Hirota–Satsuma equation

The Hirota–Satsuma equation is a mathematical model of interactions of two long waves with different dispersion relations, expressed as a set of three coupled KdV equations:^[1]

${\displaystyle {\begin{cases}u_{t}-{\frac {1}{2}}u_{xxx}+3uu_{x}-3(vw)_{x}=0,\\v_{t}+v_{xxx}-3uv_{x}=0,\\w_{t}+w_{xxx}-3uw_{x}=0.\end{cases}}}$

The Hirota–Satsuma equation appeared in the theory of shallow water waves, first discussed by Hirota, Ryogo and Satsuma, Junkichi in 1976.^[2] The equation has multiple soliton solutions and traveling wave solutions.