[research interests] [CV] [publications] [Contact info]

Nir Gavish

Postdoc
Department of Mathematics
Michigan State University

I'm a postdoc working with Prof. Keith Promislow and Andrew Christlieb.

Research interests:

My research interests lie in applied mathematics. In particular, I use a combination of tools such as mathematical modeling, asymptotic and spectral analysis, scientific computing, and rigorous analysis, and collaborate with experimentalists in order to solve problems that arise from material science, nonlinear Optics and auction theory.

Complex materials in energy conversation devices:

The need for clean and reliable renewable energy is widely recognized across the world. Polyelectrolyte membrane fuel cells have attracted much interest as one of the most promising nonpolluting power sources capable of producing electrical energy with high thermodynamic efficiencies. A vital component of these fuel cells are selectively conductive polymer electrolyte membranes that enable the charge separation, which is harvested as useful voltage at the drive level. The industry standard membrane for this type of fuel cells is Nafion ®.

Nafion absorbs water, which creates a network of selective water nano-pores in the membrane. Clearly, the structural organization of the water pores in Nafion is of great importance for the performance of the membrane. Nafion morphology was extensively studied during the past 30 years via a host of experimental and molecular-level computational approaches. These approaches, however, have given rise to various contradicting Nafion morphologies models, e.g., cylindrical pores or clustered chain structures.

In my research, we present a novel phase-field model for the self-assembly of pore networks. This model is very different from classical Cahn-Hilliard phase-field models, as it gives rise to a network formation process rather than a coarsening process. Our analysis at the sharp-interface limit gives rise to high-order Ricci-curvature flows, coupled to interfacial dynamics.

We validate our model by simulating a scattering curve for the pore network obtained numerically. Our results show an excellent agreement between numerical and experimental scattering curves of Nafion. Based on this results, we are currently working to establish a continuum morphology model for Nafion.

Our numerical Realization of Nafion morphology
composed of crystalline (green) and water pore (blue) networks.

Auction theory:

Auctions are important monetary tools, central to economic backbone. For example, in 2010 the US treasury auctioned securities in a total sum of 8.4 trillion dollars! Most of the literature on mathematical auction theory assumes that all players behave the same. Although this assumption is restrictive and often unrealistic, the symmetric case has the advantage that the optimal bidding strategy is given by a single ODE and hence relatively easy to analyze.

I consider a class of auctions in which no symmetry between the bidders is assumed, and, moreover, focus on large auction which are most relevant to online auctions. We developed a novel numerical method for computing the equilibrium bidding strategies in such auctions, and studied the nonlinear ODE system for the bidding strategies using a novel dynamical system approach.

Theory of the non linear Schrodinger equation and nonlinear Optics:

My main research topic of the PhD is the theory of singular solutions of the nonlinear Schrodinger equation (NLS) in the context of nonlinear Optics. In nonlinear Optics, the NLS models the propagation of intense laser beams in a medium such as air, water or glass. Such intense laser beams undergo catastrophic self-focusing up to a point of collapse where, mathematically, the solution becomes singular. One of the open questions in this field for more than 40 years is what is the behavior of NLS solutions as they collapse. Since the 80s, the common perception is that all singular solutions of the NLS collapse with a single profile known as the Townes profile. .

In my research I show that this common perception is false, by showing that there are also NLS solutions that do not collapse with the Townes profile, but rather collapse with a ring profile. The nonlinear Optics laboratory in Cornell University, in collaboration with us, later confirmed our prediction experimentally. Our research also included a comprehensive numerical study of NLS solutions as they approach singularity for which we developed a novel numerical method for solving such singular problems.

Scientific publications:

Self assembly of nano-structured materials:

1. Nir Gavish, Gurgen Hayrapetyan, Keith Promislow and Li Yang
Curvature driven flow of bi-layer interfaces,
Physica D, 2011.

Theory of the nonlinear Schrodinger equation:

2. G. Fibich, N. Gavish and X.P. Wang

New singular solutions of the nonlinear Schrodinger equation
Physica D 211: 193-220, 2005

3. G. Fibich, N. Gavish, and X.P. Wang

Singular ring solutions of critical and supercritical nonlinear Schrodinger equations
Physica D 231: 55-86, 2007

paper has been selected as one of the winners of the 2007 SIAM Student Paper Competitio n

4. G. Fibich and N. Gavish

Theory of singular vortex solutions of the nonlinear Schrodinger equation
Physica D 237: 2696-2730, 2008

5. G. Baruch, G. Fibich and N. Gavish

Singular standing-ring solutions of nonlinear partial differential equations

Physica D, 2010

Nonlinear Optics:

6. T. D. Grow, A. A. Ishaaya, L. T. Vuong, A. L. Gaeta, N. Gavish, and G. Fibich

Collapse dynamics of super-Gaussian Beams
Optics Express 14: 5468-5475, 2006

7. G. Fibich and N. Gavish

Critical power of collapsing vortices

Phys. Rev. A 77: 045803, 2008

8. N. Gavish, G. Fibich, L. T. Vuong and A. L. Gaeta

Predicting the filamentation of high-power beams and pulses without numerical integration: A nonlinear Geometrical Optics method
Physical Review A 78: 043807, 2008

9. Samuel Schrauth, Luat Vuong, Alexander Gaeta, Nir Gavish and Gadi Fibich,
Pulse splitting in the anomalous group-velocity-dispersion regime,
Optics Express 19, 2011

Numerical methods:

10. A. Ditkowski, G. Fibich and N. Gavish

Efficient solution of Ax⁽^k) = b^(k) using A^-1
Journal of Scientific Computing 32: 29-44, 2007

11. A. Ditkowski and N. Gavish
A grid redistribution method for singular problems
Journal of Computational Physics 228: 2354-2365, 2009

Auction theory:

12. G. Fibich and N. Gavish.

Asymmetric first-price auctions: A dynamical systems approach,
accepted for publication in Mathematics of research operations, 2011.

13. G. Fibich and N. Gavish

Numerical simulations of asymmetric first-price auctions
Games and Economic Behavior, 2011

Matlab code for computing equilibrium strategies of asymmetric first-price auctions

When using the codes below please cite the following article:

G. Fibich and N. Gavish, Numerical simulations of asymmetric first-price auction, GEB, 2011 (doi:10.1016/j.geb.2011.02.01)

Fixed point iterations code for two players

Newton iterations code for two players

Fixed point iterations code for three players

CV

Here are my CV and list of publications

Contact info

My work email is gavish@msu.edu.
Tel. (517) 353-4486

Office:

Room A334, Wells Hall.

Mailing address:

Nir Gavish
Michigan State University
Department of Mathematics
A334 Wells Hall
East Lansing, MI 48824-1027
USA