GKdV-4 equation

Quintic gKdV

Description
Equation	$u_{t}+u_{xxx}=\pm u^{4}u_{x}$
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	$L^{2}(\mathbb {R} )$
Criticality	mass-critical, energy-subcritical
Covariance	-
Theoretical results
LWP	$H^{s}(\mathbb {R} )$ for $s\geq 0$
GWP	$H^{s}(\mathbb {R} )$ for $s\geq 0$, small
Related equations
Parent class	gKdV
Special cases	-
Other 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.