GKdV-3 equation

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 in H^s for s > -1/6 [Gr-p3]
  • Was shown for s>=1/12 references.html#KnPoVe1993 KnPoVe1993
  • Was shown for s>3/2 in references.html#GiTs1989 GiTs1989
  • The result s >= 1/12 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 >= 0 [Gr-p3]
  • For s>=1 this is in references.html#KnPoVe1993 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 [CoKe-p]
• Solitons are H^1-stable references.html#CaLo1982 CaLo1982, references.html#Ws1986 Ws1986, references.html#BnSouSr1987 BnSouSr1987 and asymptotically H^1 stable [MtMe-p3], [MtMe-p]

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 references.html#CoKeStTaTk-p3 CoKeStTkTa-p3
  • Was shown for s >= 1 in references.html#St1997c St1997c
  • One has analytic ill-posedness for s<1/2 references.html#CoKeStTaTk-p3 CoKeStTkTa-p3 by a modification of the example in references.html#KnPoVe1996 KnPoVe1996.
• GWP in H^s for s>5/6 references.html#CoKeStTaTk-p3 CoKeStTkTa-p3
  • Was shown for s >= 1 in references.html#St1997c St1997c
  • This result may well be improvable by the "damping correction term" method in [[references.html#CoKeStTaTk-p2 CoKeStTkTa-p2]].
• Remark: For this equation it is convenient to make a "gauge transformation'' to subtract off the mean of P(u).