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sine-Gordon equation

Sine–Gordon equation
Order2nd
SolvableExactly solvable
The Cauchy problemSolvable by inverse scattering transform

Contents

Definition

φ tt φ xx +sinφ=0 or u ξτ =sinu

Solutions

Traveling wave

Let us look for solutions of the sine-Gordon equation

φ tt φ xx =sinφ

in the form of traveling wave

φ( x,t )=U( θ ),θ=x c 0 t.

Then the sine-Gordon equation will take the form

( c 0 2 1 ) U θθ +sinU=0.

Multyplying the latter equation by U θ and integrating with respect to θ one obtains

( c 0 2 1 ) U θ 2 +4 sin 2 U 2 =2A,

where A is a constant of integration.

Under a condition c 0 2 >1 and 0<A<2 one obtains

sin A 2 sin U 2 dz ( 1 z 2 )( 1 k 2 z ) = A 2( c 0 2 1 ) ( θ θ 0 ),

where k 2 = 2 A .

There is an elliptic integral in the left hand side of the latter equation. In this case solution U( θ ) is a periodic and oscillating function near U=0 in the interval

2arcsin A 2 <U<2arcsin A 2 .

Under a condition c 0 2 >1 and A>2 solution of the equation

U( θ )=± [ 2 c 0 2 1 ( A2 sin 2 U 2 ) ] 1 2

gives spiral waves.

Under a condition c 0 2 <1 and 0<A<2 solution U( θ ) is a periodic wave in the interval

π2arcsin A 2 <U<π+2arcsin A 2 .

Under a condition c 0 2 <1 and A<0 we have the following equation for U( θ )

U( θ )=± [ 2 1 c 0 2 ( | A |+2 sin 2 U 2 ) ] 1 2 .

The latter equation has solutions in form of spiral waves for monotonous change of U( θ ).

In a limiting case of A=0 and c 0 2 <1 solution U(θ) is given by the following formula

tg( U 4 )=±exp[ ± θ θ 0 1 c 0 2 ]

and corresponds to two solitary waves.

In the another limiting case of A=2 and c 0 2 >1 solution U(θ) is given by the following formula

tg( U+π 4 )=±exp[ ± θ θ 0 c 0 2 1 ]

and describes a shock-wave transition between π and π .

Topological soliton

Kink:

φ( x,t )=4arctg( exp( x c 0 t+ φ 0 1 c 0 2 ) )

Antikink:

φ( x,t )=4arctg( exp( x c 0 t+ φ 0 1 c 0 2 ) )

Kink and antikink are examples of the topological solitons because φ x has the form of solitary wave.

Integrals of motion

The sine-Gordon equation has conserved quantity

E 1 = 1 2π + φ x dx

which equals integer number. This conservation law is called topological charge of solution φ(x,t) . For instance, topological charge of kink is 1, topological charge of antikink is -1. Topological solitons interact as particle and antiparticle resulting in zero-charge state.

Lax pair

The Lax pair for the sine-Gordon equation

u xt =sinu

is

Ψ x = P ^ Ψ=( iλ 1 2 u x 1 2 u x iλ )Ψ,

Ψ t = Q ^ Ψ=( i 4 cosu i 4 sinu i 4 sinu i 4 cosu )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 sine-Gordon equation can be solved using the inverse scattering transform.

Applications and connections

The sine-Gordon equation is used in fields as wide as:
  • differential geometry
  • dislocations in solids
  • self-induced transparency
and more.

References

  1. Ablowitz M.J., Clarkson P.A. Solitons, nonlinear evolution equations and inverse scattering // Cambridge University Press, Cambridge, 1991. — 532 p. DOI:10.2277/0521387302
  2. Ablowitz M.J., Kaup D.J., Newell A.C., Segur H. Nonlinear-evolution equations of physical significance // Phys. Rev. Lett., 1973. 31:2. Pp.125-127. DOI:10.1103/PhysRevLett.31.125
  3. Ablowitz M.J., Kaup D.J., Newell A.C., Segur H. The inverse scattering transform-Fourier analysis for nonlinear problems // Stud. Appl. Math., 1974. 53:4. Pp.249–315.
  4. Ablowitz M.J., Segur H. Solitons and the inverse scattering transform // Society for Industrial and Applied Mathematics (SIAM Studies in Applied Mathematics, No. 4), Philadelphia, PA, 1981. — 434 p.
  5. Kudryashov N.A. Analytical theory of nonlinear differential equations (in Russian) // 2nd ed., Institute of Computer Investigation, Moscow-Izhevsk, 2004. — 360 p.
  6. Kudryashov N.A. Methods of nonlinear mathematical physics (in Russian) // Moscow Engineering Physics Institute, Moscow, 2008. — 352 p.
  7. Maymistov A.I. Optical solitons (in Russian) // Soros Educational Journal, 1999. 11. Pp.97-102.
  8. McCall S.L., Hahn E.L. Self-induced transparency by pulsed coherent light // Phys. Rev. Lett., 1967. 18:21. Pp.908-911. DOI:10.1103/PhysRevLett.18.908
  9. 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
  10. Newell A.C. Solitons in mathematics and physics // Society for Industrial and Applied Mathematics, Pennsylvania, 1985. — 244 p.
  11. Novokshenov V.Y. Introduction to soliton theory (in Russian) // Institute of Computer Investigation, Moscow-Izhevsk, 2002. — 96 p.
  12. Perring J.K., Skyrme T.H.R. A model unified field equation // Nucl. Phys., 1962. 31. Pp.550-555. DOI:10.1016/0029-5582(62)90774-5
  13. Weiss J. Bäcklund transformation and the Painlevé property // J. Math. Phys., 1986. 27:5 Pp.1293–1305. DOI:10.1063/1.527134
  14. Weiss J. The sine-Gordon equations: complete and partial integrability // J. Math. Phys., 1984. 25:7 Pp.2226–2235. DOI:10.1063/1.526415
  15. Whitham G. Linear and nonlinear waves // Wiley-Interscience, 1999. — 660 p.

External links

  1. Sine-Gordon equation in Encyclopaedia of Mathematics
  2. Sine-Gordon equation in EqWorld, the world of mathematical equations
  3. Sine–Gordon equation in Wikipedia, the free encyclopedia
  4. Sine-Gordon equation in Wolfram MathWorld

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