1. Solitons in the sine-Gordon equation
1. Phenomenology
Pororoca tidal bore (mascaret in french ) on the Amazon
It is a phenomenon of sudden elevation of the water of a river or estuary caused by the wave of the tide during spring tides. It occurs in the mouth and the lower course of some rivers when the current is impeded by the flow of the tide. Invisible most of the time, it occurs at the time of the equinoxes. The most spectacular bores occur at the mouths of Qiantang (China), Severn (UK) and Amazon (Brazil).
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2. soliton
Soliton on the Scott Russell Aqueduct on the Union Canal near Heriot-Watt University, 12 July 1995.
Historical observation by John-Scott Russell
I was observing the motion of a boat which was rapidly drawn along a narrow channel by a pair of horses, when the boat suddenly stopped—not so the mass of water in the channel which it had put in motion; it accumulated round the prow of the vessel in a state of violent agitation, then suddenly leaving it behind, rolled forward with great velocity, assuming the form of a large solitary elevation, a rounded, smooth and well-defined heap of water, which continued its course along the channel apparently without change of form or diminution of speed. I followed it on horseback, and overtook it still rolling on at a rate of some eight or nine miles an hour [14 km/h], preserving its original figure some thirty feet [9 m] long and a foot to a foot and a half [300−450 mm] in height. Its height gradually diminished, and after a chase of one or two miles [2–3 km] I lost it in the windings of the channel. Such, in the month of August 1834, was my first chance interview with that singular and beautiful phenomenon which I have called the Wave of Translation.
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3. Solitons are observed in many different fields of science
A soliton wave off the coast of Hawaii
solitons in optical fibers
Experimental demonstration of a 2D discrete spatial soliton in an optically induced photonic lattice. (a) Input, (b) diffraction output without the lattice, (c) discrete diffraction at low nonlinearity and (d) discrete soliton formation at high nonlinearity. Top panels: 3D intensity plots; bottom panels: corresponding 2D transverse intensity patterns
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4. Solitons may occur in proteins[9] and DNA
Solitons may occur in proteins[9] and DNA.[10] They are related to the low-frequency collective motion in proteins and DNA.
Self-localized soliton wavefunction of trans-polyacetylene doped by a counter ion (red ball). The red and blue lobes indicate the alternating phases of the soliton wavefunction, which has exponentially decaying tails (therefore localied) and a half-width of seven CH sites (therefore mobile). The strong bending field in the carbon backbone is due to the fundamental soliton-conformation coupling.
A matter-wave soliton train, i.e. a succession of special waves made of self-attracting ultracold atoms. Each peak in the train is a Bose-Einstein condensate, a localized collection of atoms cooled to nearly absolute zero temperature (from RICE group )
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5. Orientation of spins in a skyrmion
In quantum fields, the topological solitons are topologically nontrivial classical solutions. They have different names depending on whether they minimize the action, then they are called instantons or the energy -that is based on the respective topologies of space and the gauge group- then they are named monopole vortex skyrmion, toron, ... There exists also an other class of stable compact objects: non-topological solitons. They represent an unusual coherent states of matter, called also bulk matter. Models were suggested for the NTS to exist in forms of stars, quasars, the dark matter and nuclear matter.
Example 2D electron gas
The skyrmion are topological objects which distort the ferromagnetic ordering over large distances. The presence of these objects dominate the physics of 2D electron gas in the regime of the integer (and odd)quantum Hall effect.
Orientation of spins in a skyrmion
We observed magnetic order associated with the skyrmion. This order breaks the symmetry of the spin rotation in the plane of the 2D electron gas. This symmetry breaking is manifested by spin waves without gap.
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6. 2.1 Coupled chain pendulum and the sine-Gordon equation
We focus our attention on the sine-Gordon equation which is one of the most interesting equation that
exhibits solitons* especially in condensed matter physics.
2.1 Coupled chain pendulum and the sine-Gordon equation
Consider a chain of pendulum around a common axis, two consecutive pendulum are coupled by
a torsion spring. The deviation from equilibrium for the nth pendulum is given by angle qn . The
Hamiltonian of the system is given by a sum over all the pendulum of the chain of 3 quantities
(1)
rotational
kinetic energy
I=inertial
momentum
relative to the axis of
the pendulum
elastic coupling
term between 2
adjacent pendulum
(torsion spring of stiffness C)
gravitational potential energy
of the pendulum
l (distance to the center
of mass)
Experimental device g a
m mass of the
pendulum
* Other equations in the bibliography
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7. discrete non-linear differential equation
generalized coordinates : qn and pn= I qn
→ Hamilton equations →
(2)
discrete non-linear differential equation
We investigate the continuum limit for which exact solutions exist
It is valid in the « strong limit coupling » , such that q does not vary too much between two
successive pendulum. We note by a the distance between the pendulum.
discrete variable qn → continuous variable q(x,t) where qn (x= na,t)
A Taylor expansion of qn±1 leads to
Taking into account of the fast decreasing character of successive terms containing derivatives of higher
order of the function q that slowly decreases in space. The first non trivial order is sufficient to catch all
the interesting physical and mathematical properties as we will see.
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8. 2.2 Topology of the energy landscape
we introduce the frequency w0 and the velocity c0
(3)
(4)
Finally the equation is given by
(5)
Sine-Gordon equation
In this approximation: we do not incorporate the dissipation due to the friction
between the pendulum and the rotation axis (and also at the level of the in the torsion spring).
Sine-Gordon equation is nevertheless very interesting, it exhibits solitons solutions and other
phenomena that are related to special relativity and quantum mechanics.
2.2 Topology of the energy landscape
elastic line
Consider the gravitational potential energy V(q) as a
function of q and the space coordinate x along the chain.
∀ x: V(q) takes the same value V(q) = m g l (1 – Cos(q))
gravitational part of the
energy landscape = sinusoidal surface
We have to add the harmonic coupling between the pendulum due to the spring torsion.
In the continuum limit, the chain line is considered as a massive elastic line along V(q) .
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9. If q is perturbed at a point x, the elastic response of the system tends to induce a similar perturbation
for q in the neighbourhood of x dq
x0-dx x0
x0 +dx
Therefore it exists many different fundamental states of the system. We can put each pendulum of the
chain at q=0 or equivalently, i.e. with the same energy, at q=2 p p (where p is an integer).
→ solutions with the chain inside the same potentiel valley – line 1
→ solutions for which the chain passes from a valley to the other – line 2
(6)
(7)
We have 2 topologically different solutions. Indeed a local analysis of the solution around |x| →∞
can not distinguish between these two solutions that are in the same minimum energy state.
It is only at the level of the system that we can distinguish them.
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10. q q x x Same asymptotic behaviour |x| →∞
localized region where the phase jumps from 0 to 2p
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11. 2.3 Linear limit: the small amplitude solutions
Assume q<<2p (type 1 solution) → the Taylor expansion the sin(q) =q around q=0 at the first order
leads to
(8)
Klein-Gordon equation
linear wave equation
This equation admits plane wave solution given by
(9)
inserting (9) into (8) we obtain the dispersion relation between the frequency w and the wavenumber q
for the linear wave equation
(10) w
w is not linear in q
→ dispersive waves
due to w0 sin q
In the SG eq dispersion
and nonlinearity come
from the same term
w0 sin q
in the limit of large wavenumber
q→∞ the velocity of the wave → c0
w=c0q
cutoff frequency w0 q
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12. Linear waves q=2p/l w=2pn (11)
As dispersion and nonlinearity come from w0 sin q, we can, in the continuous limit, take only the
second derivative term 𝜕 2  𝜕 𝑥 2 in the expansion of qn+1 + qn-1 – 2qn
Linear waves
q=2p/l w=2pn
(11)
phase velocity vφ = 𝜔/q corresponds to the
moving of the wave front.
group velocity vg= 𝜕𝜔 𝜕𝑥 corresponds to the
moving of the envelope of the wave, i.e. of the energy
non-dispersive medium vφ is independent of q, i.e.
𝜔 is proportional to q → vφ = vg
 dispersive medium → vφ ≠ vg
Modulations within the wave packet
move without deformation
Modulations pass by inside the package, because they move
slower than the shape of the package.
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13. for w<w0 → q is imaginary → evanescent wave with exponential decay
2.4 Soliton solutions
Spectrum property of the SG equation
for w<w0 → q is imaginary →
evanescent wave with exponential decay w0
forbidden
band-gap in the
excitations spectrum
New types of solutions are possible since dispersion is compensated by nonlinearity
only nonlinear terms
only diffusion terms
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14. (12) (13) 𝜕 𝜕𝑥 → 1 1− 𝑢 2 ( 𝜕 𝜕 𝑥 ′ - u 𝜕 𝜕 𝑡 ′ )
 Invariance of SG eq. under Lorentz transformation
we first introduce the non-dimensional variables
distance is measured in units of d=c0/w0
time is measured in units of 1/w0
(12)
If we apply the Lorentz transformation
(13)
𝜕 𝜕𝑥 → − 𝑢 2 ( 𝜕 𝜕 𝑥 ′ - u 𝜕 𝜕 𝑡 ′ )
𝜕 𝜕𝑡 → − 𝑢 2 (−𝑢 𝜕 𝜕 𝑥 ′ + 𝜕 𝜕 𝑡 ′ )
with the rules
(14)
(15)
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15. = E (19) + = E (20) Veff q then we have
potential energy
of a fictitious particle
kinetic energy
in terms of « time » z z +
= E
q(z) corresponds to the motion of a fictitious particle with zero energy in the effective potential
(20)
Veff
A possible motion for the particle
is to start at q=0 with dq/dz=0 (zero velocity)
and to reach q=2p with dq/dz=0. This motion
takes an infinite « time » z (solution goes from
z=-∞ to z= ∞). q
c02 –v2 >0
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16. c02 –v2 <0
Veff
If c02 –v2 <0, the particle is at rest initially in a local
minima of the potential and no motion is possible.
→ There is no possible propagation of a soliton
for velocity greater than c0 q
In the first case we have v2 < c02 solution is obtained from (18)
where z0 is a constant of integration, and we can integrate the second member
where
(21)
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17. (22)
with z=x- vt. The constant of integration is fixed by the initial position of the soliton in the chain.
anti-soliton or anti-kink
(- sign in (22))
soliton or kink (+ sign in (22))
see animation on Mathematica
those solutions connect states that
have the same energy.
Notes: solitons are exact solutions of
the continuum approximation of the
chain, not of the discreted model !
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18.  Topological charge of a soliton
The topological charge is defined by
(23)
In an infinite system
Q=+1 soliton
Q=-1 antisoliton
On the energy lanscape we show that , a zero topological charge would require to let a infinite length
of chain pendulum across the potential barrier.
The topological charge of a soliton/antisoliton
is an invariant and its conservation provides a
« strong » stability of the soliton solution. A
perturbation can changes its velocity even to stop it
but not destroy it, because this would change the
valueof its topological charge.
soliton
solution
For solitons, the boundary on which the boundary conditions are specified has a non-trivial
homotopy group which is preserved in the SG equation. The solutions to the SG eq are then
topologically distinct, and are classified by their homotopy class. Topological defects, i.e. solitons
solutions are not only stable against small perturbations, but cannot decay or be, precisely because
there is no continuous transformation that will map them (homotopically) to a uniform or "trivial“
solution.
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19.  Lorentz contraction (24) We write (22) under the form with
L is a measure of the width of the soliton, as we have seen the equation is invariant under the Lorentz
transformation, we see on (24) that when the velocity v tends to c0 which is the velocity of the linear
wave (the sound velocity), the solutions remains constant, but its width L gets narrower owing of
the Lorentz contraction of its profile, given by 1− 𝑣 2 𝑐 its wave form approachs a step function.
The soliton width decreases when the velocity v increases. The amplitude of the kink solution is
independent of the velocity v.
The system of chain pendulum
allows to do a “relativity experience”
In the limit for which v is equal
to 0 we have L0=c0/w0 = a 𝐶 (𝑚 𝑔 𝑙)
The continuum limit is valid when
L0/a >>1, i.e. when C>> mg l
The torsion spring energy has to be
much greater than the gravitational
energy that acts on each pendulum
C large provides that qn(t) – qn+1 (t) remains
small,that corresponds to a strong coupling approximation
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20. The Hamiltonian density (per unit length) is obtained from
2.5 Energy of the Soliton
The Hamiltonian density (per unit length) is obtained from
energy for 1 mesh
energy by unit length (we divide by mesh length a) for the chain pendulum
(25)
to obtain the soliton energy, we introduce the moving frame coordinate z= x- vt
such that q(x,t) → q(z)
(26)
as we have that
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21. (27) H localized energy density around the center of the soliton (here
we replace the derivative of the solution by its expression
(27) H
localized
energy density
around the center
of the soliton (here
z=0) z
In physics, quasiparticles and collective excitations (which are closely related) are
emergent phenomena that occur when a microscopically complicated system such as a
solid behaves as if it contained different (fictitious) weakly interacting particles in free space.
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22. For example, as an electron travels through a semiconductor, its motion is disturbed in a
complex way by its interactions with all of the other electrons and nuclei; however it
approximately behaves like an electron with a different mass traveling unperturbed through free
space. This "electron" with a different mass is called an "electron quasiparticle Other
quasiparticles or collective excitations include phonons (particles derived from the vibrations
of atoms in a solid), plasmons (particles derived from plasma oscillations), and many others.
These fictitious particles are typically called "quasiparticles" if they are fermions
(like electrons and holes), and called "collective excitations" if they are bosons
(like phonons and plasmons), although the precise distinction is not universally agreed.
As we have 𝑑𝑥 𝑠𝑒𝑐 ℎ 2 𝑥=2, the energy of the soliton is obtained by a spatial
integration
(28)
This expression has de form of the energy of a relativistic particle (the velocity of light here
is c02 ) with a mass
m0 = 8 I w0/c0
(29)
The SG soliton corresponds to a quasi-particle of mass m0, energy E and momentum p
with
(30)
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23. It is interesting to note that, it’s not needed to have an explicit form of the soliton solution to obtain its
energy, we start from
(31)
as we have an expression for dq/dz:
(32) =
(33)
4 2
this method is sometimes uuseful when we only have the potential of the system
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24. 2.6 Muti-Soliton solutions
We can find other classes of solutions that correspond to superposition of kink and anti-kink. Those
are obtained either directly or by the application of the inverse scattering method (see after). They
exhibit important properties of the interaction between kink/antikink profiles. Starting from the non
dimensional equation
(34)
(35)
We have proof that SG eq admits solution under the form
where F and G are arbitrary functions. We can write (35) as
(36)
We make use of the following trigonometry identity
(37)
(34) takes the form
(38)
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25. (39) (40) (41) (42) (43) (44) we substitue (35) -(37) into (38)
This equation is simplifed and takes the form
(40)
When y is expressed in terms of F and G by (36) we obtain
(41)
Substitution of (41) into (40) gives
(42)
Derivating (42) with respect to X and T leads respectively to
(43)
(44)
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26. separation of variables gives in equation (44) gives
(42)
where A is a constant of separation of variables, we integrate eq (42) and obtain
(43)
(44)
where B1 and B2 are constants of integration, we mutiply (43) by 2 Fx and (44) by 2 Gt
and we integrate respectively onX and T→
(45)
(46)
where C1 and C2 are two others constants of integration
To obtain relations between those constants, we put (45-46) into (42)
(47)
setting the following notation
(48)
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27. kink-soliton q=0 b >0 n=0 kink-kink collision q=0 b2 >1 n≠0
Equations (45) and (46) finally become
(49)
(50)
where q b and n are arbitrary constants whose values will determine the kinds of
solutions of (35)
kink-soliton q=0 b >0 n=0
kink-kink collision q=0 b2 >1 n≠0
Breather-soliton q≠0 b2 <1 n=0
kink anti-kink collision q≠0 b >1 n=0
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28. Breather solution (51) (52) (53) (54) (55)
q≠0 b 2 <1, n=0 Equations (49 -50) reduce to
(51)
where we have introduced
(52)
In the table of integrals we find that
(53)
(54)
where we have
we apply this relation with B=0
We put those relations into eq (51) and obtain
(55)
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29. Constants C1 and C2 correspond to initial space and time coordinates of the solution, they are
taken equal to zero.
(56)
by substitution of (56) into (35) we obtain the Breather solution
(57)
The breather is a stationary wave solution, it’s envelope exhibit a sech form that is modulated in time at the frequency W. This pulse looks like breathing, therefore we call it breather
envelope
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30. As we have shown that the SG eq is invariant under the Lorentz transformation of the variables (X,T)→(X’,T’)
we apply this transformation on the breather solution to obtain a boosted solution given by
(58)
Using relations u=v/c0 d=c0/w0 and g=1/ 1− 𝑢 and setting W=wB/w0 we can express this breather
solution into the laboratory coordinates (x,t)
(59)
Here the internal oscillations wB is weighted by the Lorentz term g. We have that 0≤ wB ≤ w0. The
breather velocity is restricted in the range 0≤ |v| ≤ c0 we recover here, the same condition as for the
kink soliton. The breather undergoes, like the kink a Lorentz contraction when the velocity v tends to c0.
The characteristic width dB of the profile is
(60)
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31. (61) (62) (63) (64) The maximum amplitude of the breather is given by
The energy of the breather EB is obtained by substitution of (58) into
(62)
we obtain EB
(63)
By using the energy of the kink soliton (28), which is now noted by EK, we can express EB as
(64)
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32. can be considered as a bounded soliton-antisoliton pair.
For wB → 0, the breather energy EB → 2 EK AB → p/2, dB → d/g in this limit the breather
can be considered as a bounded soliton-antisoliton pair.
For wB → w0, the breather energy EB → 0, AB → 0, dB → ∞ in this limit the breather
approach a low amplitude linear wave.
Depending on its internal frequency wB the breather goes from a large amplitude nonlinear waves
to a low amplitude linear waves. Moreover it can be considered as a relativistic particle with an
internal oscillations of mass mB= 2 m0 and momentum PB = 2PK.
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33. For wB → w0, the breather energy EB → 0, AB → 0, dB → ∞ in this limit the breather
approach a low amplitude linear wave.
wB/ w0 =0.9
wB/ w0 =0.9999
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34. Lamb, G. L. Jr. Elements of Soliton Theory. New York: Wiley, 1980.
Dauxois T , Peyrard M, Physics of Solitons, Cambridge University Press 2006
Remoissenet M, Waves Called Solitons, concepts and experiments, 2 ed Springer 1999
Drazin, P. G. and Johnson, R. S. Solitons: An Introduction. Cambridge, England: Cambridge University Press, 1988.
Infeld, E. and Rowlands, G. Nonlinear Waves, Solitons, and Chaos, 2nd ed. Cambridge, England: Cambridge University Press, 2000.
Kaup, D. J. Method for Solving the Sine-Gordon Equation in Laboratory Coordinates. Stud. Appl. Math. 54, , 1975.
Lamb, G. L. Jr. Elements of Soliton Theory. New York: Wiley, 1980.
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