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Modified Kortewegde Vries equation (mKdV)
Modified Kortewegde Vries equation  

Order  3rd 
Solvable  Exactly solvable 
The Cauchy problem  Solvable by inverse scattering transform 
Hierarchy  Modified Kortewegde Vries hierarchy 

Definition
$${u}_{t}6{u}^{2}{u}_{x}+{u}_{xxx}=0$$
Solutions
Traveling wave
Let us show that solutions of the modified Kortewegde Vries equation$${u}_{t}6{u}^{2}{u}_{x}+{u}_{xxx}=0$$
in traveling wave variables$$u\left(x,t\right)=y\left(z\right),\text{\hspace{1em}}z=x{c}_{0}t$$
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.$${y}_{z}^{2}={y}^{4}+{c}_{0}{y}^{2}{c}_{1}y+{c}_{2}.$$
Let $\alpha ,\beta ,\gamma $ and $\delta $ $(\delta \le \gamma \le \beta \le \alpha )$ be the solutions of the 4^{th} order algebraic equation$${y}^{4}{c}_{0}{y}^{2}+{c}_{1}y{c}_{2}=0.$$
$$y\left(z\right)=\frac{\beta \left(\delta \alpha \right){q}^{2}\left(z\right)+\alpha \left(\beta \delta \right)}{\left(\delta \alpha \right){q}^{2}\left(z\right)+\beta \delta},$$
$${m}^{2}=\frac{\left(\beta \gamma \right)\left(\alpha \delta \right)}{\left(\alpha \gamma \right)\left(\beta \delta \right)}.$$
$$\underset{0}{\overset{q}{\int}}\frac{dx}{\sqrt{\left(1{x}^{2}\right)\left(1{m}^{2}{x}^{2}\right)}}}=\frac{1}{2}\sqrt{\left(\beta \delta \right)\left(\alpha \gamma \right)}\left(z{z}_{0}\right).$$
There is an elliptic integral of the first kind in the left hand side of the latter equation.$$q\left(z\right)=sn\left(\frac{1}{2}\sqrt{\left(\beta \delta \right)\left(\alpha \gamma \right)}\left(z{z}_{0}\right),m\right).$$
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$$y\left(z\right)=\frac{k}{2s{h}^{2}\left(\frac{1}{2}kz+{\phi}_{0}\right)+1}.$$
$$c{h}^{2}\left(kz\right)s{h}^{2}\left(kz\right)=\mathrm{1,}$$
$$c{h}^{2}\left(kz\right)+s{h}^{2}\left(kz\right)=ch\left(2kz\right),$$
$$u\left(x,t\right)=\pm \frac{k}{ch\left(kx{k}^{3}t+{\chi}_{0}\right)}\text{},$$
where ${\chi}_{0}$ is an arbitrary constant. Thus we see that the modified Kortewegde Vries equation as well as the Kortewegde Vries equation also possesses soliton solutions.Selfsimilar solutions
Let us look for solutions of the mKdV equation$${u}_{t}6{u}^{2}{u}_{x}+{u}_{xxx}=0$$
in the form$$u\left(x,t\right)=\frac{1}{{\left(3t\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{1ex}{$3$}\right.}}w\left(z\right)\text{},\text{\hspace{1em}}z=\frac{x}{{\left(3t\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{1ex}{$3$}\right.}}.$$
We then obtain reduction of the mKdV equation to the following 2^{nd} order ordinary differential equation$${w}_{zz}2{w}^{3}zw+\alpha =0.$$
This is the second Painlevé equation. Solutions of this equation generally can be expressed via the socalled Painlevé transcendents. For integer $\alpha $ the second Painlevé equation possesses rational solutions; for halfinteger $\alpha $ its solutions can be expressed via Airy functions. Thus we've found out that selfsimilar 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$${\Psi}_{x}=\widehat{P}\Psi =\left(\begin{array}{cc}i\lambda & v\\ v& i\lambda \end{array}\right)\mathrm{\Psi ,}$$
$${\Psi}_{t}=\widehat{Q}\Psi =\left(\begin{array}{cc}4i{\lambda}^{3}2i\lambda {v}^{2}& 4{\lambda}^{2}v+2i\lambda {v}_{x}{v}_{xx}+2{v}^{3}\\ 4{\lambda}^{2}v2i\lambda {v}_{x}{v}_{xx}+2{v}^{3}& 4i{\lambda}^{3}+2i\lambda {v}^{2}\end{array}\right)\Psi ,$$
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 Kortewegde Vries equation can be solved using the inverse scattering transform.Applications and connections
The modified Kortewegde Vries equation is used as a model in fields as wide as largeamplitude internal waves in the ocean
See also
Kortewegde Vries equation Modified Kortewegde Vries hierarchyReferences
 Ablowitz M.J., Kaup D.J., Newell A.C. and Segur H. The inverse scattering transformFourier analysis for nonlinear problems // Stud. Appl. Math., 1974. 53:4. Pp.249315.
 Clarkson P.A., Joshi N., Mazzocco M. The Lax pair for the mKdV Hierarchy // Séminaires et Congrès, 2006. 14. Pp.5364.
 Conte R. (editor) The Painlevé property: one century later // CRM series in mathematical physics, Springer, New York, 1999. — 810 p.
 Kudryashov N.A. Analytical theory of nonlinear differential equations (in Russian) // 2^{nd} ed., Institute of Computer Investigation, MoscowIzhevsk, 2004. — 360 p.
 Kudryashov N.A. Methods of nonlinear mathematical physics (in Russian) // Moscow Engineering Physics Institute, Moscow, 2008. — 352 p.
 Newell A.C. Solitons in mathematics and physics // Society for Industrial and Applied Mathematics, Pennsylvania, 1985. — 244 p.
 Musette M., Conte R. The twosingularmanifold method: I. Modified Kortewegde Vries and sineGordon equations // J. Phys. A: Math. Gen., 1994. 27:11. Pp.38953913. DOI:10.1088/03054470/27/11/036
 Ulam S. Adventures of a mathematician (autobiography) // Scribner, New York, 1976. — 317 p.
 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.240243. DOI:10.1103/PhysRevLett.15.240
External links
 Modified Kortewegde Vries equation in EqWorld, the world of mathematical equations