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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.

References
Ashok Das; Ziemowit Popowicz (2005). "A nonlinearly dispersive fifth order integrable equation and its hierarchy" (PDF). Journal of Nonlinear Mathematical Physics. 12 (1): 105–117. arXiv:nlin/0404049. Bibcode:2005JNMP...12..105D. doi:10.2991/jnmp.2005.12.1.9.