Hirota–Satsuma equation

The Hirota–Satsuma equation is a set of three coupled nonlinear partial differential equations:
 * $$u_t-\frac{1}{2}u_{xxx}+3uu_x-3(vw)_x=0$$
 * $$v_t+v_{xxx}-3uv_x=0$$
 * $$w_t+w_{xxx}-3uw_x=0$$

The Hirota–Satsuma equation appeared in the theory of shallow water waves, first discussed by Hirota, Ryogo and Satsuma, Junkichi in 1976. The equation has multiple soliton solutions and traveling wave solutions.