# GKdV-3 equation

Quartic gKdV
Description
Equation $u_t + u_{xxx} = \pm u^3 u_x$
Fields $u: \R \times \R \to \R$
Data class $u(0) \in H^s(\R)$
Basic characteristics
Structure Hamiltonian
Nonlinearity semilinear with derivatives
Linear component Airy
Critical regularity $\dot H^{-1/6}(\R)$
Criticality mass-subcritical, energy-subcritical
Covariance -
Theoretical results
LWP $H^s(\R)$ for $s \geq -1/6$
GWP $H^s(\R)$ for $s \geq -1/6$, small norm
Related equations
Parent class gKdV
Special cases -
Other 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 $\dot H^{-1/6}(\R)$ 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.