In mathematics, the KdV hierarchy is an infinite sequence of partial differential equations which contains the Korteweg–de Vries equation.
Let be translation operator defined on real valued functions as . Let be set of all analytic functions that satisfy , i.e. periodic functions of period 1. For each , define an operator on the space of smooth functions on . We define the Bloch spectrum to be the set of such that there is a nonzero function with and . The KdV hierarchy is a sequence of nonlinear differential operators such that for any we have an analytic function and we define to be and , then is independent of .
The KdV hierarchy arises naturally as a statement of Huygens' principle for the D'Alembertian.[1][2]
The first three partial differential equations of the KdV hierarchy are
The first equation identifies and as in the original KdV equation. These equations arise as the equations of motion from the (countably) infinite set of independent constants of motion by choosing them in turn to be the Hamiltonian for the system. For , the equations are called higher KdV equations and the variables higher times.
One can consider the higher KdVs as a system of overdetermined PDEs for
For , the Riemann surface is a hyperelliptic curve and the solution is given in terms of the theta function.[4] In fact all solutions to the KdV equation with periodic initial data arise from this construction (Manakov, Novikov & Pitaevskii et al. 1984).