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

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

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

References

1. Andrei D. Polyanin, Valentin F. Zaitsev, HANDBOOK of NONLINEAR PARTIAL DIFFERENTIAL EQUATIONS, SECOND EDITION p 1034, CRC PRESS
2. "Stability and instability of solitary waves of the fifth-order KdV equation: a numerical framework" (PDF). Retrieved 8 May 2015.