Student Seminar: Classical and Quantum Integrable Systems

Here ɛ 0 = ±1. This solution can be interpreted in terms of relativistic particle moving
with the velocity v. The field φ(x, t) has an important characteristic – topological
charge
Q = β ∫
dx ∂φ
2π ∂x = β (φ(∞) − φ(−∞)) .
2π
On our solutions we have
Q = β 2π
(
4
)
− ɛ 0 ( π β 2 − 0) = −ɛ 0 ,
because arctan(±∞) = ± π and arctan 0 = 0. In addition to the continuous parameters
v and x 0 , the soliton of the SG model has another important discrete
2
characteristic – topological charge Q = −ɛ 0 . Solutions with Q = 1 are called solitons
(kinks), while solutions with Q = −1 are called ani-solitons (anti-kinks).
Here we provide another useful representation for the SG soliton, namely
m(x−vt−x
2i
φ(x, t) = ɛ 0
β log 1 + ie 0 )
√
1−v 2
1 − ie
m(x−vt−x 0
.
)
√
1−v 2
Indeed, looking at the solution we found we see that we can cast it in the form arctan α = z ≡
− β
m(x−vt−x
4ɛ 0
φ(x, t) or α = tan z = −i e2iz −1
e 2iz +1 , where α = e 0 )
√
1−v 2 . From here z = 1 1+iα
2i
log
1−iα
and the
announced formula follows.
Remark. The stability of solitons stems from the delicate balance of ”nonlinearity”
and ”dispersion” in the model equations. Nonlinearity drives a solitary wave to
concentrate further; dispersion is the effect to spread such a localized wave. If one
of these two competing effects is lost, solitons become unstable and, eventually,
cease to exist. In this respect, solitons are completely different from ”linear waves”
like sinusoidal waves. In fact, sinusoidal waves are rather unstable in some model
equations of soliton phenomena.
Sine-Gordon model has even more sophisticated solutions. Consider the following
( )
φ(x, t) = 4 β arctan ω sin mω1 (t−vx)
√
2
1−v 2
+ φ 0
) .
ω 1 cosh
(
mω2 (x−vt−x 0 )
√
1−v 2
This is solution of the SG model which is called a double-soliton or breaser. Except
motion with velocity v corresponding to a relativistic particle the breaser oscillates
both in space and in time with frequencies √ mvω 1
1−v 2
and √ mω 1
1−v 2
respectively. The parameter
φ 0 plays a role of the initial phase. In particular, if v = 0 the breaser is a
time-periodic solution of the SG equation. It has zero topological charge and can be
interpreted as the bound state of the soliton and anti-soliton.
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