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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]] | ||
| | |||
| | :<math>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. </math> | ||
It is the first equation in a hierarchy of [[integrable system|integrable equations]] with [[Lax pair|Lax operator]] | |||
:<math> \partial_x^3 + 2u\partial_x + u_x, </math> . | |||
Ii has properties similar (but not identical) to those of the better-known [[Korteweg–de Vries equation|KdV hierarchy]] in which the Lax operator has order 2. | |||
{{Gallery | |||
|width=250 | |||
|height=200 | |||
|align=center | |||
|File:Kaup Kupershmidt JacobiSN method animation2.gif|Kaup Kupershmidt JacobiSN method bell solition animation2 | |||
|File:Kaup Kupershmidt JacobiSN method animation3.gif|Kaup Kupershmidt JacobiSN method bell soliton animation3 | |||
|File:Kaup Kupershmidt JacobiSN method animation4.gif|Kaup Kupershmidt JacobiSN method bell soloton animation4 | |||
|File:Kaup Kupershmidt csch method animation2.gif|Kaup Kupershmidt csch method animation2 | |||
|File:Kaup Kupershmidt csch method animation3.gif|Kaup Kupershmidt csch method animation3 | |||
|File:Kaup Kupershmidt csch method animation5.gif|Kaup Kupershmidt csch method animation5 | |||
|File:Kaup Kupershmidt sec method animation3.gif|Kaup Kupershmidt sec method animation3 | |||
|File:Kaup Kupershmidt sec method animation5.gif|Kaup Kupershmidt sec method animation5 | |||
|File:Kaup Kupershmidt sec method animation10.gif|Kaup Kupershmidt sec method animation10 | |||
|File:Kaup Kupershmidt sech method animation2.gif|Kaup Kupershmidt sech method animation2 | |||
|File:Kaup Kupershmidt sech method animation4.gif|Kaup Kupershmidt sech method animation4 | |||
|File:Kaup Kupershmidt sech method animation8.gif|Kaup Kupershmidt sech method animation8 | |||
}} | }} | ||
== References == | |||
* {{cite journal | first = Ashok | last = Das |author2=Ziemowit Popowicz | year = 2005 | title = A nonlinearly dispersive fifth order integrable equation and its hierarchy | journal = Journal of Nonlinear Mathematical Physics | volume = 12 | issue = 1 | pages = 105–117 | url = http://www.sm.luth.se/~norbert/home_journal/electronic/121art5.pdf | format = PDF | doi = 10.2991/jnmp.2005.12.1.9 }} | |||
== | == External links == | ||
* {{mathworld|urlname=KupershmidtEquation|title=Kupershmidt Equation}} | |||
{{DEFAULTSORT:Kaup-Kupershmidt equation}} | |||
{{ | [[Category:Partial differential equations]] | ||
{{ | {{mathanalysis-stub}} | ||
Revision as of 15:58, 14 August 2014
The Kaup–Kupershmidt equation (named after David J. Kaup and Boris Abram Kupershmidt) is the nonlinear fifth-order partial differential equation
It is the first equation in a hierarchy of integrable equations with Lax operator
Ii has properties similar (but not identical) to those of the better-known KdV hierarchy in which the Lax operator has order 2.
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