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The [[Korteweg-de Vries equation]] <center><math>\partial_t V + \partial_x^3 V = 6 V \partial_x V</math></center> <span class="GramE">can</span> be rewritten in the Lax Pair form <center><math>\partial_t L = [L, P]</math></center> where <math>L</math> is the second-order operator <center><math>L = -D^2 + V</math></center> <math>(D = d/dx)</math> and <math>P</math> is the third-order antiselfadjoint operator <center><math>P = 4D^3 + 3(DV + VD)</math>.</center> ''Note'' that <math>P</math> consists of the zeroth order and higher terms of the formal power series expansion of <math>4i L^{3/2}</math>). One can replace <math>P</math> with other fractional powers of L. For instance, the zeroth order and higher terms of <math>4i L^{5/2}</math> are <center><math>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)</math></center> and the Lax pair equation becomes <center><math>\partial_t V + \partial_x^5 u = \partial_x (5 V_x^2 + 10 V V_xx + 10 V^3)</math></center> with Hamiltonian <math>H(V) = \int V_{xx}^2 - 5 V^2 V_xx - 5 V^4 dx.</math> 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 <center><math>\partial_t u + \partial_x^{2j+1} u = P(u, u_x , ..., \partial_x^{2j} u)</math></center> where <math>u</math> is real-valued and <math>P</math> 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 <math>\partial_t u + u_{xxx} = u^2 u_x^2</math> is ill-posed in every <math>H^s</math> [Bo-p3] [[Category:Integrability]] [[Category:Equations]] [[Category:Airy]]

Return to KdV hierarchy.