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.

References

• Ashok Das; Ziemowit Popowicz (2005). "A nonlinearly dispersive fifth order integrable equation and its hierarchy". Journal of Nonlinear Mathematical Physics 12 (1): 105–117. doi:10.2991/jnmp.2005.12.1.9. Bibcode: 2005JNMP...12..105D. http://www.sm.luth.se/~norbert/home_journal/electronic/121art5.pdf. 

External links

• Weisstein, Eric W.. "Kupershmidt Equation". http://mathworld.wolfram.com/KupershmidtEquation.html.