arXiv:0704.0082v1  [cond-mat.other]  1 Apr 2007
Typeset with jpsj2.cls <ver.1.2> Full Paper
Matter-Wave Bright Solitons with a Finite Background in Spinor
Bose-Einstein Condensates
Tetsuo Kurosaki∗ and Miki Wadati
Department of Physics, Graduate School of Science, University of Tokyo, Tokyo 113-0033
We investigate dynamical properties of bright solitons with a finite background in the
F = 1 spinor Bose-Einstein condensate (BEC), based on an integrable spinor model which
is equivalent to the matrix nonlinear Schrödinger equation with a self-focusing nonlineality.
We apply the inverse scattering method formulated for nonvanishing boundary conditions.
The resulting soliton solutions can be regarded as a generalization of those under vanishing
boundary conditions. One-soliton solutions are derived in an explicit manner. According to
the behaviors at the infinity, they are classified into two kinds, domain-wall (DW) type and
phase-shift (PS) type. The DW-type implies the ferromagnetic state with nonzero total spin
and the PS-type implies the polar state, where the total spin amounts to zero. We also
discuss two-soliton collisions. In particular, the spin-mixing phenomenon is confirmed in a
collision involving the DW-type. The results are consistent with those of the previous studies
for bright solitons under vanishing boundary conditions and dark solitons. As a result, we
establish the robustness and the usefulness of the multiple matter-wave solitons in the spinor
BECs.
KEYWORDS: spinor Bose-Einstein condensate, internal degrees of freedom, matrix nonlinear
Schrödinger equation, integrable model, bright soliton, nonvanishing boundary
condition, multi-component Gross-Pitaevskii equation, spin exchange coupling,
spin switching
∗E-mail address: kurosaki@monet.phys.s.u-tokyo.ac.jp
http://arxiv.org/abs/0704.0082v1
J. Phys. Soc. Jpn. Full Paper
1. Introduction
In 2002, matter-wave bright solitons in quasi-one-dimensional (1D) Bose-Einstein conden-
sates (BECs) were observed experimentally.1, 2) Bright solitons propagate in most cases with
much larger amplitudes than dark solitons,3, 4) and are expected to have the potential for
various applications such as coherent transport and atom interferometry. Soliton propagation
in BEC can be described by the Gross-Pitaevskii (GP) equation. The GP equation, called the
nonlinear Schrödinger (NLS) equation in nonlinear science, is integrable and has soliton solu-
tions in a one-dimensional and uniform system. Recent experimental and theoretical advances
about matter-wave bright solitons are reviewed, for instance, in ref. 5.
The experimental creation of matter-wave solitons has been so far achieved only for single-
component BEC. It is, nevertheless, very interesting to consider soliton propagation in BEC
with internal degrees of freedom, so-called, spinor BEC. When BEC of ultracold alkali atoms
is trapped exclusively by optical means, the hyperfine spin of atoms remains liberated. The
spinor BEC was realized in such a way.6–8) Internal degrees of freedom endow solitons with
a multiplicity. The multiple solitons will show a rich variety of dynamics. Here, we focus on
the boson system in the F = 1 hyperfine spin state, exemplified by 23Na, 39K and 87Rb. The
multi-component GP equation for F = 1 spinor BEC turns to an integrable model at special
points, which is mathematically equivalent to the matrix NLS equation. An integrable model
with a self-focusing nonlineality enables one to perform exact analysis via the inverse scatter-
ing method (ISM) for the matrix NLS equation.9) In particular, bright soliton solutions under
vanishing boundary conditions (VBC) are obtained, whose properties are investigated in refs.
10 and 11. Recently, the ISM for the matrix NLS equation under nonvanishing boundary con-
ditions (NVBC) is formulated.12) Dark solitons in the F = 1 spinor BEC can be investigated
by applying the ISM under NVBC to an integrable model with a self-defocusing nonlineal-
ity.13) Although the ISM under NVBC is dedicated mainly to the self-defocusing case, we note
that this technique is also applicable to an integrable model with a self-focusing nonlineality,
which makes us available to bright soliton solutions with a finite background.
In this paper, the detail of matter-wave bright solitons in the quasi-1D F = 1 spinor BEC
is further investigated, based on an integrable model. We consider matter-wave spinor bright
solitons traveling on a finite background of the condensate. We write down explicitly new
soliton solutions, and verify that the obtained soliton solutions have the similar properties
compared to those without a background. In the usual experimental setups, the condensates
are confined in a finite-size regime, and the matter-wave bright solitons will accompany a
finite background. The study given in this paper is meaningful in such realistic circumstances.
The paper is organized as follows. In § 2, the GP equation for quasi-1D F = 1 spinor BEC
is introduced. In particular, the integrable model is presented. There, the interactions between
two atoms are supposed to be inter-atomic attractive and ferromagnetic, which lead to bright
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solitons. In § 3, the inverse scattering method under nonvanishing boundary conditions is
applied to the integrable model. This application leads to bright soliton solutions with a finite
background. Several conserved quantities of the model are also provided. One-soliton solutions
are investigated in § 4. The spin states of one-solitons are classified, assuming that discrete
eigenvalues are purely imaginary. Two-soliton solutions are discussed in § 5. The last section,
§ 6, is devoted to the concluding remarks.
2. GP Equation for F = 1 Spinor Bose-Einstein Condensates
For BEC of ultracold alkali atoms, the mean-field theory works well, because almost all
atoms go into condensation and the condensate is dilute. In this paper, we deal with the
quasi-one-dimensional system. Atoms in the F = 1 hyperfine spin state have three magnetic
substates labeled by the magnetic quantum number mF = 1, 0,−1. The system is charac-
terized by a vectorial field operator with the components corresponding to each substate,
Φ̂ = (Φ̂1, Φ̂0, Φ̂−1)
T , satisfying equal-time commutation relations:
[Ψ̂α(x, t), Ψ̂
(x′, t)] = δαβδ(x− x′), (1)
where the subscripts α, β take on 1, 0,−1. In the framework of the mean-field theory for BEC,
the quantum field is replaced with the order parameter:
Φ(x, t) ≡ 〈Φ̂(x, t)〉 = (Φ1(x, t),Φ0(x, t),Φ−1(x, t))T . (2)
Φ(x, t) is often called the spinor condensate wavefunction, which is normalized to the total
number of atoms NT:
dxΦ†(x, t)Φ(x, t) = NT. (3)
The spinor condensate wavefunction obeys a set of coupled evolution equations, namely, the
multi-component GP equation:
i~∂tΦ1 = −
∂2xΦ1 + (c̄0 + c̄2)
|Φ1|2 + |Φ0|2
+(c̄0 − c̄2)|Φ−1|2Φ1 + c2Φ∗−1Φ20,
i~∂tΦ0 = −
∂2xΦ0 + (c̄0 + c̄2)
|Φ1|2 + |Φ−1|2
+c̄0|Φ0|2Φ0 + 2c̄2Φ∗0Φ1Φ−1,
i~∂tΦ−1 = −
∂2xΦ−1 + (c̄0 + c̄2)
|Φ−1|2 + |Φ0|2
+(c̄0 − c̄2)|Φ1|2Φ−1 + c̄2Φ∗1Φ20, (4)
where c̄0 = (ḡ0 + 2ḡ2)/3 and c̄2 = (ḡ2 − ḡ0)/3 denote effective 1D coupling constants for the
mean-field and the spin-exchange interaction, respectively. Here, the effective 1D coupling
constants ḡf are given by
ḡf =
4~2af
1− Caf/a⊥
, (5)
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where af are the s-wave scattering lengths in the total hyperfine spin f channel, a⊥ is the
size of the transverse ground state, m is the atomic mass, and C = −ζ(1/2) ≃ 1.46. Note that
one may change the values of c̄0 and c̄2 by tuning a⊥.
Equation (4) is derived as follows. The interaction between two atoms in the F = 1
hyperfine spin state has a form,15, 16)
V̂ (x1 − x2) = δ(x1 − x2)
c̄0 + c̄2F̂ 1 · F̂ 2
, (6)
where F̂ i is the spin operator. The Gross-Pitaevskii energy functional is thus given by
EGP[Φ] =
α∂xΦα +
α′Φα′Φα +
αβ · fα′β′Φβ′Φβ
, (7)
where repeated subscripts (α, β, α′, β′ = 1, 0,−1) should be summed up and f = (fx, fy, fz)T
with fi(i = x, y, z) being 3× 3 spin-1 matrices. Then, the variational principle: i~∂tΦα(x, t) =
δEGP[Φ]/δΦ
α(x, t), for α = 1, 0,−1, yields eq. (4).
An important fact is that eq. (4) possesses a completely integrable point when c̄0 = c̄2 ≡
−c < 0, equivalently, 2ḡ0 = −ḡ2 > 0.10, 11) This condition is realized when
a⊥ = 3C
2a0 + a2
, (8)
assuming that a0a2(a2−a0) > 0 holds. The situation corresponds to attractive (c̄0 < 0) and fer-
romagnetic (c̄2 < 0) interaction. When we change the wavefunction by Φ = (φ1,
2φ0, φ−1)
and measure time and length in units of t̄ = ~a⊥/c and x̄ = ~
a⊥/2mc, respectively, we can
rewrite eq. (4) with c̄0 = c̄2 ≡ −c < 0 into the dimensionless form,
i∂tφ1 = −∂2xφ1 − 2(|φ1|2 + 2|φ0|2)φ1 − 2φ∗−1φ20,
i∂tφ0 = −∂2xφ0 − 2(|φ−1|2 + |φ0|2 + |φ1|2)φ0 − 2φ∗0φ1φ−1,
i∂tφ−1 = −∂2xφ−1 − 2(|φ−1|2 + 2|φ0|2)φ−1 − 2φ∗1φ20. (9)
Then, eq. (9) is found to be equivalent to a 2× 2 matrix version of the NLS equation with a
self-focusing nonlinearity:
i∂tQ+ ∂
xQ+ 2QQ
†Q = O, (10)
with an identification,
φ1 φ0
φ0 φ−1
. (11)
The matrix NLS equation (10) is integrable in the sense that the initial value problem can
be solved via the inverse scattering method.9, 12) The integrability of the reduced equations
(9) is thus proved automatically. Thus, we have derived the integrable spinor model. Another
integrable point of eq. (4) is c̄0 = c̄2 ≡ c > 0, i.e., the matrix NLS equation with a self-
defocusing nonlineality.13) Special solutions for generic coupling constants c̄0, c̄2 are given in
ref. 17.
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3. Bright Solitons with a Finite Background
We consider bright soliton solutions of the integrable model (9) under NVBC, whereas
those under VBC are studied in refs. 10 and 11. We summarize briefly the results of the inverse
scattering method for eq. (9) with NVBC.12)
We define the nonvanishing boundary conditions as
Q(x, t) → Q±, x→ ±∞,
±Q± = Q±Q
± = λ
0I, (12)
where λ0 is a positive real constant and I denotes a 2 × 2 unit matrix. Note that vanishing
boundary conditions are recovered as λ0 → 0. The analysis of the ISM under NVBC yields
the standard form of the multiple soliton solutions of the 2 × 2 matrix NLS equation with a
self-focusing nonlineality (10) as
Q(x, t) = λ0 e
I + 2i(I · · · I
︸ ︷︷ ︸
. (13)
Here, a 2 × 2 complex matrix Πj is called the polarization matrix. S is a 2N × 2N matrix
defined by
Sij =
ζj + λj
δijI +
i(ζi + ζj)
ζi + λi
ζj + λj
1 ≤ i, j ≤ N, (14)
where λj is a complex discrete eigenvalue for the bound state and ζj = (λ
j + λ
1/2 with
Im ζj > 0 for j = 1, . . . , N . It is required for the ISM under NVBC that a two-sheet Riemann
surface is introduced appropriately, due to a double-valued function ζj. The phase of the
carrier wave, φ(x, t), is given by
φ(x, t) = kx− (k2 − 2λ20)t+ δ, (15)
and the coordinate function is given by
χj ≡ χj(x, t) = 2iζj(x− 2(λj + k)t). (16)
The above solution is the M(= N/2)-soliton solution. The ISM under NVBC for the self-
focusing case results in pairs of discrete eigenvalues corresponding to each Riemann sheet.
The constraint should be imposed on λj and ζj (j = 1, · · · , N) such that λ2l−1 = λ∗2l and
ζ2l−1 = −ζ∗2l for l = 1, · · · , N/2. At the same time, Πj must satisfy that Π2l−1 = Π
For our reduction to the integrable model for F = 1 spinor BEC, we must make the
potential Q symmetric, noting eq. (11). The symmetry of Q is naturally reflected in Πj.
When we take every Πj to be symmetric in eq. (13), soliton solutions of the integrable model
(9) under NVBC are obtained.
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The form (13) is the standard form of soliton solutions in the sense that the boundary
value at x→ ∞ ⇔ eχj → 0 is supposed to be fixed as
Q(x, t) e−iφ(x,t) → λ0I, x→ ∞. (17)
The spinor model, however, allows the SU(2) transformation of the solutions, if they are kept
symmetric. To be concrete, let U be a 2× 2 unitary matrix. When Q is a solution of eq. (10)
with eq. (11), then
Q′ = UQUT , (18)
is also a solution. Assuming that Q is the standard form (13), the limit x→ ∞ of Q′ becomes
Q′ e−iφ(x,t) → λ0 UUT ≡ λ0Q′+, x→ ∞. (19)
Q′+ = UUT is the so-called Cholesky decomposition. The arbitrary boundary conditions Q′+
other than eq. (17) are thus realized via the SU(2) transformation. On the other hand, the
behavior in the limit x → −∞ varies depending on whether detΠ = 0 or not, which will be
discussed later for the one-soliton case.
There is another important concept about an integrable model. Due to the integrability,
the model has the infinite conservation laws, which restrict the dynamics of the system in an
essential way. Several conserved quantities, related to the physical quantities of the system,
are listed below:
Total number : N̄T =
dx n̄(x, t),
n̄(x, t) = tr (Q†Q)− tr (Q†±Q±). (20)
Total spin : FT =
dx f(x, t),
f(x, t) = tr (Q†σQ). (21)
Total momentum : P̄T =
dx p̄(x, t),
p̄(x, t) = −i~[tr (Q†Qx)− tr (Q†±Q±,x)]. (22)
Total energy : ĒT =
dx ē(x, t),
ē(x, t) = c[tr (Q†xQx −Q†QQ†Q)
−tr (Q†±,xQ±,x −Q
±Q±)]. (23)
Here, σ = (σx, σy, σz)
T are the Pauli matrices. To avoid the divergence of integrals, we should
subtract the contribution of the background from the physical quantities, except the total spin,
to which the background does not contribute explicitly. These subtractions are emphasized
by the bars on the conserved densities and quantities. The local spin density f = (fx, fy, fz)
is covariant under the SU(2) transformation (18), whereas the other densities such as n̄(x, t),
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p̄(x, t) and ē(x, t) are invariant. The total macroscopic spin is directed to face to an arbitrary
direction by the global spin rotation. The SU(2) symmetry of the system causes the energy
degenerated states of solitons for this spin rotation.
4. One-Soliton Solutions
In this section, one-soliton solutions of the integrable spinor model (9) are investigated in
detail. We can derive the explicit form of one-soliton solutions by setting N = 2 (M = 1) in
the formula (13). The calculation is complicated but straightforward. The result is as follows:
Q = λ0 e
iφ(x,t)
I + 2i
, (24)
where detS is given by
detS = κ21κ
2 − eχ1ν1κ1κ22 trΠ− ex2ν2κ21κ2 trΠ†
+eχ1+χ2κ1κ2
̟ + ν1ν2(|tr Π|2 − 1)
+e2χ1ν21κ
2 detΠ + e
2χ2ν22κ
1 detΠ
− e2χ1+χ2ν1κ2̟ trΠ† detΠ− eχ1+2χ2ν2κ1̟ trΠdetΠ†
+e2χ1+2χ2̟2|detΠ|2, (25)
and T is a 2× 2 matrix such that
T = eχ1κ1κ
2 Π+ e
χ2κ21κ2 Π
− eχ1+χ2κ1κ2
ς1 trΠ · Π† + ς2 trΠ† ·Π+ µ(|trΠ|2 − 1)I
− e2χ1µ1κ22 detΠ · I − e2χ2µ2κ21 detΠ† · I
+e2χ1+χ2κ2 detΠ
ς21 Π
† +̟ trΠ† · I
+eχ1+2χ2κ1 detΠ
ς22 Π+̟ trΠ · I
− e2χ1+2χ2̟ (ς1 + ς2) |detΠ|2 · I. (26)
We explain physical meanings of notations. The phase of the carrier wave is given by
φ(x, t) = kx− (k2 − 2λ20)t+ δ. (27)
Let λj and ζj = (λ
j + λ
1/2 for j = 1, 2 be complex constants satisfying λ1 = λ
2 and
ζ1 = −ζ∗2 . Without loss of generality, we assume that (λ1, ζ1) ((λ2, ζ2)) belongs to the upper
(lower) Riemann sheet, which is characterized such that Im ζ Im λ > 0 (Im ζ Im λ < 0).
χj(x, t) is expressed in terms of them as
χ1 ≡ χ1(x, t) = 2iζ1(x− 2(λ1 + k)t), (28)
χ2 ≡ χ2(x, t) = 2iζ2(x− 2(λ2 + k)t). (29)
Note that χ1 = χ
2 ≡ χ holds. Reχ thus denotes the coordinate of the envelope soliton,
whereas Imχ implies the self-modulation phase. The polarization matrix Π is a 2×2 symmetric
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matrix. Here, the normalization in a sense of the square norm is imposed on Π as a matter of
convenience:
, 2|α|2 + |β|2 + |γ|2 = 1. (30)
The other parameters are expedient functions of λ0, λ1 and λ2:
ζj + λj
, µ = iλ0
κ1 + κ2
ζ1 + ζ2
, νj =
iλ0κj
, ̟ = ν1ν2 − µ2, ςj = νj − µ, (31)
for j = 1, 2. We list the meaning of each parameter as follows:
k : wave number of soliton’s carrier wave.
λ0 : amplitude of soliton’s carrier wave.
φ(x, t) : phase of soliton’s carrier wave.
Re χ(x, t) : coordinate of soliton’s envelope.
Im χ(x, t) : self-modulation phase of soliton.
Π : symmetric polarization matrix of soliton.
Equations (24)-(26) are new soliton solutions which had never been written down explicitly
in the literatures. If we take the vanishing limit λ0 → 0, ζ1 and ζ2 converge at λ1 and −λ2,
respectively. Then,
· 2kR
Πe−(χR+ρ/2) + (σyΠ†σy) eχR+ρ/2detΠ
e−(2χR+ρ) + 1 + e2χR+ρ|detΠ|2
eiχI , (32)
with notations
eρ/2 ≡ 1
, (33)
χR ≡ χR(x, t) = kR(x− 2kI t)− ǫ, (34)
χI ≡ χI(x, t) = kIx+ (k2R − k2I )t, (35)
each of which holds the following correspondence respectively:
kR = −2Imλ1, (36)
kI = 2Reλ1 + k, (37)
ǫ = − ln(4|λ1|). (38)
Equations (32)-(35) are the same forms as those in ref. 11, except a phase factor. This con-
sequence is natural but non-trivial, because the formula of solitons under VBC9) is quite
different from that under NVBC (13), in particular, in the form of the matrix S. Actually, the
initial displacement ǫ can be arbitrarily changed, regardless of eq. (38), by the parallel shift
of the position x. We have shown that the soliton solutions (24)-(26) can be regarded as a
general form of bright soliton solutions, including the case of VBC.
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4.1 Classification by the boundary conditions
We shall show that there are two kinds of one-soliton solutions depending on the boundary
conditions, detΠ = 0 or detΠ 6= 0. The similar classification about the boundary conditions
also exists for dark solitons.13) The examples of snapshots of one-soliton density profiles are
shown in Fig. 1. The upper row is for detΠ = 0, and the lower row is for detΠ 6= 0. The
shape of envelope solitons looks a locally-oscilating wave rather than, literally, a solitary wave,
because of the self-modulation due to the complex velocity.
For detΠ = 0, the boundary conditions of the standard form (24)-(26) are
Q e−iφ → λ0I, x→ ∞,
Q e−iφ → λ0
I − 2i ς1 trΠ · Π
† + ς2 trΠ
† · Π+ µ(|trΠ|2 − 1)I
̟ + ν1ν2(|tr Π|2 − 1)
, x→ −∞. (39)
The left and right boundary values differ in not only the global phase but also the population
of each component, in general. That is, those are the SU(2) rotated boundary conditions.
In the upper row of Fig. 1, we see that the envelope soliton of each component forms the
domain-wall (DW) shape, although it does not manifest in the total number density.
On the other hand, for detΠ 6= 0, the boundary conditions are
Q e−iφ → λ0I, x→ ∞,
Q e−iφ → λ0
1− 2i
ς1 + ς2
I, x→ −∞. (40)
In contrast to the case that detΠ = 0, both boundary values are diagonal matrices, and only
the phase-shift (PS) occurs. That is, those are the U(1) rotated boundary conditions.
For the above reasons, we call one-soliton solutions the DW-type for detΠ = 0, and the
PS-type for detΠ 6= 0. Remark the following; the spin density profile of DW-type suggests
that the total spin is nonzero, whereas that of PS-type is dipole-shape, implying that the total
spin amounts to zero. See the right panel of Fig. 1. This observation will be solidified in § 4.3.
4.2 Case of purely imaginary discrete eigenvalues
The ISM performed on Riemann sheets involves a double-valued function of the spectral
parameter, and it usually renders a very complicated representation of N -soliton solutions
even for N = 1, as is seen from eqs. (24)-(26). To simplify an explicit representation, it is
convenient to assume that λj and ζj are purely imaginary.
18) The similar approach is employed
for the analysis of N -soliton solutions of the derivative NLS equation under NVBC.20)
If we take a pair of discrete eigenvalues as
(λ1, ζ1) ≡ (iλ0λ, iλ0ζ), (λ2, ζ2) ≡ (−iλ0λ, iλ0ζ), (41)
where λ and ζ are positive real numbers such that
λ > 1, ζ =
λ2 − 1, (42)
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(a) (b) (c)
Fig. 1. Snapshots of one-soliton density profiles. The upper row is plotted for detΠ = 0 at the moment
t = 0, with k = 0, λ0 = 1, λ1 = 1+i, ξ1 = 1.27+0.79i (χ(x, t) = −(1.57−2.54i)x− (8.23−1.94i)t)
and Π =
4/5 2/5
2/5 1/5
. The lower row is plotted for detΠ 6= 0 at the moment t = 0, with the
same parameters except for Π =
2 2/5
2/5 3/(5
. The left panel (a) depicts the local density
for each component, |φ1|2 (solid line), |φ0|2 (chain line) and |φ−1|2 (dotted line). The center panel
(b) depicts the local number density n, where the contribution of the background is included. The
right panel (c) depicts the local spin densities, fx (solid line) and fz (dotted line). fy vanishes
identically due to a choice of a real matrix Π.
we obtain a relatively simple form of one-soliton solutions as
Q = λ0 e
iφ(x,t)
I + 2i
, (43)
detS = 1−
(tr Π(t) + trΠ†(t)) + e2χP
|trΠ(t)|2
(detΠ(t) + detΠ†(t))−
trΠ†(t)detΠ(t) + trΠ(t)detΠ†(t)
e4χP |detΠ(t)|2, (44)
2i T =
2(|tr Π(t)|2 − 1) e2χP + 2e2χP
detΠ(t) +
detΠ†(t)
tr Π†(t)detΠ(t) +
trΠ(t)detΠ†(t)
10/18
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−2(ζ + λ) eχP + 2
e2χP trΠ†(t) + 2
e3χP detΠ†(t)
−2(ζ − λ) eχP − 2
e2χP trΠ(t) + 2
e3χP detΠ(t)
Π†(t), (45)
where the coordinate of the envelope soliton is given by
χP (x, t) = −2λ0ζ(x− 2kt), (46)
and the time dependence of the phase modulation is embedded in the polarization matrix,
namely,
Π(t) ≡ Πe4iλ20ζλt. (47)
When we take the limit λ0 → 0 with λ0λ and λ0ζ kept finite in eqs. (43)-(45), Q converges
to the form of eqs. (32)-(35), accompanying the parameters kR = −2λ0λ, kI = k and ǫ =
− ln(4λ0λ). Here, kR and kI are independent free parameters, apart from the trivial initial
displacement ǫ. In this sense, in spite of the reduction, we can still regard the form of one-
soliton solutions (43)-(45) as a general form of those under VBC.
We can also take another limit. That is, we consider the reduction to the single-component
case. If we set
k = 0, Π =
eiθ 0
, (48)
the (1,1)-component of Q becomes
Q11 · e−i(2λ
t+δ) = λ0 − 2λ0ζ
ζ cos(4λ20ζλt+ θ) + iλ sin(4λ
0ζλt+ θ)
λ cosh(2λ0ζx+ ψ)− cos(4λ20ζλt+ θ)
, (49)
where eψ = ζ/λ. The form (49) was given in ref. 18. We thus verify that our soliton solutions
are also generalization of those for the single-component NLS equation under NVBC.19)
4.3 Spin states
In this subsection, we discuss the spin states of one-soliton solutions with a finite back-
ground, by calculating the total spin. The conservation law guarantees that we obtain the
total spin FT from integrating at arbitrary time. Therefore, we can select the time so that the
calculation becomes easier. We concentrate on the case of purely imaginary discrete eigenval-
ues. As a result, we see that the DW-type is associated with the ferromagnetic state, whereas
the PS-type is associated with the polar state. One-soliton solutions for purely imaginary dis-
crete eigenvalues (43)-(45) include those under VBC apart from an initial displacement, and
therefore the classification about the spin states presented below is wider than that performed
before.10, 11)
4.3.1 Ferromagnetic state
For detΠ = 0 (DW-type), we substitute eqs. (43)-(45) into eq. (21) and calculate the total
spin. The time t′ such that trΠ(t′) + trΠ†(t′) = 0 is suitable for the calculation. The result is
11/18
J. Phys. Soc. Jpn. Full Paper
as follows:
FT = 4λ0τ
2λRe{α∗(β + γ)}
−2ζIm{α∗(β − γ)}
λ(|β|2 − |γ|2)
, τ ≡
|β + γ|2
, (50)
with the modulus
|F T|2 = (4λλ0)2τ. (51)
The total number of the particles transformed into a soliton, N̄T, is calculated by eq. (20),
N̄T = 4λ0ζ, (52)
and the range of the value taken by |F T|2 is expressed in terms of N̄T,
N̄2T ≤ |F T|2 ≤ N̄2T + (4λ0)2. (53)
Remark that, in the vanishing limit λ0 → 0, the modulus of the total spin is always equal to
the total number of the particles, namely, |F T| → NT.
With nonzero total spin, the DW-type of solitons belongs to the ferromagnetic state.
Since inter-atomic ferromagnetic interactions are supposed here, solitons tend to take the
ferromagnetic state or DW-type. In various contexts of physics, the domain-walls are topo-
logical solitons related with the symmetry breaking. Here, resulting from the domain-walls,
the magnetic entity emerges as the spontaneously broken symmetry. It is worthy to notice
that the case |N̄T| < |FT| may happen. The background is spinless, but its internal spin state
appears to be affected on the ground that the ferromagnetic soliton runs over the background.
Thereby, the background contributes to the total spin.
4.3.2 Polar state
If detΠ 6= 0 (PS-type), the solitons show the other magnetic property. The time t′ such
that detΠ(t′) = detΠ†(t′) > 0 is suitable for the analysis. After lengthy calculations, the local
spin density at such time is derived as
= 8λ20 e
−3υΞ−2(χP ′)
β − γ
ζ e2υΞ(χP ′) sinh(χP ′)
(ζ2 − λ2)tr Π(t′) + (ζ2 + λ2)trΠ†(t′)
sinh(2χP ′)
λ2/ζ · ((tr Π†(t′))2 − |tr Π(t′)|2) + 4ζdetΠ(t′) + 2ζ(|tr Π(t′)|2 − 1)
sinh(χP ′)
+h.c., (54)
fy = 32λ
−3υIm{α∗(β − γ)}Ξ−2(χP ′)
2ζ eυ sinh(2χP ′)− (trΠ(t′) + trΠ†(t′)) sinh(χP ′)
, (55)
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J. Phys. Soc. Jpn. Full Paper
where Ξ(χP ′) is an even function of χP ′ , χP ′(x, t) is a parallel-shifted coordinate, and υ is a
constant:
Ξ(χP ′) ≡ 2 cosh(2χP ′)−
2 e−υ
(trΠ(t′) + tr Π†(t′)) cosh(χP ′)
ζ2 + |trΠ(t′)|2
λ2detΠ(t′)
, (56)
χP ′ ≡ χP + υ = −2λ0ζ(x− 2kt′) + υ, (57)
υ ≡ ln
(detΠ(t′))1/2
. (58)
Note that fx and fz share the same functional form. Each component of the above local spin
density is an odd function of χP ′ and, in particular, it has the node at the same point x0 such
that χP ′(x0, t
′) = 0, namely, x0 = 2kt
′ + (2λ0ζ)
−1υ. Consequently, the total spin amounts to
zero:
dxf(χP ′) = (0, 0, 0)
T . (59)
For this reason, the PS-type of solitons, on the average, belongs to the polar state.15)
5. Two-Soliton Collision
We proceed to the discussion of two-soliton collisions in the integrable spinor model (9).
Two-soliton solutions are obtained by setting N = 4 (M = 2) in the formula (13). There
exist two independent discrete eigenvalues and symmetric polarization matrices, respectively,
i.e., λ1 = λ
2 and Π1 = Π
2 for one of solitons, λ3 = λ
4 and Π3 = Π
4 for the other. Each
soliton is separated at t = ±∞. Then, a two-soliton is asymptotically two one-solitons. The
classification of one-soliton solutions based on the values of the determinants of polarization
matrices, discussed in the previous section, is thus valid for two-soliton solutions.
The derivation of the explicit form is more complicated than that in the case for one-soliton
solutions. For the derivative NLS equation under NVBC, explicit two-soliton solutions and
shifts of soliton positions due to collisions between solitons have been analytically obtained,
in the case of purely imaginary eigenvalues, where complexity of calculation is considerably
reduced.20) This strategy, however, does not stand in our NLS equation under NVBC. The
reason is understood from eq. (46). In the spinor model, purely imaginary eigenvalues give
two solitons with the same velocity 2k, and they do not collide with each other. Accordingly,
we can not investigate the properties of collisions for purely imaginary discrete eigenvalues.
No one has studied explicit multi-soliton solutions of the NLS equation under NVBC, even in
the single-component case, because of the computational complexity.
Here, we graphically show the characteristic behaviors of two-soliton collisions in the
spinor model, by use of the exact solutions given by the ISM. Referring to them, we carry
out the qualitative discussions. Although the presented graphs are depicted for the specific
parameters, much the same behaviors are observed for arbitrarily selected parameters.
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J. Phys. Soc. Jpn. Full Paper
Figure 2 illustrates the behavior of a mutual collision between two PS-types, where
detΠ1 6= 0 and detΠ3 6= 0. One can see that, in all three components, Both solitons re-
tain their shapes before and after the collision, which is the common property with solitons
in the single-component case. In this sense, PS-PS soliton collision is essentially equivalent to
two-soliton collision of the single-component NLS equation.
Figure 3 illustrates the behavior of a mutual collision between DW-type and PS-type,
where detΠ1 = 0 and detΠ3 6= 0. The behavior of collisions between DW-type and PS-type is
qualitatively alien from that between two PS-types. One observes that, in PS-type, much of
the population initially inhabiting the hyperfine substate |F = 1,mF = ±1〉 is transferred into
the hyperfine substate |F = 1,mF = 0〉 due to the collision. In contrast, in DW-type, such
spin transfer does not occur, and the domain-wall shape is preserved against the collision. This
phenomenon can be interpreted as follows. DW-type, with nonzero spin, can affect the internal
spin state of PS-type, whereas PS-type, which is expected to have zero spin in total, does not
affect the internal spin state of DW-type. This kind of spin-transfer phenomenon, called the
spin-switching, has been first predicted for the case of VBC.11) Due to the conservation laws,
the total number, the total spin and so on are invariant throughout the collision process.
Population-mixing among internal degrees of freedom is permitted, as far as the conservation
laws are not violated. The spin-switching is one of the dynamical processes which make the
spinor solitons more interesting.
Finally, for a mutual collision between two DW-types, where detΠ1 = detΠ3 = 0, the
shapes of both solitons are expected to be deformed by the collision since each soliton carries
nonzero total spin. In fact, drastic population-mixing is seen in Fig. 4, which shows an example
of this kind of collisions. One finds that domain-walls ”repel” at the collision region, rather
than collide.
(a) (b) (c)
Fig. 2. Density plots of |φ1|2 (a), |φ0|2 (b) and |φ−1|2 (c) for a mutual collision between two PS-types.
The parameters used here are k = 1, λ0 = 1, λ1 = 1.03i, λ3 = 1.05 + i, Π1 =
2 i/2
i/2 0
0 i/2
i/2 1/
. The velocity of the right (left) moving soliton is 2.00 (−3.41). The collision
takes place at t = 0.
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J. Phys. Soc. Jpn. Full Paper
(a) (b) (c)
Fig. 3. Density plots of |φ1|2 (a), |φ0|2 (b) and |φ−1|2 (c) for a mutual collision between DW-
type and PS-type. The parameters used here are the same as those of Fig. 2, except for
2i/3 −1/3
, Π3 =
0 −1/
. The right (left) moving soliton is DW-type
(PS-type).
(a) (b) (c)
|φ1|2 |φ0|2 |φ−1|2
t t t
x x x
Fig. 4. Density plots of |φ1|2 (a), |φ0|2 (b) and |φ−1|2 (c) for a mutual collision between two DW-
types. The parameters used here are the same as those of Fig. 2, except for Π1 =
1/2 i/2
i/2 −1/2
. The values more than 2 are colored white.
6. Concluding Remarks
In this paper, we have investigated dynamical properties of matter-wave bright solitons
with a finite background in the F = 1 spinor Bose-Einstein condensate. To perform our anal-
ysis concretely, we have exploited an integrable spinor model with a self-focusing nonlinearity
and the inverse scattering method under nonvanishing boundary conditions. The situation
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J. Phys. Soc. Jpn. Full Paper
that matter-wave solitons are located on a finite background fits to the experiments.
One-soliton solutions are derived explicitly and studied in detail. From the point of the
mathematical view, they offer general forms of bright soliton solutions of the NLS equation.
We have confirmed that our one-soliton solutions include those obtained in the previous works.
One-soliton solutions are classified into two kinds by the difference of boundary conditions;
DW-type and PS-type. The spin density profiles of one-solitons vary depending on the bound-
ary conditions. In the case of purely imaginary discrete eigenvalues, we have analytically shown
that DW-type is in the ferromagnetic state, whereas PS-type is in the polar state. The exis-
tence of two distinct magnetic properties for one-soliton solutions also gives rise to fascinating
phenomena in the case for two-soliton collisions, for example, the spin-switching. The above
results for bright solitons with a finite background agree with those for bright solitons under
VBC10, 11) and dark solitons.13)
Several problems still remain. It is desirable to extend our analysis to the case of general
discrete eigenvalues. The computations of the conserved quantities other than the total spin
are also required. (One approach is given in Appendix.) In addition, we wish to investigate
analytical properties of general N -soliton solutions under NVBC in the spinor model. Needless
to say, too complicated calculations are inevitable for the above problems. The remaining
problems should be discussed elsewhere.
We conclude that the properties of the multiple matter-wave solitons in the spinor BECs
are interesting and should be useful in various applications. Bright solitons are preferable to
dark solitons for applications, because of the advantage in the propagation distance. We hope
that our analysis contributes to illuminating dynamical properties of solitons in the spinor
BECs, which should be demonstrated experimentally in near future.
Acknowledgment
One of the authors (TK) acknowledges Dr. J. Ieda and Dr. M. Uchiyama for valuable
comments and discussions.
Appendix: Several Conserved Quantities of One-Soliton Solutions
The conserved quantities help us to understand the dynamics of the system. In this ap-
pendix, we calculate the total number, the total spin, the total momentum and the total
energy of the spinor model. We assume that, in addition to purely imaginary discrete eigen-
values, Π is a real symmetric 2× 2 matrix. The condition that Π is a real symmetric matrix
is inherent in the self-defocusing case, i.e., dark solitons.12)
For Π = Π†, one-soliton solutions of purely imaginary discrete eigenvalues (43)-(45) be-
come the following form at t = t′ = (4n− 1)π/8λ20ξλ for n = 0,±1, . . . :
Q = λ0 e
iφ(x,t)
I + 4iζ
Πe−(χP+ρ
′/2) + (σyΠσy) eχP+ρ
′/2detΠ
e−(2χP+ρ
′) + 1 + e2χP+ρ
(detΠ)2
, (A·1)
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J. Phys. Soc. Jpn. Full Paper
where eρ
′/2 ≡ λ/ζ. This form is suitable for calculations, since the imaginary part is separated
from the real one. One can see clearly that the one-soliton solutions under VBC (32) are located
on a finite background in the form (A·1). Note that the domain-wall shape is lost even for
detΠ = 0 there.
Several conserved quantities of the solitons (A·1) are calculated by use of eqs. (20)-(23).
The results for detΠ = 0 are given by
N̄T = 4λ0ζ, (A·2)
FT = N̄T
2α(β + γ)
β2 − γ2
, |F T| = N̄T, (A·3)
P̄T = N̄T~k, (A·4)
ĒT = N̄Tc
(k2 − 2λ20)−
, (A·5)
and those for detΠ 6= 0 are given by
N̄T = 8λ0ζ, (A·6)
FT = (0, 0, 0)
T , (A·7)
P̄T = N̄T~k, (A·8)
ĒT = N̄Tc
(k2 − 2λ20)−
. (A·9)
It is intriguing that, for fixed amplitude and discrete eigenvalue, N̄T, P̄T and ĒT of the
PS-type (detΠ 6= 0) have just twice values as those of the DW-type (detΠ = 0), respectively.
This enables us to interpret that the PS-type of solitons is a bound state of the two DW-types
of solitons. Additionally, for fixed amplitude and total number, the total energy ĒT of the
DW-type is lower than that of the PS-type: ĒDWT − ĒPST = −N̄3Tc/16 < 0, which suggests that
the DW-type is physically preferable. This result is consistent with inter-atomic ferromagnetic
interaction, i.e., c̄2 < 0.
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J. Phys. Soc. Jpn. Full Paper
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18/18

  We investigate dynamical properties of bright solitons with a finite
background in the F=1 spinor Bose-Einstein condensate (BEC), based on an
integrable spinor model which is equivalent to the matrix nonlinear
Schr\"{o}dinger equation with a self-focusing nonlineality. We apply the
inverse scattering method formulated for nonvanishing boundary conditions. The
resulting soliton solutions can be regarded as a generalization of those under
vanishing boundary conditions. One-soliton solutions are derived in an explicit
manner. According to the behaviors at the infinity, they are classified into
two kinds, domain-wall (DW) type and phase-shift (PS) type. The DW-type implies
the ferromagnetic state with nonzero total spin and the PS-type implies the
polar state, where the total spin amounts to zero. We also discuss two-soliton
collisions. In particular, the spin-mixing phenomenon is confirmed in a
collision involving the DW-type. The results are consistent with those of the
previous studies for bright solitons under vanishing boundary conditions and
dark solitons. As a result, we establish the robustness and the usefulness of
the multiple matter-wave solitons in the spinor BECs.

Introduction
In 2002, matter-wave bright solitons in quasi-one-dimensional (1D) Bose-Einstein conden-
sates (BECs) were observed experimentally.1, 2) Bright solitons propagate in most cases with
much larger amplitudes than dark solitons,3, 4) and are expected to have the potential for
various applications such as coherent transport and atom interferometry. Soliton propagation
in BEC can be described by the Gross-Pitaevskii (GP) equation. The GP equation, called the
nonlinear Schrödinger (NLS) equation in nonlinear science, is integrable and has soliton solu-
tions in a one-dimensional and uniform system. Recent experimental and theoretical advances
about matter-wave bright solitons are reviewed, for instance, in ref. 5.
The experimental creation of matter-wave solitons has been so far achieved only for single-
component BEC. It is, nevertheless, very interesting to consider soliton propagation in BEC
with internal degrees of freedom, so-called, spinor BEC. When BEC of ultracold alkali atoms
is trapped exclusively by optical means, the hyperfine spin of atoms remains liberated. The
spinor BEC was realized in such a way.6–8) Internal degrees of freedom endow solitons with
a multiplicity. The multiple solitons will show a rich variety of dynamics. Here, we focus on
the boson system in the F = 1 hyperfine spin state, exemplified by 23Na, 39K and 87Rb. The
multi-component GP equation for F = 1 spinor BEC turns to an integrable model at special
points, which is mathematically equivalent to the matrix NLS equation. An integrable model
with a self-focusing nonlineality enables one to perform exact analysis via the inverse scatter-
ing method (ISM) for the matrix NLS equation.9) In particular, bright soliton solutions under
vanishing boundary conditions (VBC) are obtained, whose properties are investigated in refs.
10 and 11. Recently, the ISM for the matrix NLS equation under nonvanishing boundary con-
ditions (NVBC) is formulated.12) Dark solitons in the F = 1 spinor BEC can be investigated
by applying the ISM under NVBC to an integrable model with a self-defocusing nonlineal-
ity.13) Although the ISM under NVBC is dedicated mainly to the self-defocusing case, we note
that this technique is also applicable to an integrable model with a self-focusing nonlineality,
which makes us available to bright soliton solutions with a finite background.
In this paper, the detail of matter-wave bright solitons in the quasi-1D F = 1 spinor BEC
is further investigated, based on an integrable model. We consider matter-wave spinor bright
solitons traveling on a finite background of the condensate. We write down explicitly new
soliton solutions, and verify that the obtained soliton solutions have the similar properties
compared to those without a background. In the usual experimental setups, the condensates
are confined in a finite-size regime, and the matter-wave bright solitons will accompany a
finite background. The study given in this paper is meaningful in such realistic circumstances.
The paper is organized as follows. In § 2, the GP equation for quasi-1D F = 1 spinor BEC
is introduced. In particular, the integrable model is presented. There, the interactions between
two atoms are supposed to be inter-atomic attractive and ferromagnetic, which lead to bright
J. Phys. Soc. Jpn. Full Paper
solitons. In § 3, the inverse scattering method under nonvanishing boundary conditions is
applied to the integrable model. This application leads to bright soliton solutions with a finite
background. Several conserved quantities of the model are also provided. One-soliton solutions
are investigated in § 4. The spin states of one-solitons are classified, assuming that discrete
eigenvalues are purely imaginary. Two-soliton solutions are discussed in § 5. The last section,
§ 6, is devoted to the concluding remarks.
2. GP Equation for F = 1 Spinor Bose-Einstein Condensates
For BEC of ultracold alkali atoms, the mean-field theory works well, because almost all
atoms go into condensation and the condensate is dilute. In this paper, we deal with the
quasi-one-dimensional system. Atoms in the F = 1 hyperfine spin state have three magnetic
substates labeled by the magnetic quantum number mF = 1, 0,−1. The system is charac-
terized by a vectorial field operator with the components corresponding to each substate,
Φ̂ = (Φ̂1, Φ̂0, Φ̂−1)
T , satisfying equal-time commutation relations:
[Ψ̂α(x, t), Ψ̂
(x′, t)] = δαβδ(x− x′), (1)
where the subscripts α, β take on 1, 0,−1. In the framework of the mean-field theory for BEC,
the quantum field is replaced with the order parameter:
Φ(x, t) ≡ 〈Φ̂(x, t)〉 = (Φ1(x, t),Φ0(x, t),Φ−1(x, t))T . (2)
Φ(x, t) is often called the spinor condensate wavefunction, which is normalized to the total
number of atoms NT:
dxΦ†(x, t)Φ(x, t) = NT. (3)
The spinor condensate wavefunction obeys a set of coupled evolution equations, namely, the
multi-component GP equation:
i~∂tΦ1 = −
∂2xΦ1 + (c̄0 + c̄2)
|Φ1|2 + |Φ0|2
+(c̄0 − c̄2)|Φ−1|2Φ1 + c2Φ∗−1Φ20,
i~∂tΦ0 = −
∂2xΦ0 + (c̄0 + c̄2)
|Φ1|2 + |Φ−1|2
+c̄0|Φ0|2Φ0 + 2c̄2Φ∗0Φ1Φ−1,
i~∂tΦ−1 = −
∂2xΦ−1 + (c̄0 + c̄2)
|Φ−1|2 + |Φ0|2
+(c̄0 − c̄2)|Φ1|2Φ−1 + c̄2Φ∗1Φ20, (4)
where c̄0 = (ḡ0 + 2ḡ2)/3 and c̄2 = (ḡ2 − ḡ0)/3 denote effective 1D coupling constants for the
mean-field and the spin-exchange interaction, respectively. Here, the effective 1D coupling
constants ḡf are given by
ḡf =
4~2af
1− Caf/a⊥
, (5)
J. Phys. Soc. Jpn. Full Paper
where af are the s-wave scattering lengths in the total hyperfine spin f channel, a⊥ is the
size of the transverse ground state, m is the atomic mass, and C = −ζ(1/2) ≃ 1.46. Note that
one may change the values of c̄0 and c̄2 by tuning a⊥.
Equation (4) is derived as follows. The interaction between two atoms in the F = 1
hyperfine spin state has a form,15, 16)
V̂ (x1 − x2) = δ(x1 − x2)
c̄0 + c̄2F̂ 1 · F̂ 2
, (6)
where F̂ i is the spin operator. The Gross-Pitaevskii energy functional is thus given by
EGP[Φ] =
α∂xΦα +
α′Φα′Φα +
αβ · fα′β′Φβ′Φβ
, (7)
where repeated subscripts (α, β, α′, β′ = 1, 0,−1) should be summed up and f = (fx, fy, fz)T
with fi(i = x, y, z) being 3× 3 spin-1 matrices. Then, the variational principle: i~∂tΦα(x, t) =
δEGP[Φ]/δΦ
α(x, t), for α = 1, 0,−1, yields eq. (4).
An important fact is that eq. (4) possesses a completely integrable point when c̄0 = c̄2 ≡
−c < 0, equivalently, 2ḡ0 = −ḡ2 > 0.10, 11) This condition is realized when
a⊥ = 3C
2a0 + a2
, (8)
assuming that a0a2(a2−a0) > 0 holds. The situation corresponds to attractive (c̄0 < 0) and fer-
romagnetic (c̄2 < 0) interaction. When we change the wavefunction by Φ = (φ1,
2φ0, φ−1)
and measure time and length in units of t̄ = ~a⊥/c and x̄ = ~
a⊥/2mc, respectively, we can
rewrite eq. (4) with c̄0 = c̄2 ≡ −c < 0 into the dimensionless form,
i∂tφ1 = −∂2xφ1 − 2(|φ1|2 + 2|φ0|2)φ1 − 2φ∗−1φ20,
i∂tφ0 = −∂2xφ0 − 2(|φ−1|2 + |φ0|2 + |φ1|2)φ0 − 2φ∗0φ1φ−1,
i∂tφ−1 = −∂2xφ−1 − 2(|φ−1|2 + 2|φ0|2)φ−1 − 2φ∗1φ20. (9)
Then, eq. (9) is found to be equivalent to a 2× 2 matrix version of the NLS equation with a
self-focusing nonlinearity:
i∂tQ+ ∂
xQ+ 2QQ
†Q = O, (10)
with an identification,
φ1 φ0
φ0 φ−1
. (11)
The matrix NLS equation (10) is integrable in the sense that the initial value problem can
be solved via the inverse scattering method.9, 12) The integrability of the reduced equations
(9) is thus proved automatically. Thus, we have derived the integrable spinor model. Another
integrable point of eq. (4) is c̄0 = c̄2 ≡ c > 0, i.e., the matrix NLS equation with a self-
defocusing nonlineality.13) Special solutions for generic coupling constants c̄0, c̄2 are given in
ref. 17.
J. Phys. Soc. Jpn. Full Paper
3. Bright Solitons with a Finite Background
We consider bright soliton solutions of the integrable model (9) under NVBC, whereas
those under VBC are studied in refs. 10 and 11. We summarize briefly the results of the inverse
scattering method for eq. (9) with NVBC.12)
We define the nonvanishing boundary conditions as
Q(x, t) → Q±, x→ ±∞,
±Q± = Q±Q
± = λ
0I, (12)
where λ0 is a positive real constant and I denotes a 2 × 2 unit matrix. Note that vanishing
boundary conditions are recovered as λ0 → 0. The analysis of the ISM under NVBC yields
the standard form of the multiple soliton solutions of the 2 × 2 matrix NLS equation with a
self-focusing nonlineality (10) as
Q(x, t) = λ0 e
I + 2i(I · · · I
︸ ︷︷ ︸
. (13)
Here, a 2 × 2 complex matrix Πj is called the polarization matrix. S is a 2N × 2N matrix
defined by
Sij =
ζj + λj
δijI +
i(ζi + ζj)
ζi + λi
ζj + λj
1 ≤ i, j ≤ N, (14)
where λj is a complex discrete eigenvalue for the bound state and ζj = (λ
j + λ
1/2 with
Im ζj > 0 for j = 1, . . . , N . It is required for the ISM under NVBC that a two-sheet Riemann
surface is introduced appropriately, due to a double-valued function ζj. The phase of the
carrier wave, φ(x, t), is given by
φ(x, t) = kx− (k2 − 2λ20)t+ δ, (15)
and the coordinate function is given by
χj ≡ χj(x, t) = 2iζj(x− 2(λj + k)t). (16)
The above solution is the M(= N/2)-soliton solution. The ISM under NVBC for the self-
focusing case results in pairs of discrete eigenvalues corresponding to each Riemann sheet.
The constraint should be imposed on λj and ζj (j = 1, · · · , N) such that λ2l−1 = λ∗2l and
ζ2l−1 = −ζ∗2l for l = 1, · · · , N/2. At the same time, Πj must satisfy that Π2l−1 = Π
For our reduction to the integrable model for F = 1 spinor BEC, we must make the
potential Q symmetric, noting eq. (11). The symmetry of Q is naturally reflected in Πj.
When we take every Πj to be symmetric in eq. (13), soliton solutions of the integrable model
(9) under NVBC are obtained.
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The form (13) is the standard form of soliton solutions in the sense that the boundary
value at x→ ∞ ⇔ eχj → 0 is supposed to be fixed as
Q(x, t) e−iφ(x,t) → λ0I, x→ ∞. (17)
The spinor model, however, allows the SU(2) transformation of the solutions, if they are kept
symmetric. To be concrete, let U be a 2× 2 unitary matrix. When Q is a solution of eq. (10)
with eq. (11), then
Q′ = UQUT , (18)
is also a solution. Assuming that Q is the standard form (13), the limit x→ ∞ of Q′ becomes
Q′ e−iφ(x,t) → λ0 UUT ≡ λ0Q′+, x→ ∞. (19)
Q′+ = UUT is the so-called Cholesky decomposition. The arbitrary boundary conditions Q′+
other than eq. (17) are thus realized via the SU(2) transformation. On the other hand, the
behavior in the limit x → −∞ varies depending on whether detΠ = 0 or not, which will be
discussed later for the one-soliton case.
There is another important concept about an integrable model. Due to the integrability,
the model has the infinite conservation laws, which restrict the dynamics of the system in an
essential way. Several conserved quantities, related to the physical quantities of the system,
are listed below:
Total number : N̄T =
dx n̄(x, t),
n̄(x, t) = tr (Q†Q)− tr (Q†±Q±). (20)
Total spin : FT =
dx f(x, t),
f(x, t) = tr (Q†σQ). (21)
Total momentum : P̄T =
dx p̄(x, t),
p̄(x, t) = −i~[tr (Q†Qx)− tr (Q†±Q±,x)]. (22)
Total energy : ĒT =
dx ē(x, t),
ē(x, t) = c[tr (Q†xQx −Q†QQ†Q)
−tr (Q†±,xQ±,x −Q
±Q±)]. (23)
Here, σ = (σx, σy, σz)
T are the Pauli matrices. To avoid the divergence of integrals, we should
subtract the contribution of the background from the physical quantities, except the total spin,
to which the background does not contribute explicitly. These subtractions are emphasized
by the bars on the conserved densities and quantities. The local spin density f = (fx, fy, fz)
is covariant under the SU(2) transformation (18), whereas the other densities such as n̄(x, t),
J. Phys. Soc. Jpn. Full Paper
p̄(x, t) and ē(x, t) are invariant. The total macroscopic spin is directed to face to an arbitrary
direction by the global spin rotation. The SU(2) symmetry of the system causes the energy
degenerated states of solitons for this spin rotation.
4. One-Soliton Solutions
In this section, one-soliton solutions of the integrable spinor model (9) are investigated in
detail. We can derive the explicit form of one-soliton solutions by setting N = 2 (M = 1) in
the formula (13). The calculation is complicated but straightforward. The result is as follows:
Q = λ0 e
iφ(x,t)
I + 2i
, (24)
where detS is given by
detS = κ21κ
2 − eχ1ν1κ1κ22 trΠ− ex2ν2κ21κ2 trΠ†
+eχ1+χ2κ1κ2
̟ + ν1ν2(|tr Π|2 − 1)
+e2χ1ν21κ
2 detΠ + e
2χ2ν22κ
1 detΠ
− e2χ1+χ2ν1κ2̟ trΠ† detΠ− eχ1+2χ2ν2κ1̟ trΠdetΠ†
+e2χ1+2χ2̟2|detΠ|2, (25)
and T is a 2× 2 matrix such that
T = eχ1κ1κ
2 Π+ e
χ2κ21κ2 Π
− eχ1+χ2κ1κ2
ς1 trΠ · Π† + ς2 trΠ† ·Π+ µ(|trΠ|2 − 1)I
− e2χ1µ1κ22 detΠ · I − e2χ2µ2κ21 detΠ† · I
+e2χ1+χ2κ2 detΠ
ς21 Π
† +̟ trΠ† · I
+eχ1+2χ2κ1 detΠ
ς22 Π+̟ trΠ · I
− e2χ1+2χ2̟ (ς1 + ς2) |detΠ|2 · I. (26)
We explain physical meanings of notations. The phase of the carrier wave is given by
φ(x, t) = kx− (k2 − 2λ20)t+ δ. (27)
Let λj and ζj = (λ
j + λ
1/2 for j = 1, 2 be complex constants satisfying λ1 = λ
2 and
ζ1 = −ζ∗2 . Without loss of generality, we assume that (λ1, ζ1) ((λ2, ζ2)) belongs to the upper
(lower) Riemann sheet, which is characterized such that Im ζ Im λ > 0 (Im ζ Im λ < 0).
χj(x, t) is expressed in terms of them as
χ1 ≡ χ1(x, t) = 2iζ1(x− 2(λ1 + k)t), (28)
χ2 ≡ χ2(x, t) = 2iζ2(x− 2(λ2 + k)t). (29)
Note that χ1 = χ
2 ≡ χ holds. Reχ thus denotes the coordinate of the envelope soliton,
whereas Imχ implies the self-modulation phase. The polarization matrix Π is a 2×2 symmetric
J. Phys. Soc. Jpn. Full Paper
matrix. Here, the normalization in a sense of the square norm is imposed on Π as a matter of
convenience:
, 2|α|2 + |β|2 + |γ|2 = 1. (30)
The other parameters are expedient functions of λ0, λ1 and λ2:
ζj + λj
, µ = iλ0
κ1 + κ2
ζ1 + ζ2
, νj =
iλ0κj
, ̟ = ν1ν2 − µ2, ςj = νj − µ, (31)
for j = 1, 2. We list the meaning of each parameter as follows:
k : wave number of soliton’s carrier wave.
λ0 : amplitude of soliton’s carrier wave.
φ(x, t) : phase of soliton’s carrier wave.
Re χ(x, t) : coordinate of soliton’s envelope.
Im χ(x, t) : self-modulation phase of soliton.
Π : symmetric polarization matrix of soliton.
Equations (24)-(26) are new soliton solutions which had never been written down explicitly
in the literatures. If we take the vanishing limit λ0 → 0, ζ1 and ζ2 converge at λ1 and −λ2,
respectively. Then,
· 2kR
Πe−(χR+ρ/2) + (σyΠ†σy) eχR+ρ/2detΠ
e−(2χR+ρ) + 1 + e2χR+ρ|detΠ|2
eiχI , (32)
with notations
eρ/2 ≡ 1
, (33)
χR ≡ χR(x, t) = kR(x− 2kI t)− ǫ, (34)
χI ≡ χI(x, t) = kIx+ (k2R − k2I )t, (35)
each of which holds the following correspondence respectively:
kR = −2Imλ1, (36)
kI = 2Reλ1 + k, (37)
ǫ = − ln(4|λ1|). (38)
Equations (32)-(35) are the same forms as those in ref. 11, except a phase factor. This con-
sequence is natural but non-trivial, because the formula of solitons under VBC9) is quite
different from that under NVBC (13), in particular, in the form of the matrix S. Actually, the
initial displacement ǫ can be arbitrarily changed, regardless of eq. (38), by the parallel shift
of the position x. We have shown that the soliton solutions (24)-(26) can be regarded as a
general form of bright soliton solutions, including the case of VBC.
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4.1 Classification by the boundary conditions
We shall show that there are two kinds of one-soliton solutions depending on the boundary
conditions, detΠ = 0 or detΠ 6= 0. The similar classification about the boundary conditions
also exists for dark solitons.13) The examples of snapshots of one-soliton density profiles are
shown in Fig. 1. The upper row is for detΠ = 0, and the lower row is for detΠ 6= 0. The
shape of envelope solitons looks a locally-oscilating wave rather than, literally, a solitary wave,
because of the self-modulation due to the complex velocity.
For detΠ = 0, the boundary conditions of the standard form (24)-(26) are
Q e−iφ → λ0I, x→ ∞,
Q e−iφ → λ0
I − 2i ς1 trΠ · Π
† + ς2 trΠ
† · Π+ µ(|trΠ|2 − 1)I
̟ + ν1ν2(|tr Π|2 − 1)
, x→ −∞. (39)
The left and right boundary values differ in not only the global phase but also the population
of each component, in general. That is, those are the SU(2) rotated boundary conditions.
In the upper row of Fig. 1, we see that the envelope soliton of each component forms the
domain-wall (DW) shape, although it does not manifest in the total number density.
On the other hand, for detΠ 6= 0, the boundary conditions are
Q e−iφ → λ0I, x→ ∞,
Q e−iφ → λ0
1− 2i
ς1 + ς2
I, x→ −∞. (40)
In contrast to the case that detΠ = 0, both boundary values are diagonal matrices, and only
the phase-shift (PS) occurs. That is, those are the U(1) rotated boundary conditions.
For the above reasons, we call one-soliton solutions the DW-type for detΠ = 0, and the
PS-type for detΠ 6= 0. Remark the following; the spin density profile of DW-type suggests
that the total spin is nonzero, whereas that of PS-type is dipole-shape, implying that the total
spin amounts to zero. See the right panel of Fig. 1. This observation will be solidified in § 4.3.
4.2 Case of purely imaginary discrete eigenvalues
The ISM performed on Riemann sheets involves a double-valued function of the spectral
parameter, and it usually renders a very complicated representation of N -soliton solutions
even for N = 1, as is seen from eqs. (24)-(26). To simplify an explicit representation, it is
convenient to assume that λj and ζj are purely imaginary.
18) The similar approach is employed
for the analysis of N -soliton solutions of the derivative NLS equation under NVBC.20)
If we take a pair of discrete eigenvalues as
(λ1, ζ1) ≡ (iλ0λ, iλ0ζ), (λ2, ζ2) ≡ (−iλ0λ, iλ0ζ), (41)
where λ and ζ are positive real numbers such that
λ > 1, ζ =
λ2 − 1, (42)
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(a) (b) (c)
Fig. 1. Snapshots of one-soliton density profiles. The upper row is plotted for detΠ = 0 at the moment
t = 0, with k = 0, λ0 = 1, λ1 = 1+i, ξ1 = 1.27+0.79i (χ(x, t) = −(1.57−2.54i)x− (8.23−1.94i)t)
and Π =
4/5 2/5
2/5 1/5
. The lower row is plotted for detΠ 6= 0 at the moment t = 0, with the
same parameters except for Π =
2 2/5
2/5 3/(5
. The left panel (a) depicts the local density
for each component, |φ1|2 (solid line), |φ0|2 (chain line) and |φ−1|2 (dotted line). The center panel
(b) depicts the local number density n, where the contribution of the background is included. The
right panel (c) depicts the local spin densities, fx (solid line) and fz (dotted line). fy vanishes
identically due to a choice of a real matrix Π.
we obtain a relatively simple form of one-soliton solutions as
Q = λ0 e
iφ(x,t)
I + 2i
, (43)
detS = 1−
(tr Π(t) + trΠ†(t)) + e2χP
|trΠ(t)|2
(detΠ(t) + detΠ†(t))−
trΠ†(t)detΠ(t) + trΠ(t)detΠ†(t)
e4χP |detΠ(t)|2, (44)
2i T =
2(|tr Π(t)|2 − 1) e2χP + 2e2χP
detΠ(t) +
detΠ†(t)
tr Π†(t)detΠ(t) +
trΠ(t)detΠ†(t)
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−2(ζ + λ) eχP + 2
e2χP trΠ†(t) + 2
e3χP detΠ†(t)
−2(ζ − λ) eχP − 2
e2χP trΠ(t) + 2
e3χP detΠ(t)
Π†(t), (45)
where the coordinate of the envelope soliton is given by
χP (x, t) = −2λ0ζ(x− 2kt), (46)
and the time dependence of the phase modulation is embedded in the polarization matrix,
namely,
Π(t) ≡ Πe4iλ20ζλt. (47)
When we take the limit λ0 → 0 with λ0λ and λ0ζ kept finite in eqs. (43)-(45), Q converges
to the form of eqs. (32)-(35), accompanying the parameters kR = −2λ0λ, kI = k and ǫ =
− ln(4λ0λ). Here, kR and kI are independent free parameters, apart from the trivial initial
displacement ǫ. In this sense, in spite of the reduction, we can still regard the form of one-
soliton solutions (43)-(45) as a general form of those under VBC.
We can also take another limit. That is, we consider the reduction to the single-component
case. If we set
k = 0, Π =
eiθ 0
, (48)
the (1,1)-component of Q becomes
Q11 · e−i(2λ
t+δ) = λ0 − 2λ0ζ
ζ cos(4λ20ζλt+ θ) + iλ sin(4λ
0ζλt+ θ)
λ cosh(2λ0ζx+ ψ)− cos(4λ20ζλt+ θ)
, (49)
where eψ = ζ/λ. The form (49) was given in ref. 18. We thus verify that our soliton solutions
are also generalization of those for the single-component NLS equation under NVBC.19)
4.3 Spin states
In this subsection, we discuss the spin states of one-soliton solutions with a finite back-
ground, by calculating the total spin. The conservation law guarantees that we obtain the
total spin FT from integrating at arbitrary time. Therefore, we can select the time so that the
calculation becomes easier. We concentrate on the case of purely imaginary discrete eigenval-
ues. As a result, we see that the DW-type is associated with the ferromagnetic state, whereas
the PS-type is associated with the polar state. One-soliton solutions for purely imaginary dis-
crete eigenvalues (43)-(45) include those under VBC apart from an initial displacement, and
therefore the classification about the spin states presented below is wider than that performed
before.10, 11)
4.3.1 Ferromagnetic state
For detΠ = 0 (DW-type), we substitute eqs. (43)-(45) into eq. (21) and calculate the total
spin. The time t′ such that trΠ(t′) + trΠ†(t′) = 0 is suitable for the calculation. The result is
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as follows:
FT = 4λ0τ
2λRe{α∗(β + γ)}
−2ζIm{α∗(β − γ)}
λ(|β|2 − |γ|2)
, τ ≡
|β + γ|2
, (50)
with the modulus
|F T|2 = (4λλ0)2τ. (51)
The total number of the particles transformed into a soliton, N̄T, is calculated by eq. (20),
N̄T = 4λ0ζ, (52)
and the range of the value taken by |F T|2 is expressed in terms of N̄T,
N̄2T ≤ |F T|2 ≤ N̄2T + (4λ0)2. (53)
Remark that, in the vanishing limit λ0 → 0, the modulus of the total spin is always equal to
the total number of the particles, namely, |F T| → NT.
With nonzero total spin, the DW-type of solitons belongs to the ferromagnetic state.
Since inter-atomic ferromagnetic interactions are supposed here, solitons tend to take the
ferromagnetic state or DW-type. In various contexts of physics, the domain-walls are topo-
logical solitons related with the symmetry breaking. Here, resulting from the domain-walls,
the magnetic entity emerges as the spontaneously broken symmetry. It is worthy to notice
that the case |N̄T| < |FT| may happen. The background is spinless, but its internal spin state
appears to be affected on the ground that the ferromagnetic soliton runs over the background.
Thereby, the background contributes to the total spin.
4.3.2 Polar state
If detΠ 6= 0 (PS-type), the solitons show the other magnetic property. The time t′ such
that detΠ(t′) = detΠ†(t′) > 0 is suitable for the analysis. After lengthy calculations, the local
spin density at such time is derived as
= 8λ20 e
−3υΞ−2(χP ′)
β − γ
ζ e2υΞ(χP ′) sinh(χP ′)
(ζ2 − λ2)tr Π(t′) + (ζ2 + λ2)trΠ†(t′)
sinh(2χP ′)
λ2/ζ · ((tr Π†(t′))2 − |tr Π(t′)|2) + 4ζdetΠ(t′) + 2ζ(|tr Π(t′)|2 − 1)
sinh(χP ′)
+h.c., (54)
fy = 32λ
−3υIm{α∗(β − γ)}Ξ−2(χP ′)
2ζ eυ sinh(2χP ′)− (trΠ(t′) + trΠ†(t′)) sinh(χP ′)
, (55)
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where Ξ(χP ′) is an even function of χP ′ , χP ′(x, t) is a parallel-shifted coordinate, and υ is a
constant:
Ξ(χP ′) ≡ 2 cosh(2χP ′)−
2 e−υ
(trΠ(t′) + tr Π†(t′)) cosh(χP ′)
ζ2 + |trΠ(t′)|2
λ2detΠ(t′)
, (56)
χP ′ ≡ χP + υ = −2λ0ζ(x− 2kt′) + υ, (57)
υ ≡ ln
(detΠ(t′))1/2
. (58)
Note that fx and fz share the same functional form. Each component of the above local spin
density is an odd function of χP ′ and, in particular, it has the node at the same point x0 such
that χP ′(x0, t
′) = 0, namely, x0 = 2kt
′ + (2λ0ζ)
−1υ. Consequently, the total spin amounts to
zero:
dxf(χP ′) = (0, 0, 0)
T . (59)
For this reason, the PS-type of solitons, on the average, belongs to the polar state.15)
5. Two-Soliton Collision
We proceed to the discussion of two-soliton collisions in the integrable spinor model (9).
Two-soliton solutions are obtained by setting N = 4 (M = 2) in the formula (13). There
exist two independent discrete eigenvalues and symmetric polarization matrices, respectively,
i.e., λ1 = λ
2 and Π1 = Π
2 for one of solitons, λ3 = λ
4 and Π3 = Π
4 for the other. Each
soliton is separated at t = ±∞. Then, a two-soliton is asymptotically two one-solitons. The
classification of one-soliton solutions based on the values of the determinants of polarization
matrices, discussed in the previous section, is thus valid for two-soliton solutions.
The derivation of the explicit form is more complicated than that in the case for one-soliton
solutions. For the derivative NLS equation under NVBC, explicit two-soliton solutions and
shifts of soliton positions due to collisions between solitons have been analytically obtained,
in the case of purely imaginary eigenvalues, where complexity of calculation is considerably
reduced.20) This strategy, however, does not stand in our NLS equation under NVBC. The
reason is understood from eq. (46). In the spinor model, purely imaginary eigenvalues give
two solitons with the same velocity 2k, and they do not collide with each other. Accordingly,
we can not investigate the properties of collisions for purely imaginary discrete eigenvalues.
No one has studied explicit multi-soliton solutions of the NLS equation under NVBC, even in
the single-component case, because of the computational complexity.
Here, we graphically show the characteristic behaviors of two-soliton collisions in the
spinor model, by use of the exact solutions given by the ISM. Referring to them, we carry
out the qualitative discussions. Although the presented graphs are depicted for the specific
parameters, much the same behaviors are observed for arbitrarily selected parameters.
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Figure 2 illustrates the behavior of a mutual collision between two PS-types, where
detΠ1 6= 0 and detΠ3 6= 0. One can see that, in all three components, Both solitons re-
tain their shapes before and after the collision, which is the common property with solitons
in the single-component case. In this sense, PS-PS soliton collision is essentially equivalent to
two-soliton collision of the single-component NLS equation.
Figure 3 illustrates the behavior of a mutual collision between DW-type and PS-type,
where detΠ1 = 0 and detΠ3 6= 0. The behavior of collisions between DW-type and PS-type is
qualitatively alien from that between two PS-types. One observes that, in PS-type, much of
the population initially inhabiting the hyperfine substate |F = 1,mF = ±1〉 is transferred into
the hyperfine substate |F = 1,mF = 0〉 due to the collision. In contrast, in DW-type, such
spin transfer does not occur, and the domain-wall shape is preserved against the collision. This
phenomenon can be interpreted as follows. DW-type, with nonzero spin, can affect the internal
spin state of PS-type, whereas PS-type, which is expected to have zero spin in total, does not
affect the internal spin state of DW-type. This kind of spin-transfer phenomenon, called the
spin-switching, has been first predicted for the case of VBC.11) Due to the conservation laws,
the total number, the total spin and so on are invariant throughout the collision process.
Population-mixing among internal degrees of freedom is permitted, as far as the conservation
laws are not violated. The spin-switching is one of the dynamical processes which make the
spinor solitons more interesting.
Finally, for a mutual collision between two DW-types, where detΠ1 = detΠ3 = 0, the
shapes of both solitons are expected to be deformed by the collision since each soliton carries
nonzero total spin. In fact, drastic population-mixing is seen in Fig. 4, which shows an example
of this kind of collisions. One finds that domain-walls ”repel” at the collision region, rather
than collide.
(a) (b) (c)
Fig. 2. Density plots of |φ1|2 (a), |φ0|2 (b) and |φ−1|2 (c) for a mutual collision between two PS-types.
The parameters used here are k = 1, λ0 = 1, λ1 = 1.03i, λ3 = 1.05 + i, Π1 =
2 i/2
i/2 0
0 i/2
i/2 1/
. The velocity of the right (left) moving soliton is 2.00 (−3.41). The collision
takes place at t = 0.
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(a) (b) (c)
Fig. 3. Density plots of |φ1|2 (a), |φ0|2 (b) and |φ−1|2 (c) for a mutual collision between DW-
type and PS-type. The parameters used here are the same as those of Fig. 2, except for
2i/3 −1/3
, Π3 =
0 −1/
. The right (left) moving soliton is DW-type
(PS-type).
(a) (b) (c)
|φ1|2 |φ0|2 |φ−1|2
t t t
x x x
Fig. 4. Density plots of |φ1|2 (a), |φ0|2 (b) and |φ−1|2 (c) for a mutual collision between two DW-
types. The parameters used here are the same as those of Fig. 2, except for Π1 =
1/2 i/2
i/2 −1/2
. The values more than 2 are colored white.
6. Concluding Remarks
In this paper, we have investigated dynamical properties of matter-wave bright solitons
with a finite background in the F = 1 spinor Bose-Einstein condensate. To perform our anal-
ysis concretely, we have exploited an integrable spinor model with a self-focusing nonlinearity
and the inverse scattering method under nonvanishing boundary conditions. The situation
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that matter-wave solitons are located on a finite background fits to the experiments.
One-soliton solutions are derived explicitly and studied in detail. From the point of the
mathematical view, they offer general forms of bright soliton solutions of the NLS equation.
We have confirmed that our one-soliton solutions include those obtained in the previous works.
One-soliton solutions are classified into two kinds by the difference of boundary conditions;
DW-type and PS-type. The spin density profiles of one-solitons vary depending on the bound-
ary conditions. In the case of purely imaginary discrete eigenvalues, we have analytically shown
that DW-type is in the ferromagnetic state, whereas PS-type is in the polar state. The exis-
tence of two distinct magnetic properties for one-soliton solutions also gives rise to fascinating
phenomena in the case for two-soliton collisions, for example, the spin-switching. The above
results for bright solitons with a finite background agree with those for bright solitons under
VBC10, 11) and dark solitons.13)
Several problems still remain. It is desirable to extend our analysis to the case of general
discrete eigenvalues. The computations of the conserved quantities other than the total spin
are also required. (One approach is given in Appendix.) In addition, we wish to investigate
analytical properties of general N -soliton solutions under NVBC in the spinor model. Needless
to say, too complicated calculations are inevitable for the above problems. The remaining
problems should be discussed elsewhere.
We conclude that the properties of the multiple matter-wave solitons in the spinor BECs
are interesting and should be useful in various applications. Bright solitons are preferable to
dark solitons for applications, because of the advantage in the propagation distance. We hope
that our analysis contributes to illuminating dynamical properties of solitons in the spinor
BECs, which should be demonstrated experimentally in near future.
Acknowledgment
One of the authors (TK) acknowledges Dr. J. Ieda and Dr. M. Uchiyama for valuable
comments and discussions.
Appendix: Several Conserved Quantities of One-Soliton Solutions
The conserved quantities help us to understand the dynamics of the system. In this ap-
pendix, we calculate the total number, the total spin, the total momentum and the total
energy of the spinor model. We assume that, in addition to purely imaginary discrete eigen-
values, Π is a real symmetric 2× 2 matrix. The condition that Π is a real symmetric matrix
is inherent in the self-defocusing case, i.e., dark solitons.12)
For Π = Π†, one-soliton solutions of purely imaginary discrete eigenvalues (43)-(45) be-
come the following form at t = t′ = (4n− 1)π/8λ20ξλ for n = 0,±1, . . . :
Q = λ0 e
iφ(x,t)
I + 4iζ
Πe−(χP+ρ
′/2) + (σyΠσy) eχP+ρ
′/2detΠ
e−(2χP+ρ
′) + 1 + e2χP+ρ
(detΠ)2
, (A·1)
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J. Phys. Soc. Jpn. Full Paper
where eρ
′/2 ≡ λ/ζ. This form is suitable for calculations, since the imaginary part is separated
from the real one. One can see clearly that the one-soliton solutions under VBC (32) are located
on a finite background in the form (A·1). Note that the domain-wall shape is lost even for
detΠ = 0 there.
Several conserved quantities of the solitons (A·1) are calculated by use of eqs. (20)-(23).
The results for detΠ = 0 are given by
N̄T = 4λ0ζ, (A·2)
FT = N̄T
2α(β + γ)
β2 − γ2
, |F T| = N̄T, (A·3)
P̄T = N̄T~k, (A·4)
ĒT = N̄Tc
(k2 − 2λ20)−
, (A·5)
and those for detΠ 6= 0 are given by
N̄T = 8λ0ζ, (A·6)
FT = (0, 0, 0)
T , (A·7)
P̄T = N̄T~k, (A·8)
ĒT = N̄Tc
(k2 − 2λ20)−
. (A·9)
It is intriguing that, for fixed amplitude and discrete eigenvalue, N̄T, P̄T and ĒT of the
PS-type (detΠ 6= 0) have just twice values as those of the DW-type (detΠ = 0), respectively.
This enables us to interpret that the PS-type of solitons is a bound state of the two DW-types
of solitons. Additionally, for fixed amplitude and total number, the total energy ĒT of the
DW-type is lower than that of the PS-type: ĒDWT − ĒPST = −N̄3Tc/16 < 0, which suggests that
the DW-type is physically preferable. This result is consistent with inter-atomic ferromagnetic
interaction, i.e., c̄2 < 0.
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J. Phys. Soc. Jpn. Full Paper
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