KdV hierarchy

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$\partial_t V + \partial_x^3 V = 6 V \partial_x V$

can be rewritten in the Lax Pair form

$\partial_t L = [L, P]$

where $L$ is the second-order operator

L = - D 2 + V

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

P = 4 D 3 + 3(D V + V D)

.

Note that $P$ consists of the zeroth order and higher terms of the formal power series expansion of $4 i L 3 / 2$ ).

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

$P = 4D^5 + 5(D^3 V + V D^3) - 5/4 (D \partial^2_x V + \partial_x^2 V D) + 15/4 (D V^2 + V^2 D)$

and the Lax pair equation becomes

$\partial_t V + \partial_x^5 u = \partial_x (5 V_x^2 + 10 V V_xx + 10 V^3)$

with Hamiltonian

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

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</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 $\partial_t u + u_{xxx} = u^2 u_x^2$ is ill-posed in every $H s$ [Bo-p3]

KdV hierarchy

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