# Difference between revisions of "Generalized Korteweg-de Vries equation"

From DispersiveWiki

Jump to navigationJump to searchm (More bib cleaning) |
|||

Line 23: | Line 23: | ||

* On R with any k, <span class="SpellE">gKdV</span>-k is GWP in <math>H^s</math> for s >= 1 [[Bibliography#KnPoVe1993|KnPoVe1993]], though for k >= 4 one needs the <math>L^2 </math>norm to be small; global weak solutions were constructed much earlier, with the same smallness assumption when k >= 4. This should be improvable below <math>H^1 </math>for all k. | * On R with any k, <span class="SpellE">gKdV</span>-k is GWP in <math>H^s</math> for s >= 1 [[Bibliography#KnPoVe1993|KnPoVe1993]], though for k >= 4 one needs the <math>L^2 </math>norm to be small; global weak solutions were constructed much earlier, with the same smallness assumption when k >= 4. This should be improvable below <math>H^1 </math>for all k. | ||

* On R with any k, <span class="SpellE">gKdV</span>-k has the <math>H^s</math> norm growing like <math>t^{(s-1)} </math>in time for any integer s >= 1 [[Bibliography#St1997b|St1997b]] | * On R with any k, <span class="SpellE">gKdV</span>-k has the <math>H^s</math> norm growing like <math>t^{(s-1)} </math>in time for any integer s >= 1 [[Bibliography#St1997b|St1997b]] | ||

− | * On R with any non-linearity, a non-zero solution to <span class="SpellE">gKdV</span> cannot be supported on the half-line <math>R^+ </math>(or <math>R^-</math>) for two different times [[ | + | * On R with any non-linearity, a non-zero solution to <span class="SpellE">gKdV</span> cannot be supported on the half-line <math>R^+ </math>(or <math>R^-</math>) for two different times [[Bibliography#KnPoVe-p3 |KnPoVe-p3]], [KnPoVe-p4]. |

** In the completely <span class="SpellE">integrable</span> cases k=1,2 this is in [[Bibliography#Zg1992|Zg1992]] | ** In the completely <span class="SpellE">integrable</span> cases k=1,2 this is in [[Bibliography#Zg1992|Zg1992]] | ||

** Also, a non-zero solution to <span class="SpellE">gKdV</span> cannot vanish on a rectangle in <span class="SpellE">spacetime</span> [[Bibliography#SauSc1987|SauSc1987]]; see also [[Bibliography#Bo1997b|Bo1997b]]. | ** Also, a non-zero solution to <span class="SpellE">gKdV</span> cannot vanish on a rectangle in <span class="SpellE">spacetime</span> [[Bibliography#SauSc1987|SauSc1987]]; see also [[Bibliography#Bo1997b|Bo1997b]]. | ||

Line 35: | Line 35: | ||

* On T with any k, <span class="SpellE">gKdV</span>-k has the <math>H^s</math> norm growing like <math>t^{2(s-1)+}</math> in time for any integer s >= 1 [[Bibliography#St1997b|St1997b]] | * On T with any k, <span class="SpellE">gKdV</span>-k has the <math>H^s</math> norm growing like <math>t^{2(s-1)+}</math> in time for any integer s >= 1 [[Bibliography#St1997b|St1997b]] | ||

− | * On T with k >= 3, <span class="SpellE">gKdV</span>-k is LWP for s >= 1/2 [[ | + | * On T with k >= 3, <span class="SpellE">gKdV</span>-k is LWP for s >= 1/2 [[Bibliography#CoKeStTaTk-p3 |CoKeStTkTa-p3]] |

** Was shown for s >= 1 in [[Bibliography#St1997c|St1997c]] | ** Was shown for s >= 1 in [[Bibliography#St1997c|St1997c]] | ||

− | ** Analytic well-<span class="SpellE">posedness</span> fails for s < 1/2 [[ | + | ** Analytic well-<span class="SpellE">posedness</span> fails for s < 1/2 [[Bibliography#CoKeStTaTk-p3 |CoKeStTkTa-p3]], [[Bibliography#KnPoVe1996|KnPoVe1996]] |

− | ** For arbitrary smooth non-<span class="SpellE">linearities</span>, weak <math>H^1 </math>solutions were constructed in [[ | + | ** For arbitrary smooth non-<span class="SpellE">linearities</span>, weak <math>H^1 </math>solutions were constructed in [[Bibliography#Bo1993b |Bo1993b]]. |

* On T with k >= 3, <span class="SpellE">gKdV</span>-k is GWP for s >= 1 except in the <span class="SpellE">focussing</span> case [[Bibliography#St1997c|St1997c]] | * On T with k >= 3, <span class="SpellE">gKdV</span>-k is GWP for s >= 1 except in the <span class="SpellE">focussing</span> case [[Bibliography#St1997c|St1997c]] | ||

− | ** The estimates in [[ | + | ** The estimates in [[Bibliography#CoKeStTaTk-p3 |CoKeStTkTa-p3]] suggest that this is improvable to 13/14 - 2/7k, but this has only been proven in the sub-critical case k=3 [[Bibliography#CoKeStTaTk-p3 |CoKeStTkTa-p3]]. In the critical and super-critical cases there are some low-frequency issues which may require the techniques in [[Bibliography#KeTa-p |KeTa-p]]. |

[[Category:Equations]] | [[Category:Equations]] | ||

[[Category:Airy]] | [[Category:Airy]] |

## Revision as of 16:45, 31 July 2006

## Half-line theory

The gKdV Cauchy-boundary problem on the half-line is

The sign of is important (it makes the influence of the boundary x=0 mostly negligible), the sign of is not. The drift term is convenient for technical reasons; it is not known whether it is truly necessary.

- LWP is known for initial data in and boundary data in when .
- The techniques are based on KnPoVe1993 and a replacement of the IVBP with a forced IVP.
- This has been improved to when .
- More specific results are known for KdV, mKdV, gKdV-3, and gKdV-4.

## Miscellaneous gKdV results

- On R with k > 4, is LWP down to scaling: KnPoVe1993
- Was shown for s>3/2 in GiTs1989
- One has ill-posedness in the supercritical regime BirKnPoSvVe1996
- For small data one has scattering KnPoVe1993c.Note that one cannot have scattering in except in the critical case k=4 because one can scale solitons to be arbitrarily small in the non-critical cases.
- Solitons are -unstable BnSouSr1987
- If one considers an arbitrary smooth non-linearity (not necessarily a power) then one has LWP for small data in St1995

- On R with any k, gKdV-k is GWP in for s >= 1 KnPoVe1993, though for k >= 4 one needs the norm to be small; global weak solutions were constructed much earlier, with the same smallness assumption when k >= 4. This should be improvable below for all k.
- On R with any k, gKdV-k has the norm growing like in time for any integer s >= 1 St1997b
- On R with any non-linearity, a non-zero solution to gKdV cannot be supported on the half-line (or ) for two different times KnPoVe-p3, [KnPoVe-p4].
- On R with non-integer k, one has decay of for small decaying data if CtWs1991
- In the subcritical case 0 < k < 4, multisoliton solutions are asymptotically -stable [MtMeTsa-p]
- A dissipative version of gKdV-k was analyzed in MlRi2001

- On T with any k, gKdV-k has the norm growing like in time for any integer s >= 1 St1997b
- On T with k >= 3, gKdV-k is LWP for s >= 1/2 CoKeStTkTa-p3
- Was shown for s >= 1 in St1997c
- Analytic well-posedness fails for s < 1/2 CoKeStTkTa-p3, KnPoVe1996
- For arbitrary smooth non-linearities, weak solutions were constructed in Bo1993b.

- On T with k >= 3, gKdV-k is GWP for s >= 1 except in the focussing case St1997c
- The estimates in CoKeStTkTa-p3 suggest that this is improvable to 13/14 - 2/7k, but this has only been proven in the sub-critical case k=3 CoKeStTkTa-p3. In the critical and super-critical cases there are some low-frequency issues which may require the techniques in KeTa-p.