GKdV-3 equation: Difference between revisions

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


[[Category:Equations]]
[[Category:Equations]]
[[Category:Airy]]
[[Category:Airy]]

Revision as of 14:47, 10 August 2006

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 KnPoVe1993
    • Was shown for s>3/2 in 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 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 CaLo1982, Ws1986, 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 CoKeStTkTa-p3
  • 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 "damping correction term" method in CoKeStTkTa-p2.
  • Remark: For this equation it is convenient to make a "gauge transformation'' to subtract off the mean of P(u).