Difference between revisions of "KdV hierarchy"

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<center><span class="SpellE">u_t</span> + <span class="SpellE">partial_x</span><span class="GramE">^{</span>2j+1} u = P(u, <span class="SpellE">u_x</span>, ..., <span class="SpellE">partial_x</span>^{2j} u)</center>
 
<center><span class="SpellE">u_t</span> + <span class="SpellE">partial_x</span><span class="GramE">^{</span>2j+1} u = P(u, <span class="SpellE">u_x</span>, ..., <span class="SpellE">partial_x</span>^{2j} u)</center>
  
<span class="GramE">where</span> u is real-valued and P is a polynomial with no constant or linear terms; thus <span class="SpellE">KdV</span> and <span class="SpellE">gKdV</span> 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 <span class="SpellE">Sobolev</span> spaces is in [[references.html#KnPoVe1994 KnPoVe1994]], and independently by <span class="SpellE"><span class="GramE">Cai</span></span> (ref?); see also [[references.html#CrKpSr1992 CrKpSr1992]].The case j=2 was studied by <span class="SpellE">Choi</span> (ref?).The non-scalar diagonal case was treated in [[references.html#KnSt1997 KnSt1997]]; the periodic case was studied in [Bo-p3].Note in the periodic case it is possible to have ill-<span class="SpellE">posedness</span> for every regularity, for instance <span class="SpellE">u_t</span> + <span class="SpellE">u_xxx</span> = u^2 u_x^2 is ill-posed in every <span class="SpellE">H^s</span> [Bo-p3]
+
<span class="GramE">where</span> u is real-valued and P is a polynomial with no constant or linear terms; thus <span class="SpellE">KdV</span> and <span class="SpellE">gKdV</span> 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 <span class="SpellE">Sobolev</span> spaces is in [[Bibliography#KnPoVe1994|KnPoVe1994]], and independently by <span class="SpellE"><span class="GramE">Cai</span></span> (ref?); see also [[Bibliography#CrKpSr1992|CrKpSr1992]].The case j=2 was studied by <span class="SpellE">Choi</span> (ref?).The non-scalar diagonal case was treated in [[Bibliography#KnSt1997|KnSt1997]]; the periodic case was studied in [Bo-p3].Note in the periodic case it is possible to have ill-<span class="SpellE">posedness</span> for every regularity, for instance <span class="SpellE">u_t</span> + <span class="SpellE">u_xxx</span> = u^2 u_x^2 is ill-posed in every <span class="SpellE">H^s</span> [Bo-p3]
  
 
[[Category:Equations]]
 
[[Category:Equations]]

Revision as of 19:48, 28 July 2006

The Korteweg-de Vries equation

can be rewritten in the Lax Pair form

L_t = [L, P]

where L is the second-order operator

L = -D^2 + V

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

P = 4D^3 + 3(DV + VD).

(note that P consists of the zeroth order and higher terms of the formal power series expansion of 4i L^{3/2}).

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

P = 4D^5 + 5(D^3 V + V D^3) - 5/4 (D V_xx + V_xx D) + 15/4 (D V^2 + V^2 D)

and the Lax pair equation becomes

V_t + u_xxxxx = (5 V_x^2 + 10 V V_xx + 10 V^3)_x

with Hamiltonian

H(V) = \int V_xx^2 - 5 V^2 V_xx - 5 V^4.

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

u_t + partial_x^{2j+1} u = P(u, u_x, ..., partial_x^{2j} u)

where u is real-valued and P 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 u_t + u_xxx = u^2 u_x^2 is ill-posed in every H^s [Bo-p3]