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The Cauchy problem for first-order PDE with the initial data which have apiecewise discontinuities localized in different spatial points is completelysolved. The interactions between discontinuities arising after breakup ofinitial discontinuities are studied with the help of the hodograph method. Thesolution is constructed in analytical implicit form. To recovery the explicitform of solution we propose the transformation of the PDEs into some ODEs onthe level lines (isochrones) of implicit solution. In particular, this methodallows us to solve the Goursat problem with initial data on characteristics.The paper describes a specific problem for zone electrophoresis (method of themixture separation). However, the method proposed allows to solve any system oftwo first-order quasilinear PDEs for which the second order linear PDE, arisingafter the hodograph transformation, has the Riemann-Green function in explicitform.

Interactions between discontinuities for binary mixture separation problem and hodograph method