Interactions between discontinuities for binary mixture separation   problem and hodograph method
  The Cauchy problem for first-order PDE with the initial data which have a piecewise discontinuities localized in different spatial points is completely solved. The interactions between discontinuities arising after breakup of initial discontinuities are studied with the help of the hodograph method. The solution is constructed in analytical implicit form. To recovery the explicit form of solution we propose the transformation of the PDEs into some ODEs on the level lines (isochrones) of implicit solution. In particular, this method allows us to solve the Goursat problem with initial data on characteristics. The paper describes a specific problem for zone electrophoresis (method of the mixture separation). However, the method proposed allows to solve any system of two first-order quasilinear PDEs for which the second order linear PDE, arising after the hodograph transformation, has the Riemann-Green function in explicit form. 
M. S. Elaeva, E. V. Shiryaeva, Zhukov M. Yu
math.AP math-ph math.MP math.NA physics.chem-ph