# Modified Korteweg-de Vries hierarchy (mKdV hierarchy)

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

## Definition

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

${u}_{t}+\frac{\partial }{\partial x}\left(\frac{\partial }{\partial x}+2u\right){L}_{n}\left[{u}_{x}-{u}^{2}\right]=0,$

where $L$ is Lenard operator defined by

$\frac{\partial }{\partial x}{L}_{n+1}\left[u\right]=\left(\frac{{\partial }^{3}}{\partial {x}^{3}}+4u\frac{\partial }{\partial x}+2\frac{\partial u}{\partial x}\right){L}_{n}\left[u\right],\text{ }{L}_{0}\left[u\right]=\frac{1}{2}.$

There are four first operators

${L}_{1}\left[u\right]=u,$

${L}_{2}\left[u\right]={u}_{xx}+3{u}^{2},$

${L}_{3}\left[u\right]={u}_{xxxx}+10u{u}_{xx}+5{u}_{x}^{2}+10{u}^{3},$

${L}_{4}\left[u\right]={u}_{xxxxxx}+14u{u}_{xxxx}+28{u}_{x}{u}_{xxx}+21{u}_{xx}^{2}+70{u}^{2}{u}_{xx}+70u{u}_{x}^{2}+35{u}^{4}.$

Corresponding equations of the modified Korteweg-de Vries hierarchy are

${u}_{t}-6{u}^{2}{u}_{x}+{u}_{xxx}=0,$

${u}_{t}-10{u}^{2}{u}_{xxx}-40{u}_{x}{u}_{xx}-10{u}_{x}^{3}+30{u}^{4}{u}_{x}+{u}_{xxxxx}=0,$

$\begin{array}{c}{u}_{t}-14{u}^{2}{u}_{5x}-84u{u}_{x}{u}_{4x}-140u{u}_{xx}{u}_{xxx}-126{u}_{x}^{2}{u}_{xxx}-182{u}_{x}{u}_{xx}^{2}+\\ +70{u}^{4}{u}_{xxx}+560{u}^{3}{u}_{x}{u}_{xx}+420{u}^{2}{u}_{x}^{3}-140{u}^{6}{u}_{x}+{u}_{7x}=0.\end{array}$

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

${u}_{t}+\frac{\partial }{\partial x}{L}_{n+1}\left[u\right]=0$

via the Miura transformation

$u={v}_{x}-{v}^{2}$

and can be written as

${v}_{t}+\frac{\partial }{\partial x}\left(\frac{\partial }{\partial x}+2v\right){L}_{n}\left[{v}_{x}-{v}^{2}\right]=0.$

## Lax pair

For each integer $n\ge 1$, the Lax pair for that nth equation of the mKdV hierarchy

${u}_{t}+\frac{\partial }{\partial x}\left(\frac{\partial }{\partial x}+2u\right){L}_{n}\left[{u}_{x}-{u}^{2}\right]=0$

is (Clarkson, et al., 2006)

${\Psi }_{x}=\stackrel{^}{P}\Psi =\left(\begin{array}{cc}-i\lambda & u\\ u& i\lambda \end{array}\right)\Psi ,$

${\Psi }_{t}=\stackrel{^}{Q}\Psi =\left(\begin{array}{cc}\sum _{j=0}^{2n+1}{A}_{j}{\left(i\lambda \right)}^{j}& \sum _{j=0}^{2n}{B}_{j}{\left(i\lambda \right)}^{j}\\ \sum _{j=0}^{2n}{C}_{j}{\left(i\lambda \right)}^{j}& -\sum _{j=0}^{2n+1}{A}_{j}{\left(i\lambda \right)}^{j}\end{array}\right)\Psi ,$

where

$\Psi =\left(\begin{array}{c}{\psi }_{1}\left(x,t\right)\\ {\psi }_{2}\left(x,t\right)\end{array}\right),$

${A}_{2n+1}={4}^{n},\text{ }{A}_{2k}=0,\text{ }\forall k=0,\dots ,n,$

${A}_{2k+1}=\frac{{4}^{k+1}}{2}\left\{{L}_{n-k}\left[{u}_{x}-{u}^{2}\right]-\frac{\partial }{\partial x}\left(\frac{\partial }{\partial x}+2u\right){L}_{n-k-1}\left[{u}_{x}-{u}^{2}\right]\right\},\text{ }k=0,\dots ,n-1,$

${B}_{2k+1}=\frac{{4}^{k+1}}{2}\frac{\partial }{\partial x}\left(\frac{\partial }{\partial x}+2u\right){L}_{n-k-1}\left[{u}_{x}-{u}^{2}\right],\text{ }k=0,\dots ,n-1,$

${B}_{2k}=-{4}^{k}\left(\frac{\partial }{\partial x}+2u\right){L}_{n-k}\left[{u}_{x}-{u}^{2}\right],\text{ }k=0,\dots ,n,$

${C}_{2k+1}=-{B}_{2k+1},\text{ }k=0,\dots ,n-1,$

${C}_{2k}={B}_{2k},\text{ }k=0,\dots ,n.$

Modified Korteweg-de Vries equation

Korteweg-de Vries hierarchy

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