# KdV hierarchy

The Korteweg-de Vries equation

can be rewritten in the Lax Pair form

where L is the second-order operator

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

(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

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 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]