Kaup–Kupershmidt equation

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

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

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

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

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.

External links

• Weisstein, Eric W. "Kupershmidt Equation". MathWorld.