KdV hierarchy: Difference between revisions

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The [[Korteweg-de Vries equation]]
The [[Korteweg-de Vries equation]]


<center><math>\partial_t V   \partial_x^3 V  = 6 V \partial_x V</math></center>
<center><math>\partial_t V + \partial_x^3 V  = 6 V \partial_x V</math></center>


<span class="GramE">can</span> be rewritten in the Lax Pair form
<span class="GramE">can</span> be rewritten in the Lax Pair form
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where <math>L</math> is the second-order operator
where <math>L</math> is the second-order operator


<center><math>L = -D^2   V</math></center>
<center><math>L = -D^2 + V</math></center>


<math>(D = d/dx)</math> and <math>P</math> is the third-order antiselfadjoint operator
<math>(D = d/dx)</math> and <math>P</math> is the third-order antiselfadjoint operator


<center><math>P = 4D^3   3(DV   VD)</math>.</center>
<center><math>P = 4D^3 + 3(DV + VD)</math>.</center>


''Note''that <math>P</math> consists of the zeroth order and higher terms of the formal power series expansion of <math>4i L^{3/2}</math>).
''Note'' that <math>P</math> consists of the zeroth order and higher terms of the formal power series expansion of <math>4i L^{3/2}</math>).


One can replace <math>P</math> with other fractional powers of L. For instance, the zeroth order and higher terms of <math>4i L^{5/2}</math> are
One can replace <math>P</math> with other fractional powers of L. For instance, the zeroth order and higher terms of <math>4i L^{5/2}</math> are


<center><math>P = 4D^5   5(D^3 V   V D^3) - 5/4 (D \partial^2_x V   \partial_x^2 V  D)   15/4 (D V^2   V^2 D)</math></center>
<center><math>P = 4D^5 + 5(D^3 V + V D^3) - 5/4 (D \partial^2_x V + \partial_x^2 V  D) + 15/4 (D V^2 + V^2 D)</math></center>


and the Lax pair equation becomes
and the Lax pair equation becomes


<center><math>\partial_t V   \partial_x^5 u  = \partial_x (5 V_x^2   10 V V_xx   10 V^3)</math></center>
<center><math>\partial_t V + \partial_x^5 u  = \partial_x (5 V_x^2 + 10 V V_xx + 10 V^3)</math></center>


with Hamiltonian
with Hamiltonian
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The ''KdV hierarchy'' are examples of higher order water wave models; a general formulation is
The ''KdV hierarchy'' are examples of higher order water wave models; a general formulation is


<center><math>\partial_t u   \partial_x^{2j 1} u = P(u, u_x , ..., \partial_x^{2j} u)</math></center>
<center><math>\partial_t u + \partial_x^{2j+1} u = P(u, u_x , ..., \partial_x^{2j} u)</math></center>


where <math>u</math> is real-valued and <math>P</math> is a polynomial with no constant or linear terms; thus KdV and gKdV correspond to j=1, and the higher order equations in the hierarchy correspond to j=2,3,etc. LWP for these equations in high regularity Sobolev spaces is in [[KnPoVe1994]], and independently by Cai  (ref?); see also [[CrKpSr1992]].The case j=2 was studied by Choi</span> (ref?).The non-scalar diagonal case was treated in [[KnSt1997]]; the periodic case was studied in [Bo-p3].Note in the periodic case it is possible to have ill-posedness for every regularity, for instance <math>\partial_t u   u_{xxx} = u^2 u_x^2</math> is ill-posed in every <math>H^s</math> [Bo-p3]
where <math>u</math> is real-valued and <math>P</math> is a polynomial with no constant or linear terms; thus KdV and gKdV correspond to j=1, and the higher order equations in the hierarchy correspond to j=2,3,etc. LWP for these equations in high regularity Sobolev spaces is in [[KnPoVe1994]], and independently by Cai  (ref?); see also [[CrKpSr1992]].The case j=2 was studied by Choi</span> (ref?).The non-scalar diagonal case was treated in [[KnSt1997]]; the periodic case was studied in [Bo-p3].Note in the periodic case it is possible to have ill-posedness for every regularity, for instance <math>\partial_t u + u_{xxx} = u^2 u_x^2</math> is ill-posed in every <math>H^s</math> [Bo-p3]


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

Latest revision as of 20:10, 11 June 2007

The Korteweg-de Vries equation

can be rewritten in the Lax Pair form

where is the second-order operator

and is the third-order antiselfadjoint operator

.

Note that consists of the zeroth order and higher terms of the formal power series expansion of ).

One can replace with other fractional powers of L. For instance, the zeroth order and higher terms of are

and the Lax pair equation becomes

with Hamiltonian

These flows all commute with each other and their Hamiltonians are conserved by all the flows simultaneously.

The KdV hierarchy are examples of higher order water wave models; a general formulation is

where is real-valued and is a polynomial with no constant or linear terms; thus KdV and gKdV correspond to j=1, and the higher order equations in the hierarchy correspond to j=2,3,etc. LWP for these equations in high regularity Sobolev spaces is in KnPoVe1994, and independently by Cai (ref?); see also CrKpSr1992.The case j=2 was studied by Choi (ref?).The non-scalar diagonal case was treated in KnSt1997; the periodic case was studied in [Bo-p3].Note in the periodic case it is possible to have ill-posedness for every regularity, for instance is ill-posed in every [Bo-p3]