  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.
