Sine-Gordon equation

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The sine-Gordon equation is a partial differential equation in two dimensions. For a function φ of two real variables, x and t, it is

(\Box + \sin)\phi = \phi_{tt}- \phi_{xx} + \sin\phi = 0.

Contents

[edit] Origin of the equation and name

The name is a pun on the Klein-Gordon equation, which is

(\Box + 1)\phi = \phi_{tt}- \phi_{xx} + \phi\ = 0.

The sine-Gordon equation is the Euler-Lagrange equation of the Lagrangian

\mathcal{L}_{\mathrm{sine-Gordon}}(\phi) := \frac{1}{2}(\phi_t^2 - \phi_x^2) + \cos\phi.

If you Taylor-expand the cosine

\cos(\phi) = \sum_{n=0}^\infty \frac{(-\phi ^2)^n}{(2n)!}

and put this into the Lagrangian you get the Klein-Gordon Lagrangian plus some higher order terms

\mathcal{L}_{\mathrm{sine-Gordon}}(\phi) - 1 = \frac{1}{2}(\phi_t^2 - \phi_x^2) - \frac{\phi^2}{2} + \sum_{n=2}^\infty \frac{(-\phi^2)^n}{(2n)!}
= 2\mathcal{L}_{\mathrm{Klein-Gordon}}(\phi) + \sum_{n=2}^\infty \frac{(-\phi^2)^n}{(2n)!}

The sine-Gordon equation has the following 1-soliton solutions:

\phi_{\mathrm{soliton}}(x, t) := 4 \arctan \exp[m \gamma (x - v t) + \delta]\,

where \gamma^2 = \frac{1}{1 - v^2}

The 1-soliton solution for which we have chosen the positive root for γ is called a kink, and represents a twist in the variable φ which takes the system from one solution φ = 0 to an adjacent with φ = 2π. The states φ = 0(mod2π) are known as vacuum states as they are constant solutions of zero energy. The 1-soliton solution in which we take the negative root for γ is called an antikink.

[edit] 1-soliton solutions

The 1-soliton solutions can be visualized with the use of the elastic ribbon sine-Gordon model as discussed by Dodd et al. (1982). Here we take a clockwise twist of the elastic ribbon to be a kink with topological charge \vartheta_{\textrm{K}}=+1. The alternative counterclockwise twist with topological charge \vartheta_{\textrm{AK}}=-1will be an antikink.

Traveling kink soliton represents propagating clockwise twist.
Traveling kink soliton represents propagating clockwise twist.
Traveling antikink soliton represents propagating counterclockwise twist.
Traveling antikink soliton represents propagating counterclockwise twist.

[edit] 2-soliton solutions

Multi-soliton solutions can be obtained with the Bäcklund transform. The 2-soliton solutions of the sine-Gordon equation show some of the characteristic features of the solitons. The traveling sine-Gordon kinks and/or antikinks pass through each other as if perfectly permeable, and the only observed effect is a phase shift. Since the colliding solitons recover their velocity and shape such kind of interaction is called an elastic collision.

Antikink-kink collision.
Antikink-kink collision.
Kink-kink collision.
Kink-kink collision.

Another interesting 2-soliton solutions arise from the possibility of coupled kink-antikink behaviour known as a breather. There are known three types of breathers: standing breather, traveling large amplitude breather, and traveling small amplitude breather.

Standing breather is a swinging in time coupled kink-antikink soliton.
Standing breather is a swinging in time coupled kink-antikink soliton.
Large amplitude moving breather.
Large amplitude moving breather.
Small amplitude moving breather - looks exotically but essentially has a breather envelope.
Small amplitude moving breather - looks exotically but essentially has a breather envelope.

[edit] 3-soliton solutions

3-soliton collisions between a traveling kink and a standing breather or a traveling antikink and a standing breather results in a phase shift of the standing breather. In the process of collision between a moving kink and a standing breather, the shift of the breather ΔB is given by:

\Delta_{B}=\frac{2\textrm{arctanh}\sqrt{(1-\omega^{2})(1-v_{\textrm{K}}^{2})}}{\sqrt{1-\omega^{2}}}

where vK is the velocity of the kink, and ω is the breather's frequency. If the old position of the standing breather is x0, after the collision the new position will be x0 + ΔB.

Moving kink-standing breather collision.
Moving kink-standing breather collision.
Moving antikink-standing breather collision.
Moving antikink-standing breather collision.

[edit] Mainardi-Codazzi equation

Another equation is also called the sine-Gordon equation:

\phi_{uv} = \sin\phi\,

where φ is again a function of two real variables u and v.

The last one is better known in the differential geometry of surfaces. There it is the Mainardi-Codazzi equation, i.e. the integrability condition, of a pseudospherical surface given in (arc-length) asymptotic line parameterization, where φ is the angle between the parameter lines. A pseudospherical surface is a surface of negative constant Gaussian curvature K = − 1.

This partial differential equation has solitons.

See also Bäcklund transform.

[edit] sinh-Gordon equation

The sinh-Gordon equation is given by

\phi_{tt}- \phi_{xx} = -\sinh\phi\,

This is the Euler-Lagrange equation of the Lagrangian

\mathcal{L}={1\over 2}(\phi_t^2-\phi_x^2)-\cosh\phi\,

[edit] External links

[edit] Bibliography

  • Dodd RK, Eilbeck JC, Gibbon JD, Morris HC. Solitons and Nonlinear Wave Equations. Academic Press, London, 1982.
  • Polyanin AD, Zaitsev VF. Handbook of Nonlinear Partial Differential Equations. Chapman & Hall/CRC Press, Boca Raton, 2004.
  • Rajaraman R. Solitons and instantons. North-Holland Personal Library, 1989.
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