KdV hierarchy

The Korteweg-de Vries equation

$\partial _{t}V+\partial _{x}^{3}V=6V\partial _{x}V$

can be rewritten in the Lax Pair form

$\partial _{t}L=[L,P]$

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 $4iL^{3/2}$).

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

$P=4D^{5}+5(D^{3}V+VD^{3})-5/4(D\partial _{x}^{2}V+\partial _{x}^{2}VD)+15/4(DV^{2}+V^{2}D)$

and the Lax pair equation becomes

$\partial _{t}V+\partial _{x}^{5}u=\partial _{x}(5V_{x}^{2}+10VV_{x}x+10V^{3})$

with Hamiltonian

$H(V)=\int V_{xx}^{2}-5V^{2}V_{x}x-5V^{4}dx.$

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

$\partial _{t}u+\partial _{x}^{2j+1}u=P(u,u_{x},...,\partial _{x}^{2j}u)$

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 $\partial _{t}u+u_{xxx}=u^{2}u_{x}^{2}$ is ill-posed in every $H^{s}$ [Bo-p3]