44
8 Examples
ϕ(x, t) = ∓μ tanh(
λ/2μ(x − vt)/
1 − v 2 ), v
2
< 1.
(8.11)
The energy–momentum density is localized around the point x = vt (“center of
mass” of the kink), which moves with velocity v (moving kink solution).
Clearly, ϕ(x, t) − ϕ(x, 0) ∈ C
◦
(H
1
, R), i.e. ϕ(x, t) defines a sector. Furthermore ϕ(x, 0) ∈ L
∞
(R), ψ(x, 0) = ˙
ϕ(x, 0) ∈ L
2
(R) and obviously condition (b)
of Theorem 5.2 is satisfied; then (ϕ(x, 0), ψ(x, 0)) defines a Hilbert space sector.
This implies the stability of such solutions under H
1
⊕ L
2 perturbations (see
Chap. 5). This settles the problem of stability of the kink sector and, thanks to
Theorem 5.2, the proof does not involve expansions or linearizations.
37 It is not
difficult to see that the static kink solution, corresponding to v = 0 in (8.11), belongs
to the same sector defined by the corresponding moving kink solution.
From a physical point of view (energy–momentum localization and stability), the
kink is a candidate to describe particle-like excitations associated with (8.2). In fact,
in the past this feature has motivated attempts to use such kink-like solution as a
non-perturbative semi-classical approach to the descriptions of baryons in quantum
field theory.
38
2) The sine-Gordon Equation
The sine-Gordon equation is
ϕ = −g sin ϕ,
(8.12)
where ϕ(x, t) is a scalar field in one space dimension. It is of great interest in various
fields of theoretical physics, like propagation of crystal dislocation, magnetic flux
in Josephson lines, Bloch wall motion in magnetic crystals, fermion bosonization in
the Thirring model of elementary particle interactions, etc.
39
i) Static solutions
The simplest static solutions are the constants
ϕ = πn, n ∈ Z.
(8.13)
They all define disjoint Hilbert space sectors and for n even correspond to absolute
minima of the potential
U = g(1 − cos ϕ).
(8.14)
In this case the energy is bounded below in the corresponding Hilbert sectors.
37 See, e.g. R. Rajaraman, Phys. Rep. 21, 227 (1975), especially Sect. 3.2.
38 R.F. Dashen, B. Hasslacher and A. Neveu, Phys. Rev. D10, 4130 (1974); J. Goldstone and R.
Jackiw, Phys. Rev. D11, 1486 (1975); for a rich collection of important papers see C. Rebbi and G.
Soliani, Solitons and Particles, World Scientific 1984.
39 See A. Barone, F. Esposito and C.J. Magee, Theory and Applications of the Sine-Gordon Equation,
in Riv. Nuovo Cim. 1, 227 (1971); A.C. Scott, F.Y. Chiu, and D.W. Mclaughlin, Proc.I.E.E.E. 61,
1443 (1973); G.B. Whitham, Linear and Non-Linear Waves, J. Wiley 1974; S. Coleman, Phys. Rev.
D11, 2088 (1975); S. Coleman, Aspects of Symmetry, Cambridge Univ. Press 1985; J. Fröhlich, in
Invariant Wave Equations, G. Velo and A.S. Wightman eds., Springer-Verlag 1977.
8 Examples
ϕ(x, t) = ∓μ tanh(
λ/2μ(x − vt)/
1 − v 2 ), v
2
< 1.
(8.11)
The energy–momentum density is localized around the point x = vt (“center of
mass” of the kink), which moves with velocity v (moving kink solution).
Clearly, ϕ(x, t) − ϕ(x, 0) ∈ C
◦
(H
1
, R), i.e. ϕ(x, t) defines a sector. Furthermore ϕ(x, 0) ∈ L
∞
(R), ψ(x, 0) = ˙
ϕ(x, 0) ∈ L
2
(R) and obviously condition (b)
of Theorem 5.2 is satisfied; then (ϕ(x, 0), ψ(x, 0)) defines a Hilbert space sector.
This implies the stability of such solutions under H
1
⊕ L
2 perturbations (see
Chap. 5). This settles the problem of stability of the kink sector and, thanks to
Theorem 5.2, the proof does not involve expansions or linearizations.
37 It is not
difficult to see that the static kink solution, corresponding to v = 0 in (8.11), belongs
to the same sector defined by the corresponding moving kink solution.
From a physical point of view (energy–momentum localization and stability), the
kink is a candidate to describe particle-like excitations associated with (8.2). In fact,
in the past this feature has motivated attempts to use such kink-like solution as a
non-perturbative semi-classical approach to the descriptions of baryons in quantum
field theory.
38
2) The sine-Gordon Equation
The sine-Gordon equation is
ϕ = −g sin ϕ,
(8.12)
where ϕ(x, t) is a scalar field in one space dimension. It is of great interest in various
fields of theoretical physics, like propagation of crystal dislocation, magnetic flux
in Josephson lines, Bloch wall motion in magnetic crystals, fermion bosonization in
the Thirring model of elementary particle interactions, etc.
39
i) Static solutions
The simplest static solutions are the constants
ϕ = πn, n ∈ Z.
(8.13)
They all define disjoint Hilbert space sectors and for n even correspond to absolute
minima of the potential
U = g(1 − cos ϕ).
(8.14)
In this case the energy is bounded below in the corresponding Hilbert sectors.
37 See, e.g. R. Rajaraman, Phys. Rep. 21, 227 (1975), especially Sect. 3.2.
38 R.F. Dashen, B. Hasslacher and A. Neveu, Phys. Rev. D10, 4130 (1974); J. Goldstone and R.
Jackiw, Phys. Rev. D11, 1486 (1975); for a rich collection of important papers see C. Rebbi and G.
Soliani, Solitons and Particles, World Scientific 1984.
39 See A. Barone, F. Esposito and C.J. Magee, Theory and Applications of the Sine-Gordon Equation,
in Riv. Nuovo Cim. 1, 227 (1971); A.C. Scott, F.Y. Chiu, and D.W. Mclaughlin, Proc.I.E.E.E. 61,
1443 (1973); G.B. Whitham, Linear and Non-Linear Waves, J. Wiley 1974; S. Coleman, Phys. Rev.
D11, 2088 (1975); S. Coleman, Aspects of Symmetry, Cambridge Univ. Press 1985; J. Fröhlich, in
Invariant Wave Equations, G. Velo and A.S. Wightman eds., Springer-Verlag 1977.