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^2u_x = \frac16 (6u_{xxxx}+60uu_{xx}+45u_x^2+40u^3)_x.

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

\partial_x^3 + 2u\partial_x + u_x,

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

Kaup Kupershmidt JacobiSN method bell solition animation2

Kaup Kupershmidt JacobiSN method bell soliton animation3

Kaup Kupershmidt JacobiSN method bell soloton animation4

Kaup Kupershmidt csch method animation2

Kaup Kupershmidt csch method animation3

Kaup Kupershmidt csch method animation5

Kaup Kupershmidt sec method animation3

Kaup Kupershmidt sec method animation5

Kaup Kupershmidt sec method animation10

Kaup Kupershmidt sech method animation2

Kaup Kupershmidt sech method animation4

Kaup Kupershmidt sech method animation8

References

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