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. 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
u t + α u x x x + β u x x x x x = ∂ ∂ x f ( u , u x , u x x ) {\displaystyle u_{t}+\alpha u_{xxx}+\beta u_{xxxxx}={\frac {\partial }{\partial x}}f(u,u_{x},u_{xx})}
where f {\displaystyle f} is a smooth function and α {\displaystyle \alpha } and β {\displaystyle \beta } are real with β ≠ 0 {\displaystyle \beta \neq 0} . Unlike the KdV system, it is not integrable. It admits a great variety of soliton solutions.
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