In mathematics, the KdV hierarchy is an infinite sequence of partial differential equation

Partial differential equation

In mathematics, partial differential equations are a type of differential equation, i.e., a relation involving an unknown function of several independent variables and their partial derivatives with respect to those variables...

s which starts with the Korteweg–de Vries equation

Korteweg–de Vries equation

In mathematics, the Korteweg–de Vries equation is a mathematical model of waves on shallow water surfaces. It is particularly notable as the prototypical example of an exactly solvable model, that is, a non-linear partial differential equation whose solutions can be exactly and precisely specified...

.

Let be translation operator defined on real valued functions

Function (mathematics)

In mathematics, a function associates one quantity, the argument of the function, also known as the input, with another quantity, the value of the function, also known as the output. A function assigns exactly one output to each input. The argument and the value may be real numbers, but they can...

  as . Let be set of all analytic function

Analytic function

In mathematics, an analytic function is a function that is locally given by a convergent power series. There exist both real analytic functions and complex analytic functions, categories that are similar in some ways, but different in others...

s that satisfy , i.e. periodic function

Periodic function

In mathematics, a periodic function is a function that repeats its values in regular intervals or periods. The most important examples are the trigonometric functions, which repeat over intervals of length 2π radians. Periodic functions are used throughout science to describe oscillations,...

s of period 1. For each , define an operator

on the space of smooth function

Smooth function

In mathematical analysis, a differentiability class is a classification of functions according to the properties of their derivatives. Higher order differentiability classes correspond to the existence of more derivatives. Functions that have derivatives of all orders are called smooth.Most of...

s on . We define the Bloch spectrum to be the set of so that there is a nonzero function with and . The KdV hierarchy is a sequence of nonlinear differential operators so that for any we have an analytic function and we define to be and
,
then is independent of .

External links

• KdV hierarchy at the Dispersive PDE Wiki.