University of Wisconsin
- Department of Mathematics, The Ohio State University
- Department of Mathematics, Indiana University
- Department of Mathematics, The University of Illinois at Urbana Champaign
- School of Mathematics, The University of Minnesota
- Institute for Mathematics and its Applications, The University of Minnesota
- Department of Mathematics, The University of Pittsburgh
- Center for BioDynamics, Boston University
- Department of Mathematics, Yale University
- Department of Mathematics, Harvard University
- Department of Mathematics, The University of Utah
- Department of Mathematics, The University of Nottingham
- Department of Mathematics, Cornell University
- Department of Mathematics, Rutgers University
- Department of Mathematics, The Pennsylvania State University
- Mathematical Institute, Tohoku University, Japan
- Department of Biological Sciences, Lehigh University
- Institute of Information and Mathematical Sciences, Massey University, New Zealand
- Department of Mathematics, The University of California in Los Angeles
- Department of Mathematics, The University of Michigan
- Department of Mathematics, Princeton University
- Courant Institute of Mathematical Sciences, Ney York University
- Department of Mathematics, Stanford University
- Department of Mathematics, The University of Chicago
- Department of Mathematics, Carnegie Mellon University
- Department of Mathematics, The University of Maryland
- Department of Mathematics, Massachusetts Institute of Technology
- Department of Mathematics, Peking University, China
- Department of Mathematics, Tsinghua University, China
-
Archive for Rational Mechanics and Analysis
- Communications in Partial Differential Equations
- Journal of Differential Equations
- Journal of Functional Analysis
- Physica D
- SIAM Journal on Applied Dynamical Systems
- SIAM Journal on Applied Mathematics
- Transactions of the American Mathematical Society
- American Mathematical Society meeting. San Diego, California. January 2002.
- Workshop on Nonlocal Integral Differential Equations in Mathematics and Biology.
Mathematical Biosciences Institute, The Ohio State University. March 6-8, 2003.
- Conference on Mathematical Neuroscience
(A Satellite Activity of the International Congress of Mathematicians 2006 - NEUROMATH 2006)
(an activity of the project: Shaping new directions in Mathematics for Science and Society)
at Sant Julia de Loria, Principat d'Andorra. September 1-4, 2006.
-
Mini-symposium on Nonlinear Dynamics. Department of Mathematics and Statistics, York University. Canada. August 3-4, 2005.
- International Conference on Applied Mathematics and Interdisciplinary Research.
Nankai Institute of Mathematics, Nankai University. Tianjin, The People's Republic of China. June 12-15, 2006.
- International Conference: Nonlinear Waves - Theory and Applications.
Tsinghua University, Beijing, The People's Republic of China. June 9-12, 2008.
- First Joint International Meeting of the American Mathematical Society and
Shanghai Mathematical Society. Shanghai, Peoples Republic of China. December 17-21, 2008. Meeting Number 1045.
- Summer Workshop on Dynamical Systems and Nonlinear Studies.
Zhejiang Normal University, Jinhua, Zhejiang, People's Republic of China. May 25 - June 5, 2008.
Psalm 23
The Lord is my shepherd, I shall not be in want.
He makes me lie down in green pastures,
he leads me beside quiet waters, he restores my soul.
He guides me in paths of righteousness for his name's sake.
Even though I walk through the valley of the shadow of death,
I will fear no evil, for you are with me;
your rod and your staff, they comfort me.
You prepare a table before me in the presence of my enemies.
You anoint my head with oil; my cup overflows.
Surely goodness and love will follow me all the days of my life,
and I will dwell in the house of the Lord forever.
Last modified on Wednesday, October 1, 2008.
