Kaup–Kupershmidt equation

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

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

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

$\underset{x}{\overset{3}{\partial }}+2u{\partial }_x+u_x,$ .

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

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