Generalized Korteweg-de Vries equation: Difference between revisions

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The sign of <math>\partial_x^3 u </math> is important (it makes the influence of the boundary x=0 mostly negligible), the sign of <math>u \partial_x u</math> is not. The drift term <math>\partial_x u</math> is convenient for technical reasons; it is not known whether it is truly necessary.
The sign of <math>\partial_x^3 u </math> is important (it makes the influence of the boundary x=0 mostly negligible), the sign of <math>u \partial_x u</math> is not. The drift term <math>\partial_x u</math> is convenient for technical reasons; it is not known whether it is truly necessary.


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


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* On R with any k, <span class="SpellE">gKdV</span>-k is GWP in <math>H^s</math> for s >= 1 [[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 [[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 [[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 [[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 [[KnPoVe-p3]], [KnPoVe-p4].
* 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 [[KnPoVe2003]], [[KnPoVe-p4]].
** In the completely <span class="SpellE">integrable</span> cases k=1,2 this is in [[Zg1992]]
** In the [[completely integrable]] cases k=1,2 this is in [[Zg1992]]
** Also, a non-zero solution to <span class="SpellE">gKdV</span> cannot vanish on a rectangle in <span class="SpellE">spacetime</span> [[SauSc1987]]; see also [[Bo1997b]].
** Also, a non-zero solution to <span class="SpellE">gKdV</span> cannot vanish on a rectangle in <span class="SpellE">spacetime</span> [[SauSc1987]]; see also [[Bo1997b]].
** Extensions to higher order <span class="SpellE">gKdV</span> type equations are in [[Bo1997b]], [KnPoVe-p5].
** Extensions to higher order <span class="SpellE">gKdV</span> type equations are in [[Bo1997b]], [[KnPoVe-p5]].
* On R with non-integer k, one has decay of <math>O(t^{-1/3}) in L^\infty</math> for small decaying data if <math>k > {(19 - \sqrt(57)) \over 4} \sim 2.8625...</math> [[CtWs1991]]
* On R with non-integer k, one has decay of <math>O(t^{-1/3}) in L^\infty</math> for small decaying data if <math>k > {(19 - \sqrt(57)) \over 4} \sim 2.8625...</math> [[CtWs1991]]
** A similar result for <math> k > (5+\sqrt(73))/4 \sim 3.39... </math><span class="GramE">was</span> obtained in [[PoVe1990]].
** A similar result for <math> k > (5+\sqrt(73))/4 \sim 3.39... </math><span class="GramE">was</span> obtained in [[PoVe1990]].
** When k=2 solutions decay like <math>O(t^{-1/3})</math>, and when k=1 solutions decay generically like <math>O(t^{-2/3})</math> but like <math>O( (t/log t)^{-2/3})</math> for exceptional data [[AbSe1977]]
** When k=2 solutions decay like <math>O(t^{-1/3})</math>, and when k=1 solutions decay generically like <math>O(t^{-2/3})</math> but like <math>O( (t/log t)^{-2/3})</math> for exceptional data [[AbSe1977]]
* In the <math>L^2</math> <span class="SpellE">subcritical</span> case 0 < k < 4, <span class="SpellE">multisoliton</span> solutions are asymptotically <math>H^1</math>-stable [<span class="SpellE">MtMeTsa</span>-p]
* In the <math>L^2</math> <span class="SpellE">subcritical</span> case 0 < k < 4, <span class="SpellE">multisoliton</span> solutions are asymptotically <math>H^1</math>-stable [[MtMeTsa-p]]
** For a single <span class="SpellE">soliton</span> this is in [MtMe-p3], [<span class="SpellE">MtMe</span>-p], [[Miz2001]]; earlier work is in [[Bj1972]], [[Bn1975]], [[Ws1986]], [[PgWs1994]]
** For a single <span class="SpellE">soliton</span> this is in [[MtMe-p3]], [[MtMe-p]], [[Miz2001]]; earlier work is in [[Bj1972]], [[Bn1975]], [[Ws1986]], [[PgWs1994]]
* A dissipative version of <span class="SpellE">gKdV</span>-k was analyzed in [[MlRi2001]]
* A dissipative version of <span class="SpellE">gKdV</span>-k was analyzed in [[MlRi2001]]


Latest revision as of 20:49, 10 June 2007

Half-line theory

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

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \partial_t u + \partial_x^3 u + \partial_x u + u^k \partial_x u = 0; u(x,0) = u_0(x); u(0,t) = h(t)}

The sign of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \partial_x^3 u } is important (it makes the influence of the boundary x=0 mostly negligible), the sign of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle u \partial_x u} is not. The drift term Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \partial_x u} is convenient for technical reasons; it is not known whether it is truly necessary.

  • LWP is known for initial data in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle H^s} and boundary data in when Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle 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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle s >= \partial_c s = 1/2 - 2/k } when Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle k > 4} CoKn-p.
    • More specific results are known for KdV, mKdV, gKdV-3, and gKdV-4.

Miscellaneous gKdV results

  • On R with k > 4, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle gKdV-k } is LWP down to scaling: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle s >= \partial_c s = 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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle 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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle H^1} -unstable BnSouSr1987
    • If one considers an arbitrary smooth non-linearity (not necessarily a power) then one has LWP for small data in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle H^s, s > 1/2 } St1995
  • On R with any k, gKdV-k is GWP in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle H^s} for s >= 1 KnPoVe1993, though for k >= 4 one needs the Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle 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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle H^1 } for all k.
  • On R with any k, gKdV-k has the Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle H^s} norm growing like Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle 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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle R^+ } (or Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle R^-} ) for two different times KnPoVe2003, KnPoVe-p4.
  • On R with non-integer k, one has decay of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle O(t^{-1/3}) in L^\infty} for small decaying data if Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle k > {(19 - \sqrt(57)) \over 4} \sim 2.8625...} CtWs1991
    • A similar result for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle k > (5+\sqrt(73))/4 \sim 3.39... } was obtained in PoVe1990.
    • When k=2 solutions decay like Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle O(t^{-1/3})} , and when k=1 solutions decay generically like Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle O(t^{-2/3})} but like Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle O( (t/log t)^{-2/3})} for exceptional data AbSe1977
  • In the Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle L^2} subcritical case 0 < k < 4, multisoliton solutions are asymptotically Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle H^1} -stable MtMeTsa-p
  • A dissipative version of gKdV-k was analyzed in MlRi2001
  • On T with any k, gKdV-k has the Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle H^s} norm growing like Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle 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 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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle H^1 } 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.
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