GKdV-3 equation

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