In mathematics, the Korteweg–De Vries (KdV) hierarchy is an infinite sequence of mutually compatible nonlinear evolution equations containing the Korteweg–de Vries equation as its first nontrivial member. It is one of the central examples in the theory of integrable systems and soliton equations, because it combines several characteristic features of integrability: a Lax formulation, infinitely many commuting flows and conserved quantities, Hamiltonian and bi-Hamiltonian structures, and exact solution methods such as the inverse scattering transform and finite-gap integration. It is most commonly formulated as a family of Lax equations for the one-dimensional Schrödinger operator
L = ∂ x 2 + u ( x ) , {\displaystyle L=\partial _{x}^{2}+u(x),}
or, equivalently up to sign conventions, L = − ∂ x 2 + u ( x ) {\displaystyle L=-\partial _{x}^{2}+u(x)} . The hierarchy consists of commuting flows in auxiliary variables t 0 , t 1 , t 2 , … {\displaystyle t_{0},t_{1},t_{2},\dots } , each of which preserves the spectral data of L {\displaystyle L} . In periodic and quasiperiodic settings, this spectral interpretation leads to the theory of finite-gap and algebro-geometric solutions, while more broadly the hierarchy serves as a prototype for many later integrable hierarchies and reductions such as the modified KdV and KP hierarchies.
Definition Let
L = ∂ x 2 + u ( x , t 0 , t 1 , t 2 , … ) . {\displaystyle L=\partial _{x}^{2}+u(x,t_{0},t_{1},t_{2},\dots ).}
The KdV hierarchy may be defined by requiring that L {\displaystyle L} evolve according to the family of Lax equations
∂ L ∂ t n = [ P 2 n + 1 , L ] , n = 0 , 1 , 2 , … , {\displaystyle {\frac {\partial L}{\partial t_{n}}}=[P_{2n+1},L],\qquad n=0,1,2,\dots ,}
where each P 2 n + 1 {\displaystyle P_{2n+1}} is an odd-order differential operator determined by L {\displaystyle L} . A compact way to construct these operators uses formal pseudodifferential operators:
∂ L ∂ t n = [ ( L ( 2 n + 1 ) / 2 ) + , L ] , {\displaystyle {\frac {\partial L}{\partial t_{n}}}={\big [}(L^{(2n+1)/2})_{+},L{\big ]},}
where ( ⋅ ) + {\displaystyle (\cdot )_{+}} denotes the differential part of a formal pseudodifferential operator. This formulation implies that the flows commute:
∂ ∂ t m ∂ L ∂ t n = ∂ ∂ t n ∂ L ∂ t m , {\displaystyle {\frac {\partial }{\partial t_{m}}}{\frac {\partial L}{\partial t_{n}}}={\frac {\partial }{\partial t_{n}}}{\frac {\partial L}{\partial t_{m}}},}
so one may regard the hierarchy as a compatible overdetermined system for u {\displaystyle u} depending on infinitely many times. Equivalent descriptions can be given in terms of commuting Hamiltonian vector fields or the Lenard–Magri recursion scheme. In all of these formulations, the distinguishing feature of the hierarchy is that it produces infinitely many compatible flows, all associated with the same Schrödinger operator.
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