Modified Korteweg–De Vries equation

The modified Korteweg–de Vries (KdV) equation is an integrable nonlinear partial differential equation: u t + u x x x + α u 2 u x = 0 {\displaystyle u_{t}+u_{xxx}+\alpha u^{2}u_{x}=0\,} where α {\displaystyle \alpha } is an arbitrary (nonzero) constant. This is a special case of the Gardner equation.