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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]]
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| :<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>
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| It is the first equation in a hierarchy of [[integrable system|integrable equations]] with [[Lax pair|Lax operator]]
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| :<math> \partial_x^3 + 2u\partial_x + u_x, </math> . | |
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| 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.
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| {{Gallery
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| |File:Kaup Kupershmidt JacobiSN method animation2.gif|Kaup Kupershmidt JacobiSN method bell solition animation2
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| |File:Kaup Kupershmidt JacobiSN method animation3.gif|Kaup Kupershmidt JacobiSN method bell soliton animation3
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| |File:Kaup Kupershmidt JacobiSN method animation4.gif|Kaup Kupershmidt JacobiSN method bell soloton animation4
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| |File:Kaup Kupershmidt csch method animation2.gif|Kaup Kupershmidt csch method animation2
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| |File:Kaup Kupershmidt csch method animation3.gif|Kaup Kupershmidt csch method animation3
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| |File:Kaup Kupershmidt csch method animation5.gif|Kaup Kupershmidt csch method animation5
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| |File:Kaup Kupershmidt sec method animation3.gif|Kaup Kupershmidt sec method animation3
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| |File:Kaup Kupershmidt sec method animation5.gif|Kaup Kupershmidt sec method animation5
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| |File:Kaup Kupershmidt sec method animation10.gif|Kaup Kupershmidt sec method animation10
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| |File:Kaup Kupershmidt sech method animation2.gif|Kaup Kupershmidt sech method animation2
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| |File:Kaup Kupershmidt sech method animation4.gif|Kaup Kupershmidt sech method animation4
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| |File:Kaup Kupershmidt sech method animation8.gif|Kaup Kupershmidt sech method animation8
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| == References == | |
| * {{cite journal | first = Ashok | last = Das | coauthors = and 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 }}
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| == External links ==
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| * {{mathworld|urlname=KupershmidtEquation|title=Kupershmidt Equation}}
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| {{DEFAULTSORT:Kaup-Kupershmidt equation}}
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| [[Category:Partial differential equations]]
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| {{mathanalysis-stub}}
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