M. Zafar, Ph.D.

Wave propagation is observed in a large number of physical applications such as electromagnetic signals, flow through rocket nozzles, detonation of explosives, supersonic flight of projectiles, and bottleneck in traffic flows, etc. Problems involving the propagation of  waves   have attracted a lot of attention from both mathematical and physical points of view in different branches of science and engineering. The study of nonlinear waves started  with seminal works of Stokes (1848) and Riemann (1860) and continues to evolve rapidly to present day.  Courant and Friedrichs, in their classical book,  have systematically presented the majority of early research done in this field. After the outbreak of second world war, a wide spread interest arose in mathematical study of wave propagation in gases. A large number of physical phenomena in continuum mechanics are modeled by hyperbolic systems of partial differential equations. 

Nonlinear (or quasilinear) hyperbolic system of partial differential equations, unlike their linear counterparts, come to us in tremendous variety; each with its  own difficulties and require  special treatments for solutions. The familiar laws of superposition, reflection, and refraction, applicable for linear partial differential equations, cease to be valid in case of nonlinear hyperbolic partial differential equations. The main complexity in the nonlinear hyperbolic partial differential  equations is the break down of solutions within  a finite span of  time due to blow up of their derivatives and consequently forbidding the existence of global classical solutions.  A fascinating feature  appears  in nature due to  the breaking of solutions,  i.e., the appearance of shock waves across which the medium undergoes abrupt changes in velocity, pressure and temperature. In contrast to the linear partial differential equations where signals propagate along the characteristics, the characteristics can cross each other precipitating the onset of a shock for nonlinear hyperbolic partial differential equations. An interesting example is the supersonic boom produced by an airplane that breaks the sound barrier. Indeed, a shock is an admissible discontinuity which satisfies Rankine-Hugoniot jump conditions and the entropy condition. 
 
Riemann (1860), in his outstanding work on the mathematical theory of shock waves, introduced the Riemann problem for one-dimensional Euler equations of isentropic planar flow in gasdynamics. The initial value problem for some given set of conservation laws, in which the initial data consist of two adjacent different constant states, is called a Riemann problem. The most familiar example is the shock tube experiment of gasdynamics, where a long thin cylindrical tube has a gas in two different states, separated by a thin membrane. Solutions of Riemann problem  consisting of  rarefaction waves and shock waves (elementary waves) for an isentropic flow were extended to non-isentropic flow by Courant and Friedrichs in 1940, and a new kind of elementary wave, contact discontinuity (or slip line), was introduced . A rarefaction wave is a piecewise smooth solution across which corresponding Riemann invariants are constant and characteristics on left and right diverge while a contact discontinuity is an admissible discontinuity which satisfies Rankine-Hugoniot jump conditions and characteristic speed on each side agrees with the propagation speed of wave.  Indeed, a contact discontinuity is a discontinuity separating gas into two parts without any flow of gas  particles through the discontinuity; a shock is  discontinuity which is crossed by the gas particles. The researchers have unanimous opinion that the Riemann problem is of great significance and plays a fundamental role in all three areas of theory, application and computation due to the following facts:  (1)  it is extensively used in discussing  existence, uniqueness and asymptotic behavior of the solution for the general initial value problem; e.g., the solution of the Riemann problem is used as a foundation stone for  construction of solutions to general initial value problems using the well known random choice method established by Glimm , (2)  the solution of the Riemann problem,   being analytical one, provides an inevitable and difficult test case for any numerical scheme dealing with discontinuous solutions, and (3) the interaction of elementary waves manifests some vital traits of shock waves in the form of nonlinear waves, and is crucial not only in establishing the existence theorem for a solution, but also in furnishing  a comprehensive picture of solutions with rich geometric structure.