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.
RESEARCH INTERESTS
Mathematical Neuroscience: By using rigorous mathematical analysis,
I accomplish the existence and stability of traveling wave solutions (traveling wave fronts and traveling pulse solutions)
of nonlinear singularly perturbed systems of integral differential equations arising from delayed synaptically coupled neuronal networks.
I use speed index functions and stability index functions (Evans functions) to establish the existence and stability of the traveling wave solutions.
Moreover, I investigate the influence of biological mechanisms on the wave speeds of the traveling wave solutions.
Partial Differential Equations: I have invented a very general method to establish exact limits and decay estimates with sharp rates
of global strong solutions of n-dimensional nonlinear systems of fluid dynamical equations. The results on the long time asymptotic behaviors of
the global strong solutions of the nonlinear systems of fluid dynamical equations are strong and broad.
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.
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, Wave speed analysis of traveling wave fronts in delayed synaptically coupled neuronal networks.
Discrete and Continuous Dynamical Systems, Series A, 34(2014).
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.
Mathematics 435: Introduction to Functional Analysis Math435.pdf
PH.D STUDENTS
Melissa Anne Stoner received her Ph.D (with research interest in mathematical neuroscience) in Spring 2011.
Jie Yuan is working on her Ph.D (with research interests in dynamical systems and mathematical neuroscience).
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.