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Korteweg-de Vries hierarchy (KdV hierarchy)

Korteweg-de Vries hierarchy
SolvableExactly solvable
The Cauchy problemSolvable by inverse scattering transform
First introducedLax, 1968



The Korteweg-de Vries equation is the first equation of the so-called Korteweg-de Vries hierarchy defined as

u t + x L n+1 [ u ]=0,

where L is Lenard operator defined by

x L n+1 [u]=( 3 x 3 +4u x +2 u x ) L n [u], L 0 [u]= 1 2 .

There are four first operators

L 1 [ u ]= u ,

L 2 [ u ]= u xx +3 u 2 ,

L 3 [ u ]= u xxxx +10u u xx +5 u x 2 +10 u 3 ,

L 4 [ u ]= u xxxxxx +14u u xxxx +28 u x u xxx +21 u xx 2 +70 u 2 u xx +70u u x 2 +35 u 4 .

Corresponding equations of the Korteweg-de Vries hierarchy are

u t + u x =0,

u t +6u u x + u xxx =0,

u t +10u u xxx +30 u 2 u x +20 u x u xx + u 5x =0,

u t +14u u 5x +70 u 2 u xxx +42 u x u 4x + +70 u xx u xxx +280u u x u xx +70 u x 3 +140 u 3 u x + u 7x =0.

All equations of the Korteweg-de Vries hierarchy pass the Painlevé test, possess soliton solutions and are exactly solvable. Also all equations has odd degrees that corresponds to the lack of dissipative terms.


In 1968 P.D. Lax solved problem of finding all partial differential equations for which the Cauchy problem can be solved by the inverse scattering transform with Schrödinger steady-state equation. He suggested to look for equation class representable as a system of two linear equations

ψ xx +( λ+u )ψ=0

ψ t =B( u, u x ,,λ )ψ2E( u, u x ,,λ ) ψ x ,

where B and E are functions of λ , u and derivatives form u by x which are to be found.

Lax pair

Lax pair for the nth equation of the Korteweg-de Vries hierarchy can be written as the following system of linear partial differential equations

ψ xx +( λ+u )ψ=0,

ψ t =( c( t )+ k=0 n ( 4λ ) nk L x k [ u ] )ψ2( k=0 n ( 4λ ) nk L k [ u ] ) ψ x .

See also

Korteweg-de Vries equation

Modified Korteweg-de Vries hierarchy


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