# KdV hierarchy

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

can be rewritten in the Lax Pair form

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

where ${\displaystyle L}$ is the second-order operator

${\displaystyle L=-D^{2}+V}$

${\displaystyle (D=d/dx)}$ and ${\displaystyle P}$ is the third-order antiselfadjoint operator

${\displaystyle P=4D^{3}+3(DV+VD)}$.

Note that ${\displaystyle P}$ consists of the zeroth order and higher terms of the formal power series expansion of ${\displaystyle 4iL^{3/2}}$).

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

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

and the Lax pair equation becomes

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

with Hamiltonian

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

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

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

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