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Modified Korteweg-de Vries equation (mKdV)

Modified Korteweg-de Vries equation
Order	3rd
Solvable	Exactly solvable
The Cauchy problem	Solvable by inverse scattering transform
Hierarchy	Modified Korteweg-de Vries hierarchy

Contents 1 Modified Korteweg-de Vries equation (mKdV) 1.1 Definition 1.2 Solutions 1.2.1 Traveling wave 1.2.2 Solitons 1.2.3 Self-similar solutions 1.3 Lax pair 1.4 The Cauchy problem 1.5 Applications and connections 1.6 See also 1.7 References 1.8 External links

Definition

$$u_t-6u^2{u}_x+u_{xxx}=0$$

Solutions

Traveling wave

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

$$u_t-6u^2{u}_x+u_{xxx}=0$$

in traveling wave variables

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

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

$$y_{zz}=-2y^3+c_{0}y-c_1,$$

where $c_1$ is a constant of integration.

Let us introduce new unknown function $q(z)$ 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 $q(z)$

$$q\left(z\right)=sn\left(\frac{1}{2}\sqrt{\left(\beta -\delta \right)\left(\alpha -\gamma \right)}\left(z-z_0\right),m\right).$$

Solitons

Let us suppose $c_1=\mathrm{0,}$ $c_0=k^2,$ $c_2=0$ 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

$$u\left(x,t\right)=\pm \frac{k}{ch\left(kx-k^{3}t+{\chi }_0\right)}\text{},$$

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

$$u_t-6u^2{u}_x+u_{xxx}=0$$

in the form

$$u\left(x,t\right)=\frac{1}{{\left(3t\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}}w\left(z\right)\text{},z=\frac{x}{{\left(3t\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}}.$$

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

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

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

$$v_t-6v^2{v}_x+v_{xxx}=0$$

is

$${\Psi }_x=\widehat{P}\Psi =\left(\begin{array}{cc}-i\lambda & v\\ v& i\lambda \end{array}\right)\mathrm{\Psi ,}$$

$${\Psi }_t=\widehat{Q}\Psi =\left(\begin{array}{cc}-4i{\lambda }^3-2i\lambda v^2& 4{\lambda }^{2}v+2i\lambda v_x-v_{xx}+2v^3\\ 4{\lambda }^{2}v-2i\lambda v_x-v_{xx}+2v^3& 4i{\lambda }^3+2i\lambda v^2\end{array}\right)\Psi ,$$

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.

See also

Korteweg-de Vries equation

Modified Korteweg-de Vries hierarchy

References

1. Ablowitz M.J., Kaup D.J., Newell A.C. and Segur H. The inverse scattering transform-Fourier analysis for nonlinear problems // Stud. Appl. Math., 1974. 53:4. Pp.249-315.
2. Clarkson P.A., Joshi N., Mazzocco M. The Lax pair for the mKdV Hierarchy // Séminaires et Congrès, 2006. 14. Pp.53-64.
3. Conte R. (editor) The Painlevé property: one century later // CRM series in mathematical physics, Springer, New York, 1999. — 810 p.
4. Kudryashov N.A. Analytical theory of nonlinear differential equations (in Russian) // 2^nd ed., Institute of Computer Investigation, Moscow-Izhevsk, 2004. — 360 p.
5. Kudryashov N.A. Methods of nonlinear mathematical physics (in Russian) // Moscow Engineering Physics Institute, Moscow, 2008. — 352 p.
6. Newell A.C. Solitons in mathematics and physics // Society for Industrial and Applied Mathematics, Pennsylvania, 1985. — 244 p.
7. Musette M., Conte R. The two-singular-manifold method: I. Modified Korteweg-de Vries and sine-Gordon equations // J. Phys. A: Math. Gen., 1994. 27:11. Pp.3895-3913. DOI:10.1088/0305-4470/27/11/036
8. Ulam S. Adventures of a mathematician (autobiography) // Scribner, New York, 1976. — 317 p.
9. Zabusky N.J., Kruskal M.D. Interaction of "solitons" in a collisionless plasma and the recurrence of initial states // Phys. Rev. Lett., 1965. 15:6. Pp.240-243. DOI:10.1103/PhysRevLett.15.240

External links

1. Modified Korteweg-de Vries equation in EqWorld, the world of mathematical equations