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Modified Korteweg-de Vries hierarchy (mKdV hierarchy)

Modified Korteweg-de Vries hierarchy
SolvableExactly solvable
The Cauchy problemSolvable by inverse scattering transform

Contents

Definition

The modified Korteweg-de Vries equation is the first equation of the so called modified Korteweg-de Vries hierarchy given by

u t + x ( x +2u ) L n [ u x u 2 ]=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 modified Korteweg-de Vries hierarchy are

u t 6 u 2 u x + u xxx =0 ,

u t 10 u 2 u xxx 40 u x u xx 10 u x 3 +30 u 4 u x + u xxxxx =0 ,

u t 14 u 2 u 5x 84u u x u 4x 140u u xx u xxx 126 u x 2 u xxx 182 u x u xx 2 + +70 u 4 u xxx +560 u 3 u x u xx +420 u 2 u x 3 140 u 6 u x + u 7x =0.

All equations of the modified Korteweg-de Vries hierarchy pass the Painlevé test, possess soliton solutions and are exactly solvable.

Derivation

The mKdV hierarchy is obtained from the KdV hierarchy

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

via the Miura transformation

u= v x v 2

and can be written as

v t + x ( x +2v ) L n [ v x v 2 ]=0.

Lax pair

For each integer n1 , the Lax pair for that nth equation of the mKdV hierarchy

u t + x ( x +2u ) L n [ u x u 2 ]=0

is (Clarkson, et al., 2006)

Ψ x = P ^ Ψ=( iλ u u iλ )Ψ,

Ψ t = Q ^ Ψ=( j=0 2n+1 A j ( iλ ) j j=0 2n B j ( iλ ) j j=0 2n C j ( iλ ) j j=0 2n+1 A j ( iλ ) j )Ψ,

where

Ψ=( ψ 1 (x,t) ψ 2 (x,t) ),

A 2n+1 = 4 n , A 2k =0,k=0,,n,

A 2k+1 = 4 k+1 2 { L nk [ u x u 2 ] x ( x +2u ) L nk1 [ u x u 2 ] },k=0,,n1,

B 2k+1 = 4 k+1 2 x ( x +2u ) L nk1 [ u x u 2 ],k=0,,n1,

B 2k = 4 k ( x +2u ) L nk [ u x u 2 ],k=0,,n,

C 2k+1 = B 2k+1 ,k=0,,n1,

C 2k = B 2k ,k=0,,n.

See also

Modified Korteweg-de Vries equation

Korteweg-de Vries hierarchy

References

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