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 x x x x x + 10 u x x x u + 25 u x x u x + 20 u 2 u x = 1 6 ( 6 u x x x x + 60 u u x x + 45 u x 2 + 40 u 3 ) x . {\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

∂ x 3 + 2 u ∂ x + u x , {\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. S2CID 17610218.

External links

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


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