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{{equation
 | name = Quintic gKdV
 | equation = <math>u_t + u_{xxx} = \pm u^4 u_x</math>
 | fields = <math>u: \R \times \R \to \R</math>
 | data = <math>u(0) \in H^s(\R)</math>
 | hamiltonian = [[Hamiltonian]]
 | linear = [[Airy equation|Airy]]
 | nonlinear = [[semilinear|semilinear with derivatives]]
 | critical = <math>L^2(\R)</math>
 | criticality = mass-critical, energy-subcritical
 | covariance = -
 | lwp = <math>H^s(\R)</math> for <math>s \geq 0</math>
 | gwp = <math>H^s(\R)</math> for <math>s \geq 0</math>, small
 | parent = [[gKdV]]
 | special = -
 | related = -
}}

== Non-periodic theory ==

(Thanks to Felipe Linares for help with the references here - Ed.)A good survey for the results here is in [[Tz-p2]].

The local and global [[well-posedness]] theory for the quintic [[generalized Korteweg-de Vries equation]] on the line and half-line is as follows. 

* Scaling is s_c = 0 (i.e. L^2-critical).
* LWP in H^s for s >= 0 [[KnPoVe1993]]
** Was shown for s>3/2 in [[GiTs1989]]
** The same result s >= 0 has also been established for the half-line [[CoKe-p]], assuming boundary data is in H^{(s+1)/3} of course.
* GWP in H^s for s > 3/4 in both the focusing and defocusing cases, though one must of course have smaller L^2 mass than the ground state in the focusing case [[FoLiPo-p]].
** For s >= 1 and the defocusing case this is in [[KnPoVe1993]]
** Blowup has been shown for the focussing case for data close to a ground state with negative energy [[Me-p]]. In such a case the blowup profile must approach the ground state (modulo scalings and translations), see [[MtMe-p4]], [[MtMe2001]]. Also, the blow up rate in H^1 must be strictly faster than t^{-1/3} [[MtMe-p4]], which is the rate suggested by scaling.
** Explicit self-similar blow-up solutions have been constructed [[BnWe-p]] but these are not in L^2.
** GWP for small L^2 data in either case [[KnPoVe1993]]. In the focussing case we have GWP whenever the L^2 norm is strictly smaller than that of the ground state Q (thanks to Weinstein's sharp Gagliardo-Nirenberg inequality). It seems like a reasonable (but difficult) conjecture to have GWP for large L^2 data in the defocusing case.
** On the half-line GWP is known when s >= 1 and the boundary data is in H^{11/12}, assuming compatibility and small L^2 norm [[CoKe-p]]
* Solitons are H^1-unstable [[MtMe2001]]. However, small H^1 perturbations of a [[soliton]] must asymptotically converge weakly to some rescaled soliton shape provided that the H^1 norm stays comparable to 1 [[MtMe-p]].

== Periodic theory ==

The local and global [[well-posedness]] theory for the quintic [[generalized Korteweg-de Vries equation]] on the torus is as follows. 

* Scaling is s_c = 0.
* LWP in H^s for s>=1/2 [[CoKeStTkTa-p3]]
** Was shown for s >= 1 in [[St1997c]]
** Analytic well-posedness fails for s < 1/2; this is essentially in [[KnPoVe1996]]
* GWP in H^s for s>=1 [[St1997c]]
** This is almost certainly improvable by the techniques in [[CoKeStTkTa-p3]], probably to s > 6/7. There are some low-frequency issues which may require the techniques in [[KeTa-p]].
* ''Remark'': For this equation it is convenient to make a [[gauge transformation]] to subtract off the mean of Pu.

[[Category:Equations]]
[[Category:Airy]]