KdV hierarchy

In mathematics, the KdV hierarchy is an infinite sequence of partial differential equations which contains the Korteweg–de Vries equation.

be translation operator defined on real valued functions as

be set of all analytic functions that satisfy

, define an operator

on the space of smooth functions on

We define the Bloch spectrum

( λ , α ) ∈

such that there is a nonzero function

( ψ ) = λ ψ

( ψ ) = α ψ

The KdV hierarchy is a sequence of nonlinear differential operators

we have an analytic function

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

{\displaystyle {\begin{aligned}u_{t_{0}}&=u_{x}\\u_{t_{1}}&=6uu_{x}-u_{xxx}\\u_{t_{2}}&=10uu_{xxx}-20u_{x}u_{xx}-30u^{2}u_{x}-u_{xxxxx}.\end{aligned}}}

where each equation is considered as a PDE for

[3] The first equation identifies

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.

, 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

Then solutions which are independent of higher times above some fixed

and with periodic boundary conditions are called finite-gap solutions.

Such solutions turn out to correspond to compact Riemann surfaces, which are classified by their genus

gives the constant solution, while

corresponds to cnoidal wave solutions.

, 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).