Mark Edelman’s Statement of Research Interests

In 1982 at the department of Astrophysics of the Rostov University, while working on the evolution of gas clouds with primeval chemical composition and star formation, I read an article by Roger Chevalier and James Imamura on the linear stability of radiative shock waves. The article was on the conditions of the oscillatory instability. My immediate reaction was that the conditions for the corrugation instability must be weaker and the reason is simple – flows behind the trailing elements of the front will increase the density of gas behind those elements and as result the cooling will increase and the pressure will drop. In 1985 I published in Russian an article on corrugation instability of radiative shock waves. Most of the results were independently obtained by Edmund Bertschinger

from MIT and published in “The Astrophysical Journal” in 1986. Bertschinger’s article was cited almost one hundred times. My publication gave me an opportunity to communicate with astrophysicists from Pulkovo observatory and Ioffe Physico-Technical Institute.

The next natural step was to consider the corrugation instability of magnetized radiative and adiabatic shocks, as the astrophysical plasma, especially for the case of the accretion onto magnetized stars, is always magnetized. As the result, the whole linear theory of corrugation instability of radiative and magnetized gases was created. I ended my work on this theory in 1995 with the work done in collaboration with James Stone from Princeton University on nonlinear instability of adiabatic magnetized shock waves.

In 1995 George Zaslavsky offered me to work on the computer simulations related to chaos theory at the Courant Institute. I accepted this offer and since 1995 the chaos theory has been the subject of my research. Particular areas of my interest include research on the connection between topological properties of phase space of dynamical systems and transport coefficients, pseudochaos (with zero Lyapunov exponent), description of dynamical systems by fractional differential equations and numerical solution of such equations. Design of corresponding software (visual C++) and web design are just hobbies. An application Phase Portrait Builder can be sent upon request and I can issue a password for my educational site: http://cims.nyu.edu/~edelman/MyCourse/Precalculus0/Precalculus/

In their pioneering work Nick Laskin and George Zaslavsky (Physica A, 2006) showed that systems of long range interacting oscillators in the infrared limit (small k) can be adequately described by fractional partial differential equations. This work stimulated activities in our group related to generalization of the properties of well known PDEs like sine-Gordon equation, non-linear Shrödinger equation, Ginzburg-Landau equation etc. for the case of fractional space derivatives. This is also the area of my main interest at the present time.