Kaup–Kupershmidt equation

The Kaup–Kupershmidt equation (named after David J. Kaup and Boris Abram Kupershmidt) is the nonlinear fifth-order partial differential equation

${\displaystyle u_{t}=u_{xxxxx}+10u_{xxx}u+25u_{xx}u_{x}+20u^{2}u_{x}={\frac {1}{6}}(6u_{xxxx}+60uu_{xx}+45u_{x}^{2}+40u^{3})_{x}.}$

It is the first equation in a hierarchy of integrable equations with the Lax operator

${\displaystyle \partial _{x}^{3}+2u\partial _{x}+u_{x},}$ .

It has properties similar (but not identical) to those of the better-known KdV hierarchy in which the Lax operator has order 2.