University of Qom

Title: Legacy of Hesaaraki approach to structure of MHD and plasma shock waves

Abstract: Shock waves occur in many physical systems, such as plasma and MHD. Mathematically, shock waves are discontinuous weak solutions of conservation laws PDE’s. In order to distinguish physical shock wave solutions, one has to apply structure or viscous profile criterion, which reduces to the existence of heteroclinic orbits of singularly perturbed shock layer ODE’s. This latter problem is considered a qualitative analysis of shock layer ODE’s. The vitality and importance of a valid and rigour analysis on the¬†¬†problem of the existence of structure for MHD and plasma shock waves has been explicitly and repeatedly declared in the research papers and books in the field of MHD, plasma and even tokamaks dating back to fifties, with only scratching results, before eighties. Prof. M. Hesaaraki, in his innovative researches, applied topological and algebraic methods such as Conley Index, connection matrix, transition matrix and analytical methods in structure of shock waves in a wide variety of plasma and MHD physical configurations. Further, he has devised an unprecedented analytic method to study the behavior of structures using vanishing viscosity tests. In this paper we will review prof Hesaaraki’s research legacy in the field of structure of plasma and MHD shock waves. Further we present a comparative study of the related results in the field to contrast his works amongst the other’s.

Keywords: differential equations, structure of shock waves, viscous profile, conley index, shock waves, plasma, magnetohydrodynamics, MHD.

AMS 2010 Subject Classifications: 76L05, 76W05, 76X05, 34A12.