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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

Contents

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 c 1 is a constant of integration.

Let us introduce new unknown function q(z) and parameter 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 q(z)

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

Solitons

Let us suppose c 1 =0, c 0 = k 2 , c 2 =0 and therefore α=δ=k and 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 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 α the second Painlevé equation possesses rational solutions; for half-integer 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.

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) // 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

External links

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

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