Quasilinear Systems of Partial Differential Equations; Hyperbolic Conservation Laws and Nonlinear Waves
Brief Research Profile:
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.
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