GKdV-3 equation

Quartic gKdV

Description
Equation	$u_{t}+u_{xxx}=\pm u^{3}u_{x}$
Fields	$u:\mathbb {R} \times \mathbb {R} \to \mathbb {R}$
Data class	$u(0)\in H^{s}(\mathbb {R} )$
Basic characteristics
Structure	Hamiltonian
Nonlinearity	semilinear with derivatives
Linear component	Airy
Critical regularity	${\dot {H}}^{-1/6}(\mathbb {R} )$
Criticality	mass-subcritical, energy-subcritical
Covariance	-
Theoretical results
LWP	$H^{s}(\mathbb {R} )$ for $s\geq -1/6$
GWP	$H^{s}(\mathbb {R} )$ 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 ${\dot {H}}^{-1/6}(\mathbb {R} )$ 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.

• 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).