In mathematics, the KdV hierarchy is an infinite sequence of partial differential equations which starts with the Korteweg–de Vries equation.
Let T be translation operator defined on real valued functions as T ( g ) ( x ) = g ( x + 1 ) . Let C be set of all analytic functions that satisfy T ( g ) ( x ) = g ( x ) , i.e. periodic functions of period 1. For each g ∈ C , define an operator L g ( ψ ) ( x ) = ψ ″ ( x ) + g ( x ) ψ ( x ) on the space of smooth functions on R . We define the Bloch spectrum B g to be the set of ( λ , α ) ∈ C × C ∗ such that there is a nonzero function ψ with L g ( ψ ) = λ ψ and T ( ψ ) = α ψ . The KdV hierarchy is a sequence of nonlinear differential operators D i : C → C such that for any i we have an analytic function g ( x , t ) and we define g t ( x ) to be g ( x , t ) and D i ( g t ) = d d t g t , then B g is independent of t [NEEDS CLARIFYING EXPLANATION].
The KdV hierarchy arises naturally as a statement of Huygens' principle for the D'Alembertian.
