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{{equation
 | name = Quartic gKdV
 | equation = <math>u_t + u_{xxx} = \pm u^3 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>\dot H^{-1/6}(\R)</math>
 | criticality = mass-subcritical, energy-subcritical
 | covariance = -
 | lwp = <math>H^s(\R)</math> for <math>s \geq -1/6</math>
 | gwp = <math>H^s(\R)</math> for <math>s \geq -1/6</math>, small norm
 | parent = [[gKdV]]
 | special = -
 | related = -
}}

== Non-periodic theory ==

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

* Scaling is s_c = -1/6.
* LWP for s >= -1/6 [[Ta2007]]
** For s > -1/6 this is in [[Gr-p3]]
** Was shown for s>=1/12 [[KnPoVe1993]]
** Was shown for s>3/2 in [[GiTs1989]]
** The result s >= 1/12 has also been established for the half-line [[CoKn-p]], assuming boundary data is in H^{(s+1)/3} of course.
* GWP in H^s for s >= 0 [[Gr-p3]]
** For s>=1 this is in [[KnPoVe1993]]
** Presumably one can use either the Fourier truncation method or the [[I-method]] to go below L^2. Even though the equation is not [[completely integrable]], the one-dimensional nature of the equation suggests that [[correction term]] techniques will also be quite effective.
** On the half-line GWP is known when s >= 1 and the boundary data is in H^{5/4}, assuming compatibility and small L^2 norm [[CoKn-p]]
* [[Solitons]] are H^1-stable [[CaLo1982]], [[Ws1986]], [[BnSouSr1987]] and asymptotically H^1 stable [[MtMe-p3]], [[MtMe-p]]
** If one also assumes the error is small in the critical space <math>\dot H^{-1/6}(\R)</math> then one has asymptotic stability [[Ta2007]]

== Periodic theory ==

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

* Scaling is s_c = -1/6.
* LWP in H^s for s>=1/2 [[CoKeStTkTa-p3]]
** Was shown for s >= 1 in [[St1997c]]
** One has analytic ill-posedness for s<1/2 [[CoKeStTkTa-p3]] by a modification of the example in [[KnPoVe1996]].
* GWP in H^s for s>5/6 [[CoKeStTkTa-p3]]
** Was shown for s >= 1 in [[St1997c]]
** This result may well be improvable by the [[correction term]] method.
* ''Remark'': For this equation it is convenient to make a [[gauge transformation]] to subtract off the mean of P(u).

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