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

From DispersiveWiki

Jump to navigationJump to search ((Some) cleaning of bibliographic references) |
|||

Line 8: | Line 8: | ||

* LWP is known for initial data in <span class="SpellE">H^s</span> and boundary data in H<span class="GramE">^{</span>(s+1)/3} when s > 3/4 [<span class="SpellE">CoKn</span>-p]. | * LWP is known for initial data in <span class="SpellE">H^s</span> and boundary data in H<span class="GramE">^{</span>(s+1)/3} when s > 3/4 [<span class="SpellE">CoKn</span>-p]. | ||

− | ** The techniques are based on [[ | + | ** The techniques are based on [[Bibliography#KnPoVe1993|KnPoVe1993]] and a replacement of the IVBP with a forced IVP. |

** This has been improved to s >= <span class="SpellE">s_c</span> = 1/2 - 2/k when k > 4 [<span class="SpellE">CoKe</span>-p]. | ** This has been improved to s >= <span class="SpellE">s_c</span> = 1/2 - 2/k when k > 4 [<span class="SpellE">CoKe</span>-p]. | ||

** More specific results are known for [[Korteweg-de Vries on the half-line|KdV]], [[modified Korteweg-de Vries on the half-line|mKdV]], [[gKdV-3 on the half-line|gKdV-3]], and [[gKdV-4 on the half-line|gKdV-4]]. | ** More specific results are known for [[Korteweg-de Vries on the half-line|KdV]], [[modified Korteweg-de Vries on the half-line|mKdV]], [[gKdV-3 on the half-line|gKdV-3]], and [[gKdV-4 on the half-line|gKdV-4]]. | ||

Line 16: | Line 16: | ||

[Thanks to <span class="SpellE">Nikolaos</span> <span class="SpellE">Tzirakis</span> for some corrections - Ed.] | [Thanks to <span class="SpellE">Nikolaos</span> <span class="SpellE">Tzirakis</span> for some corrections - Ed.] | ||

− | * On R with k > 4, <span class="SpellE">gKdV</span>-k is LWP down to scaling: s >= <span class="SpellE">s_c</span> = 1/2 - 2/k [[ | + | * On R with k > 4, <span class="SpellE">gKdV</span>-k is LWP down to scaling: s >= <span class="SpellE">s_c</span> = 1/2 - 2/k [[Bibliography#KnPoVe1993|KnPoVe1993]] |

− | ** Was shown for s>3/2 in [[ | + | ** Was shown for s>3/2 in [[Bibliography#GiTs1989|GiTs1989]] |

− | ** One has ill-<span class="SpellE">posedness</span> in the supercritical regime [[ | + | ** One has ill-<span class="SpellE">posedness</span> in the supercritical regime [[Bibliography#BirKnPoSvVe1996|BirKnPoSvVe1996]] |

− | ** For small data one has scattering [[ | + | ** For small data one has scattering [[Bibliography#KnPoVe1993c|KnPoVe1993c]].Note that one cannot have scattering in L^2 except in the critical case k=4 because one can scale <span class="SpellE">solitons</span> to be arbitrarily small in the non-critical cases. |

− | ** <span class="SpellE">Solitons</span> are H^1-unstable [[ | + | ** <span class="SpellE">Solitons</span> are H^1-unstable [[Bibliography#BnSouSr1987|BnSouSr1987]] |

− | ** If one considers an arbitrary smooth non-linearity (not necessarily a power) then one has LWP for small data in <span class="SpellE">H^s</span>, s > 1/2 [[ | + | ** If one considers an arbitrary smooth non-linearity (not necessarily a power) then one has LWP for small data in <span class="SpellE">H^s</span>, s > 1/2 [[Bibliography#St1995|St1995]] |

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

− | * On R with any k, <span class="SpellE">gKdV</span>-k has the <span class="SpellE">H^s</span> norm growing like t^{(s-1)+} in time for any integer s >= 1 [[ | + | * On R with any k, <span class="SpellE">gKdV</span>-k has the <span class="SpellE">H^s</span> norm growing like t^{(s-1)+} 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 R^+ (or R^-) for two different times [[references | + | * On R with any non-linearity, a non-zero solution to <span class="SpellE">gKdV</span> cannot be supported on the half-line R^+ (or R^-) for two different times [[references:KnPoVe-p3 KnPoVe-p3]], [KnPoVe-p4]. |

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

− | ** Extensions to higher order <span class="SpellE">gKdV</span> type equations are in [[ | + | ** Extensions to higher order <span class="SpellE">gKdV</span> type equations are in [[Bibliography#Bo1997b|Bo1997b]], [KnPoVe-p5]. |

− | * On R with non-integer k, one has decay of <span class="GramE">O(</span>t^{-1/3}) in L^\<span class="SpellE">infty</span> for small decaying data if k > (19 - <span class="SpellE">sqrt</span>(57))/4 ~ 2.8625... [[ | + | * On R with non-integer k, one has decay of <span class="GramE">O(</span>t^{-1/3}) in L^\<span class="SpellE">infty</span> for small decaying data if k > (19 - <span class="SpellE">sqrt</span>(57))/4 ~ 2.8625... [[Bibliography#CtWs1991|CtWs1991]] |

− | ** A similar result for k > (5+<span class="GramE">sqrt(</span>73))/4 ~ 3.39... <span class="GramE">was</span> obtained in [[ | + | ** A similar result for k > (5+<span class="GramE">sqrt(</span>73))/4 ~ 3.39... <span class="GramE">was</span> obtained in [[Bibliography#PoVe1990|PoVe1990]]. |

− | ** When k=2 solutions decay like O(t^{-1/3}), and when k=1 solutions decay generically like O(t^{-2/3}) but like O( (t/log t)^{-2/3}) for exceptional data [[ | + | ** When k=2 solutions decay like O(t^{-1/3}), and when k=1 solutions decay generically like O(t^{-2/3}) but like O( (t/log t)^{-2/3}) for exceptional data [[Bibliography#AbSe1977|AbSe1977]] |

* In the L^2 <span class="SpellE">subcritical</span> case 0 < k < 4, <span class="SpellE">multisoliton</span> solutions are asymptotically H^1-stable [<span class="SpellE">MtMeTsa</span>-p] | * In the L^2 <span class="SpellE">subcritical</span> case 0 < k < 4, <span class="SpellE">multisoliton</span> solutions are asymptotically H^1-stable [<span class="SpellE">MtMeTsa</span>-p] | ||

− | ** For a single <span class="SpellE">soliton</span> this is in [MtMe-p3], [<span class="SpellE">MtMe</span>-p], [[ | + | ** For a single <span class="SpellE">soliton</span> this is in [MtMe-p3], [<span class="SpellE">MtMe</span>-p], [[Bibliography#Miz2001|Miz2001]]; earlier work is in [[Bibliography#Bj1972|Bj1972]], [[Bibliography#Bn1975|Bn1975]], [[Bibliography#Ws1986|Ws1986]], [[Bibliography#PgWs1994|PgWs1994]] |

− | * A dissipative version of <span class="SpellE">gKdV</span>-k was analyzed in [[ | + | * A dissipative version of <span class="SpellE">gKdV</span>-k was analyzed in [[Bibliography#MlRi2001|MlRi2001]] |

− | * On T with any k, <span class="SpellE">gKdV</span>-k has the <span class="SpellE">H^s</span> norm growing like t^{2(s-1)+} in time for any integer s >= 1 [[ | + | * On T with any k, <span class="SpellE">gKdV</span>-k has the <span class="SpellE">H^s</span> norm growing like t^{2(s-1)+} 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 [[references | + | * On T with k >= 3, <span class="SpellE">gKdV</span>-k is LWP for s >= 1/2 [[references:CoKeStTaTk-p3 CoKeStTkTa-p3]] |

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

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

− | ** For arbitrary smooth non-<span class="SpellE">linearities</span>, weak H^1 solutions were constructed in [[references | + | ** For arbitrary smooth non-<span class="SpellE">linearities</span>, weak H^1 solutions were constructed in [[references:Bo1993b Bo1993]]. |

− | * 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 [[ | + | * 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 [[references | + | ** The estimates in [[references: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 [[references:CoKeStTaTk-p3 CoKeStTkTa-p3]]. In the critical and super-critical cases there are some low-frequency issues which may require the techniques in [[references:KeTa-p <span class="SpellE">KeTa</span>-p]]. |

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

## Revision as of 19:38, 28 July 2006

## Half-line theory

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

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

- LWP is known for initial data in H^s and boundary data in H^{(s+1)/3} when s > 3/4 [CoKn-p].
- The techniques are based on KnPoVe1993 and a replacement of the IVBP with a forced IVP.
- This has been improved to s >= s_c = 1/2 - 2/k when k > 4 [CoKe-p].
- More specific results are known for KdV, mKdV, gKdV-3, and gKdV-4.

## Miscellaneous gKdV results

[Thanks to Nikolaos Tzirakis for some corrections - Ed.]

- On R with k > 4, gKdV-k is LWP down to scaling: s >= s_c = 1/2 - 2/k 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 L^2 except in the critical case k=4 because one can scale solitons to be arbitrarily small in the non-critical cases.
- Solitons are H^1-unstable BnSouSr1987
- If one considers an arbitrary smooth non-linearity (not necessarily a power) then one has LWP for small data in H^s, s > 1/2 St1995

- On R with any k, gKdV-k is GWP in H^s for s >= 1 KnPoVe1993, though for k >= 4 one needs the L^2 norm to be small; global weak solutions were constructed much earlier, with the same smallness assumption when k >= 4. This should be improvable below H^1 for all k.
- On R with any k, gKdV-k has the H^s norm growing like t^{(s-1)+} 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 R^+ (or R^-) for two different times references:KnPoVe-p3 KnPoVe-p3, [KnPoVe-p4].
- On R with non-integer k, one has decay of O(t^{-1/3}) in L^\infty for small decaying data if k > (19 - sqrt(57))/4 ~ 2.8625... CtWs1991
- In the L^2 subcritical case 0 < k < 4, multisoliton solutions are asymptotically H^1-stable [MtMeTsa-p]
- A dissipative version of gKdV-k was analyzed in MlRi2001

- On T with any k, gKdV-k has the H^s norm growing like t^{2(s-1)+} in time for any integer s >= 1 St1997b
- On T with k >= 3, gKdV-k is LWP for s >= 1/2 references:CoKeStTaTk-p3 CoKeStTkTa-p3
- Was shown for s >= 1 in St1997c
- Analytic well-posedness fails for s < 1/2 references:CoKeStTaTk-p3 CoKeStTkTa-p3, KnPoVe1996
- For arbitrary smooth non-linearities, weak H^1 solutions were constructed in references:Bo1993b Bo1993.

- On T with k >= 3, gKdV-k is GWP for s >= 1 except in the focussing case St1997c
- The estimates in references: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 references:CoKeStTaTk-p3 CoKeStTkTa-p3. In the critical and super-critical cases there are some low-frequency issues which may require the techniques in [[references:KeTa-p KeTa-p]].