# Modified Korteweg-de Vries equation (mKdV)

Modified Korteweg-de Vries equation
Order3rd
SolvableExactly solvable
The Cauchy problemSolvable by inverse scattering transform
HierarchyModified Korteweg-de Vries hierarchy

## Definition

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

## Solutions

### Traveling wave

Let us show that solutions of the modified Korteweg-de Vries equation

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

in traveling wave variables

$u\left(x,t\right)=y\left(z\right),\text{ }z=x-{c}_{0}t$

are expressed via Jacobi elliptic function. Using introduced variables one can obtain reduction of the mKdV equation to the following second order ordinary differential equation

${y}_{zz}=-2{y}^{3}+{c}_{0}y-{c}_{1},$

where ${c}_{1}$ is a constant of integration.

Let us introduce new unknown function $q\left(z\right)$ and parameter $m$ in the following way

$y\left(z\right)=\frac{\beta \left(\delta -\alpha \right){q}^{2}\left(z\right)+\alpha \left(\beta -\delta \right)}{\left(\delta -\alpha \right){q}^{2}\left(z\right)+\beta -\delta },$

${m}^{2}=\frac{\left(\beta -\gamma \right)\left(\alpha -\delta \right)}{\left(\alpha -\gamma \right)\left(\beta -\delta \right)}.$

So we obtain the following solution for $q\left(z\right)$

$q\left(z\right)=sn\left(\frac{1}{2}\sqrt{\left(\beta -\delta \right)\left(\alpha -\gamma \right)}\left(z-{z}_{0}\right),m\right).$

### Solitons

Let us suppose ${c}_{1}=0,$ ${c}_{0}={k}^{2},$ ${c}_{2}=0$ and therefore $\alpha =-\delta =k$ and $\beta =\gamma =0.$

Then we obtain

$y\left(z\right)=-\frac{k}{2s{h}^{2}\left(\frac{1}{2}kz+{\phi }_{0}\right)+1}.$

We obtain one-soliton solution of the modified Korteweg-de Vries equation in the form

$u\left(x,t\right)=±\frac{k}{ch\left(kx-{k}^{3}t+{\chi }_{0}\right)}\text{​},$

where ${\chi }_{0}$ is an arbitrary constant.

Thus we see that the modified Korteweg-de Vries equation as well as the Korteweg-de Vries equation also possesses soliton solutions.

### Self-similar solutions

Let us look for solutions of the mKdV equation

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

in the form

$u\left(x,t\right)=\frac{1}{{\left(3t\right)}^{1}{3}}}w\left(z\right)\text{​},\text{ }z=\frac{x}{{\left(3t\right)}^{1}{3}}}.$

We then obtain reduction of the mKdV equation to the following 2nd order ordinary differential equation

${w}_{zz}-2{w}^{3}-zw+\alpha =0.$

This is the second Painlevé equation. Solutions of this equation generally can be expressed via the so-called Painlevé transcendents. For integer $\alpha$ the second Painlevé equation possesses rational solutions; for half-integer $\alpha$ its solutions can be expressed via Airy functions.

Thus we've found out that self-similar solutions of the mKdV equation can be expressed via solutions of the second Painlevé equation.

## Lax pair

The well known Lax pair for the mKdV equation

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

is

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

${\Psi }_{t}=\stackrel{^}{Q}\Psi =\left(\begin{array}{cc}-4i{\lambda }^{3}-2i\lambda {v}^{2}& 4{\lambda }^{2}v+2i\lambda {v}_{x}-{v}_{xx}+2{v}^{3}\\ 4{\lambda }^{2}v-2i\lambda {v}_{x}-{v}_{xx}+2{v}^{3}& 4i{\lambda }^{3}+2i\lambda {v}^{2}\end{array}\right)\Psi ,$

where

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

This Lax pair was first given by Ablowitz, Kaup, Newell, and Segur (Ablowitz, et al., 1974).

## The Cauchy problem

The Cauchy problem for the modified Korteweg-de Vries equation can be solved using the inverse scattering transform.

## Applications and connections

The modified Korteweg-de Vries equation is used as a model in fields as wide as
• large-amplitude internal waves in the ocean
and so forth.

Korteweg-de Vries equation

Modified Korteweg-de Vries hierarchy

## References

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