Korteweg–de Vries–Burgers equation

The Korteweg–de Vries–Burgers equation is a nonlinear partial differential equation:

${\displaystyle u_{t}+\alpha u_{xxx}+uu_{x}-\beta u_{xx}=0.}$

The equation gives a description for nonlinear waves in dispersive-dissipative media by combining the nonlinear and dispersive elements from the KdV equation with the dissipative element from Burgers' equation.^[1]

The modified KdV–Burgers equation can be written as:^[2]

${\displaystyle u_{t}+au_{xxx}+u^{2}u_{x}-bu_{xx}=0.}$