# sine-Gordon equation

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

## Definition

${\phi }_{tt}-{\phi }_{xx}+\mathrm{sin}\phi =0\text{ }or\text{ }{u}_{\xi \tau }=\mathrm{sin}u$

## Solutions

### Traveling wave

Let us look for solutions of the sine-Gordon equation

${\phi }_{tt}-{\phi }_{xx}=\mathrm{sin}\phi$

in the form of traveling wave

$\phi \left(x,t\right)=U\left(\theta \right),\text{ }\theta =x-{c}_{0}t.$

Then the sine-Gordon equation will take the form

$\left({c}_{0}^{2}-1\right){U}_{\theta \theta }+\mathrm{sin}U=0.$

Multyplying the latter equation by ${U}_{\theta }$ and integrating with respect to $\theta$ one obtains

$\left({c}_{0}^{2}-1\right){U}_{\theta }^{2}+4{\mathrm{sin}}^{2}\frac{U}{2}=2A,$

where $A$ is a constant of integration.

Under a condition ${c}_{0}^{2}>1$ and $0 one obtains

$\underset{\mathrm{sin}\frac{A}{2}}{\overset{\mathrm{sin}\frac{U}{2}}{\int }}\frac{dz}{\sqrt{\left(1-{z}^{2}\right)\left(1-{k}^{2}z\right)}}=\sqrt{\frac{A}{2\left({c}_{0}^{2}-1\right)}}\left(\theta -{\theta }_{0}\right),$

where ${k}^{2}=\frac{2}{A}.$

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

$-2\mathrm{arcsin}\sqrt{\frac{A}{2}}

Under a condition ${c}_{0}^{2}>1$ and $A>2$ solution of the equation

$U\left(\theta \right)=±{\left[\frac{2}{{c}_{0}^{2}-1}\left(A-2{\mathrm{sin}}^{2}\frac{U}{2}\right)\right]}^{\frac{1}{2}}$

gives spiral waves.

Under a condition ${c}_{0}^{2}<1$ and $0 solution $U\left(\theta \right)$ is a periodic wave in the interval

$\pi -2\mathrm{arcsin}\sqrt{\frac{A}{2}}

Under a condition ${c}_{0}^{2}<1$ and $A<0$ we have the following equation for $U\left(\theta \right)$

$U\left(\theta \right)=±{\left[\frac{2}{1-{c}_{0}^{2}}\left(|A|+2{\mathrm{sin}}^{2}\frac{U}{2}\right)\right]}^{\frac{1}{2}}.$

The latter equation has solutions in form of spiral waves for monotonous change of $U\left(\theta \right).$

In a limiting case of $A=0$ and ${c}_{0}^{2}<1$ solution $U\left(\theta \right)$ is given by the following formula

$tg\left(\frac{U}{4}\right)=±\mathrm{exp}\left[±\frac{\theta -{\theta }_{0}}{\sqrt{1-{c}_{0}^{2}}}\right]$

and corresponds to two solitary waves.

In the another limiting case of $A=2$ and ${c}_{0}^{2}>1$ solution $U\left(\theta \right)$ is given by the following formula

$tg\left(\frac{U+\pi }{4}\right)=±\mathrm{exp}\left[±\frac{\theta -{\theta }_{0}}{\sqrt{{c}_{0}^{2}-1}}\right]$

and describes a shock-wave transition between $-\pi$ and $\pi$.

### Topological soliton

Kink:

$\phi \left(x,t\right)=4arctg\left(\mathrm{exp}\left(\frac{x-{c}_{0}t+{\phi }_{0}}{\sqrt{1-{c}_{0}^{2}}}\right)\right)$

Antikink:

$\phi \left(x,t\right)=4arctg\left(-\mathrm{exp}\left(-\frac{x-{c}_{0}t+{\phi }_{0}}{\sqrt{1-{c}_{0}^{2}}}\right)\right)$

Kink and antikink are examples of the topological solitons because ${\phi }_{x}$ has the form of solitary wave.

## Integrals of motion

The sine-Gordon equation has conserved quantity

${E}_{1}=\frac{1}{2\pi }\underset{-\infty }{\overset{+\infty }{\int }}{\phi }_{x}dx$

which equals integer number. This conservation law is called topological charge of solution $\phi \left(x,t\right)$. 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

${u}_{xt}=\mathrm{sin}u$

is

${\Psi }_{x}=\stackrel{^}{P}\Psi =\left(\begin{array}{cc}i\lambda & -\frac{1}{2}{u}_{x}\\ \frac{1}{2}{u}_{x}& -i\lambda \end{array}\right)\Psi ,$

${\Psi }_{t}=\stackrel{^}{Q}\Psi =\left(\begin{array}{cc}-\frac{i}{4}\mathrm{cos}u& -\frac{i}{4}\mathrm{sin}u\\ -\frac{i}{4}\mathrm{sin}u& \frac{i}{4}\mathrm{cos}u\end{array}\right)\Psi ,$

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 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

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