Student Seminar: Classical and Quantum Integrable Systems

4.2.2 Sine-Gordon cnoidal wave and soliton
Consider the Sine-Gordon equation
φ tt − φ xx + m2
β
sin βφ = 0 ,
where we assume that the functions φ(x, t) and φ(x, t) + 2π/β are assumed to be
equivalent. Make an ansatz
φ(x, t) = φ(x − vt)
which leads to
This can be integrated once
C = v2 − 1
φ 2 x − m2
2
(v 2 − 1)φ xx + m2
β
β cos βφ = v2 − 1
2 2
sin βφ = 0 .
φ 2 x + 2m2
β 2
βφ
sin2
2 − m2
β . 2
where C is an integration constant. This is nothing else as the conservation law of
energy for the mathematical pendulum in the gravitational field of the Earth! We
further bring equation to the form
φ 2 x = 2 (C + m2
v 2 − 1 β − 2m2
2 β 2
βφ sin2 2
As in the case of the pendulum we make a substitution y = sin βφ
2
(
C +
m 2
(y ′ ) 2 = m2
(v 2 − 1) (1 − y2 )
)
β 2
− y 2 .
2m 2
β 2
)
. (4.1)
which gives
This leads to solutions in terms of elliptic functions which are analogous to the cnoidal
waves of the KdV equation. However, as we know the pendulum has three phases
of motion: oscillatory (elliptic solution), rotatory (elliptic solution) and motion with
an infinite period. The later solution is precisely the one that would correspond to
the Sine-Gordon soliton we are interested in. Assuming v 2 < 1 we see 7 that such
a solution would arise from (4.1) if we take C = − m2 . In this case equation (4.1)
β 2
reduces to
2m
φ x =
β √ βφ
sin
1 − v2 2 .
This can be integrated to 8
4
( m(x − vt −
φ(x, t) = −ɛ 0
β arctan exp x0 )
)
√ .
1 − v
2
7 Restoring the speed of light c this condition for the velocity becomes v 2 < c 2 , i.e., the center
of mass of the soliton cannot propagate faster than light.
8 From the equation above we see that if φ(x, t) is a solution then −φ(x, t) is also a solution.
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