Professor Linghai Zhang
Department of Mathematics, Lehigh University, Bethlehem, Pennsylvania
Telephone: 610-758-4116. Fax: 610-758-3767.
EDUCATION
Ph.D: Department of Mathematics, The Ohio State University. 1994 - 1999
RESEARCH INTERESTS
Mathematical Neuroscience
Partial Differential Equations
Explicit Traveling Wave Solutions of Nonlinear Evolution Equations
SELECTED PUBLICATIONS
Linghai Zhang, On stability of traveling wave solutions in synaptically coupled neuronal networks.
Differential and Integral Equations, 16(2003), 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), 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), 489-522.
Linghai Zhang, Dynamics of neuronal waves. Mathematische Zeitschrift, 255(2007), 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), 597-644.
Linghai Zhang, Ping-Shi Wu and Melissa Anne Stoner, Influence of sodium currents on speeds of traveling wave fronts in
synaptically coupled neuronal networks. Physica D, 239(2010), 9-32.
Linghai Zhang, Ping-Shi Wu and Melissa Anne Stoner, Influence of neurobiological mechanisms on
speeds of traveling wave fronts in mathematical neuroscience.
Discrete and Continuous Dynamical Systems, Series B, 16(2011), 1003-1037.
Linghai Zhang, Speeds of traveling waves of some integral differential equations.
Japan Journal of Industrial and Applied Mathematics,
27(2010), 347-373. (with Eiji Yanagida)
Linghai Zhang, New results of a general n-dimensional incompressible Navier-Stokes equations.
Journal of Differential Equations, 245(2008), 3470-3502.
Linghai Zhang, Solutions to Some Open Problems in n-Dimensional Fluid Dynamics. Progress in Nonlinear Analysis Research.
Editor-in-Chief: Erik T. Hoffmann. Nova Science Publishers, INC. 2009. Pages 69-135.
Linghai Zhang, Traveling Waves Arising from Synaptically Coupled Neuronal Networks.
Advances in Mathematics Research, Volume 10.
Editor-in-Chief: Albert R. Baswell. Nova Science Publishers, INC. 2010. Pages 53-204.
SELECTED RESEARCH TOPICS
Mathematical Neuroscience: rigorous mathematical analysis of traveling wave solutions
(traveling wave fronts and traveling pulse solutions) of nonlinear singularly perturbed systems of
integral differential equations arising from synaptically coupled neuronal networks.
I investigate the influence of biological mechanisms on the existence and stability of
the traveling wave solutions and the wave speeds of nonlinear singularly perturbed systems of
integral differential equations arising from synaptically coupled neuronal networks.
I use speed index functions and stability index functions (that is, Evans functions) to
study the existence, uniqueness and stability of the traveling wave solutions.
Partial Differential Equations: long time asymptotic behaviors of global strong solutions of
the Cauchy problems for n-dimensional nonlinear nonhomogeneous dissipative partial differential equations
arising from astronomy, biology, chemistry, fluid dynamics, geography and plasma physics.
In particular, I investigate the exact limits and decay estimates with sharp rates of the global strong solutions.
I have invented a very general method to establish strong and broad results, under some mild conditions on the equations.
Exact Traveling Wave Solutions of Nonlinear Evolution Equations: By using some nonlinear transformations,
by reducing higher order ordinary differential equations to lower order ordinary differential equations,
and by using the method of undetermined coefficients, I can develop some technical lemmas
and then find the bounded explicit traveling wave solutions of many nonlinear evolution equations.
POSITIONS HELD
Associate Professor: Department of Mathematics, Lehigh University.
14 East Packer Avenue, Bethlehem, Pennsylvania 18015 USA. 2008 - present.
Assistant Professor: Department of Mathematics, Lehigh University.
14 East Packer Avenue, Bethlehem, Pennsylvania 18015 USA. 2002 - 2008.
Dunham Jackson Assistant Professor: School of Mathematics, University of Minnesota.
Minneapolis, Minnesota 55455 USA. 1999 - 2002.
Melissa Anne Stoner (with research interest in mathematical neuroscience) received her Ph.D in Spring 2011.
Jie Yuan (with research interests in dynamical systems and mathematical neuroscience) is working on her Ph.D.
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.