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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

111Modified Korteweg-de Vries equation (mKdV) 221.1Definition 321.2Solutions 431.2.1Traveling wave 531.2.2Solitons 631.2.3Self-similar solutions 721.3Lax pair 821.4The Cauchy problem 921.5Applications and connections 1021.6See also 1121.7References 1221.8External links

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( x,t )=y( z ),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 inline c 1 ]]> is a constant of integration.

Let us introduce new unknown function inline q(z) ]]> and parameter inline m ]]> in the following way y( z )= β( δα ) q 2 ( z )+α( βδ ) ( δα ) q 2 ( z )+βδ , ]]>

m 2 = ( βγ )( αδ ) ( αγ )( βδ ) . ]]>

So we obtain the following solution for inline q(z) ]]>

q( z )=sn( 1 2 ( βδ )( αγ ) ( z z 0 ),m ). ]]>

Solitons

Let us suppose inline c 1 =0, ]]> inline c 0 = k 2 , ]]> inline c 2 =0 ]]> and therefore inline α=δ=k ]]> and inline β=γ=0 . ]]>

Then we obtain y( z )= k 2s h 2 ( 1 2 kz+ φ 0 )+1 . ]]>

We obtain one-soliton solution of the modified Korteweg-de Vries equation in the form u( x,t )=± k ch( kx k 3 t+ χ 0 ) , ]]> where inline χ 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( x,t )= 1 ( 3t ) 1 3 w( z ),z= x ( 3t ) 1 3 . ]]>

We then obtain reduction of the mKdV equation to the following 2nd order ordinary differential equation w zz 2 w 3 zw+α=0. ]]>

This is the second Painlevé equation. Solutions of this equation generally can be expressed via the so-called Painlevé transcendents. For integer inline α ]]> the second Painlevé equation possesses rational solutions; for half-integer inline α ]]> 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 Ψ x = P ^ Ψ=( iλ v v iλ )Ψ, ]]> Ψ t = Q ^ Ψ=( 4i λ 3 2iλ v 2 4 λ 2 v+2iλ v x v xx +2 v 3 4 λ 2 v2iλ v x v xx +2 v 3 4i λ 3 +2iλ v 2 )Ψ, ]]> where Ψ=( ψ 1 ( x,t ) ψ 2 ( x,t ) ). ]]>

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

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) // 2nd 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