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sine-Gordon equation

Sine–Gordon equation
Order	2nd
Solvable	Exactly solvable
The Cauchy problem	Solvable by inverse scattering transform

Contents 1 sine-Gordon equation 1.1 Definition 1.2 Solutions 1.2.1 Traveling wave 1.2.2 Topological soliton 1.3 Integrals of motion 1.4 Lax pair 1.5 The Cauchy problem 1.6 Applications and connections 1.7 References 1.8 External links

Definition

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

Solutions

Traveling wave

Let us look for solutions of the sine-Gordon equation

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

in the form of traveling wave

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

Then the sine-Gordon equation will take the form

$$\left(c_0^2-1\right)U_{\theta \theta }+\sin U=0.$$

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

$$\left(c_0^2-1\right)U_{\theta }^2+4{\sin }^2\frac{U}{2}=2A,$$

where $A$ is a constant of integration.

Under a condition $c_0^2>1$ and $0<A<2$ one obtains

$${\displaystyle \underset{\sin \frac{A}{2}}{\overset{\sin \frac{U}{2}}{\int }}\frac{dz}{\sqrt{\left(1-z^2\right)\left(1-k^{2}z\right)}}}=\sqrt{\frac{A}{2\left(c_0^2-1\right)}}\left(\theta -{\theta }_0\right),$$

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

$$-2\arcsin \sqrt{\frac{A}{2}}<U<2\arcsin \sqrt{\frac{A}{2}}.$$

Under a condition $c_0^2>1$ and $A>2$ solution of the equation

$$U\left(\theta \right)=\pm {\left[\frac{2}{c_0^2-1}\left(A-2{\sin }^2\frac{U}{2}\right)\right]}^{\frac{1}{2}}$$

gives spiral waves.

Under a condition $c_0^2<1$ and $0<A<2$ solution $U\left(\theta \right)$ is a periodic wave in the interval

$$\pi -2\arcsin \sqrt{\frac{A}{2}}<U<\pi +2\arcsin \sqrt{\frac{A}{2}}.$$

Under a condition $c_0^2<1$ and $A<0$ we have the following equation for $U\left(\theta \right)$

$$U\left(\theta \right)=\pm {\left[\frac{2}{1-c_0^2}\left(\left|A\right|+2{\sin }^2\frac{U}{2}\right)\right]}^{\frac{1}{2}}.$$

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 $U(\theta )$ is given by the following formula

$$tg\left(\frac{U}{4}\right)=\pm \exp \left[\pm \frac{\theta -{\theta }_0}{\sqrt{1-c_0^2}}\right]$$

and corresponds to two solitary waves.

In the another limiting case of $A=2$ and $c_0^2>1$ solution $U(\theta )$ is given by the following formula

$$tg\left(\frac{U+\pi }{4}\right)=\pm \exp \left[\pm \frac{\theta -{\theta }_0}{\sqrt{c_0^2-1}}\right]$$

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

Topological soliton

Kink:

$$\phi \left(x,t\right)=4arctg\left(\exp \left(\frac{x-c_{0}t+{\phi }_0}{\sqrt{1-c_0^2}}\right)\right)$$

Antikink:

$$\phi \left(x,t\right)=4arctg\left(-\exp \left(-\frac{x-c_{0}t+{\phi }_0}{\sqrt{1-c_0^2}}\right)\right)$$

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

$$E_1=\frac{1}{2\pi }{\displaystyle \underset{-\infty }{\overset{+\infty }{\int }}{\phi }_{x}dx}$$

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

$$u_{xt}=\sin u$$

is

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

$${\Psi }_t=\widehat{Q}\Psi =\left(\begin{array}{cc}-\frac{i}{4}\cos u& -\frac{i}{4}\sin u\\ -\frac{i}{4}\sin u& \frac{i}{4}\cos u\end{array}\right)\Psi ,$$

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

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) // 2^nd 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