Reduction of dispersionless coupled Korteweg–de Vries equations to the Euler–Darboux equation

A quasilinear hyperbolic system of two first-order equations is introduced. The system is linearized by means of the hodograph transformation combined with Riemann’s method of characteristics. In the process of linearization, the main step is to explicitly express the characteristic velocities in terms of the Riemann invariants. The procedure is shown to be performed by quadrature only for specific combinations of the parameters in the system. We then apply the method developed here to the dispersionless versions of the typical coupled Korteweg–de Vries (cKdV) equations including the Broer–Kaup, Ito, Hirota–Satsuma, and Bogoyavlenskii equations and show that these equations are transformed into the classical Euler–Darboux equation. A more general quasilinear system of equations is also considered with application to the dispersionless cKdV equations for the Jaulent–Miodek and Nutku–Ög̃uz equations.