Fifth-order Korteweg–De Vries equation

A fifth-order Korteweg–De Vries (KdV) equation is a nonlinear partial differential equation in 1+1 dimensions related to the Korteweg–De Vries equation.^[1] Fifth order KdV equations may be used to model dispersive phenomena such as plasma waves when the third-order contributions are small. The term may refer to equations of the form

${\displaystyle u_{t}+\alpha u_{xxx}+\beta u_{xxxxx}={\frac {\partial }{\partial x}}f(u,u_{x},u_{xx})}$

where ${\displaystyle f}$ is a smooth function and ${\displaystyle \alpha }$ and ${\displaystyle \beta }$ are real with ${\displaystyle \beta \neq 0}$. Unlike the KdV system, it is not integrable. It admits a great variety of soliton solutions.^[2]