Fifth-order Korteweg–De Vries equation explained

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+\alphau_xxx+\betau_xxxxx=

	\partial
	\partialx

f(u,u_x,u_xx)

where

is a smooth function and

\alpha

and

\beta

are real with

\beta ≠ 0

. Unlike the KdV system, it is not integrable. It admits a great variety of soliton solutions.^[2]

Notes and References

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