Download Solitons of the resonant nonlinear Schrödinger equation with

Theoretical and Mathematical Physics, 152(1): 991–1003 (2007)
SOLITONS OF THE RESONANT NONLINEAR SCHRÖDINGER
EQUATION WITH NONTRIVIAL BOUNDARY CONDITIONS:
HIROTA BILINEAR METHOD
J.-H. Lee∗ and O. K. Pashaev†
We use the Hirota bilinear approach to consider physically relevant soliton solutions of the resonant
nonlinear Schrödinger equation with nontrivial boundary conditions, recently proposed for describing
uniaxial waves in a cold collisionless plasma. By the Madelung representation, the model transforms into
the reaction–diffusion analogue of the nonlinear Schrödinger equation, for which we study the bilinear
representation, the soliton solutions, and their mutual interactions.
Keywords: resonant nonlinear Schrödinger equation, quantum potential, cold plasma, magnetoacoustic
wave, soliton, Hirota method
1. Introduction
For describing low-dimensional gravity (the Jackiw–Teitelboim model) and the response of a medium
to the action of a quasimonochromatic wave with a complex amplitude ψ(x, t), which is a slowly varying
function of the coordinate and the time, a novel integrable version of the nonlinear Schrödinger (NLS)
equation was recently proposed [1]:
i
∂ψ ∂ 2 ψ
Λ 2
1 ∂ 2 |ψ|
+
|ψ|
+
ψ
=
s
ψ.
∂t
∂x2
4
|ψ| ∂x2
(1)
This has been called the resonant NLS (RNLS) equation. It can be considered a third version of the NLS
equation, intermediate between the defocusing and focusing cases. Although the RNLS model is integrable
for arbitrary values of the coefficient s, the critical value s = 1 separates two distinct regions of behavior:
for s < 1, the model is reducible to the conventional NLS equation, but for s > 1, it is reducible not to
the usual NLS equation but to a reaction–diffusion (RD) system. In the latter case, the model exhibits
resonance solitonic phenomena [1].
The RNLS equation can be interpreted as a particular realization of the NLS soliton propagating in
the so-called quantum potential UQ (x) = |ψ|xx /|ψ|. This potential, responsible for producing the quantum
behavior, was introduced by de Broglie [2] and was subsequently used by Bohm [3] to develop a hiddenvariable theory in quantum mechanics. It also appears in stochastic mechanics [4]. Connections between
such nonclassical motions with the internal spin motion and the zitterbewegung were considered in a series
of papers (see [5]). Quantum potentials also appear in proposed nonlinear extensions of quantum mechanics
with regard both to stochastic quantization [6], [7] and to corrections from quantum gravity [8]. It is noted
that the RNLS equation, like the conventional NLS equation, can also be derived in the context of capillarity
models [9], [10].
∗
Institute of Mathematics, Academia Sinica, Taipei 11529, Taiwan.
†
Department of Mathematics, Izmir Institute of Technology, Urla-Izmir 35430, Turkey,
e-mail: [email protected].
Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 152, No. 1, pp. 133–146, July, 2007.
c 2007 Springer Science+Business Media, Inc.
0040-5779/07/1521-0991 991
It was very recently shown [11] that the RNLS equation appears in plasma physics, where it describes
the propagation of one-dimensional long magnetoacoustic waves in a cold collisionless plasma subject to
a transverse magnetic field. The complex wave function satisfying the RNLS equation is a combination
of plasma density and the velocity fields as in the Madelung representation. The Bäcklund–Darboux
transformations along with a novel associated nonlinear superposition principle were presented and used to
generate solutions describing the interaction of solitonic magnetoacoustic waves. This application requires
considering a solution of the RNLS equation with nontrivial boundary conditions at infinity. Our goal here
is to derive such solutions in the Hirota bilinear approach and to study their mutual interactions.
2. Magnetoacoustic waves in cold plasma
The dynamics of a two-component cold collisionless plasma in the presence of an external magnetic
field B [12], [13] for uniaxial plasma propagation
u = u(x, t)ex ,
B = B(x, t)ez
(2)
reduces to the form [14]
∂
∂ρ
+
(ρu) = 0,
∂t
∂x
∂u B ∂B
∂u
+u
+
= 0,
∂t
∂x
ρ ∂x
∂ 1 ∂B
=B−ρ
∂x ρ ∂x
(3)
(where we set B = 1 and the plasma density ρ = 1 at infinity). This system is equivalent to the Whitham
system and was also derived by Gurevich and Meshcherkin [15]. It describes the propagation of nonlinear
magnetoacoustic waves in a cold plasma with a transverse magnetic field. El, Khodorovskii, and Tyurina
recently showed [16] that a system of type (3) also occurs in the context of hypersonic flow past slender
bodies.
3. A shallow-water approximation
Here, we consider a shallow-water approximation of magnetoacoustic system (3). Rescaling the space
and time variables via x = βx and t = βt, we have
∂ρ
∂
+ (ρu) = 0,
∂t
∂x
∂u
∂u
B ∂B
+u +
= 0,
∂t
∂x
ρ ∂x
1 ∂B
2 ∂
β
= B − ρ.
∂x ρ ∂x
(4)
(5)
(6)
Expanding B as a power series in the parameter β 2 according to
B = ρ + β 2 b2 (ρ, ρx , ρx x , . . . ) + O(β 4 )
and substituting it in (6) yields
∂
b2 =
∂x
992
1 ∂ρ
.
ρ ∂x
(7)
(8)
Substituting (7) in (5) yields
3 3
1 ∂ρ
∂u
∂u
∂ρ
2 ∂ρ ∂ 2 ρ
2 1 ∂ ρ
+
u
+
+
β
−
+
=0
∂t
∂x
∂x
ρ ∂x3
ρ2 ∂x ∂x2
ρ ∂x
(9)
up to O(β 2 ). Accordingly, we obtain the system
∂
∂ρ
+ (ρu) = 0,
∂t
∂x
(10)
2 1 ∂2ρ
∂u
∂ρ
1 1 ∂ρ
∂u
2 ∂
+u + +β
−
= 0.
∂t
∂x
∂x
∂x ρ ∂x2
2 ρ ∂x
(11)
This describes the propagation of long magnetoacoustic waves in a cold plasma of the density ρ moving with
the velocity u across the magnetic field as given by (2) and (7) (see [17], [18]). The dispersion is negative
in this system, i.e., the wave velocity decreases as the wave vector k increases.
4. Resonant nonlinear Schrödinger equation
Introducing the velocity potential
1
S(x, t) = −
2
x
u(x , t) dx
such that u = −2∂S/∂x and integrating Eq. (11) once, we obtain the system
∂
∂ρ
∂S
−2
ρ
= 0,
∂t
∂x
∂x
2
2 ∂S
β 2 1 ∂ 2 ρ 1 1 ∂ρ
1
∂S
+
+ ρ+
−
=0
−
∂t
∂x
2
2 ρ ∂x2
2 ρ ∂x
(12)
(13)
(where we omit the primes). Combining ρ and S into one complex function
ψ=
√
ρe−iS ,
(14)
we can represent system (12), (13) as the NLS equation with a quantum potential
i
∂ψ ∂ 2 ψ 1 2
1 ∂ 2 |ψ|
+
− |ψ| ψ = (1 + β 2 )
ψ,
2
∂t
∂x
2
|ψ| ∂x2
(15)
i.e., as RNLS (1), where Λ = −2, s = 1 + β 2 , and the quantum potential is expressed as
2
√
1 ∂ 2 ρ 1 1 ∂ρ
1 ∂2 ρ
−
=
2
.
√
ρ ∂x2
2 ρ ∂x
ρ ∂x2
(16)
Because the parameter s > 1, Eq. (15) cannot be transformed into the NLS equation [1]. But by combining
ρ and S into a couple of real functions
e(+) =
√
ρ eS/β ,
√
e(−) = − ρ e−S/β
(17)
993
such that e(+) > 0 and e(−) < 0, we can write Eqs. (12) and (13) as the RD system
∓
1
∂e(±) ∂ 2 e(±)
+
− 2 e(+) e(−) e(±) = 0,
∂τ
∂x2
2β
(18)
where τ = βt .
√
Linearizing (15) near the “condensate” solution ψ = ρ0 e−iρ0 t/2 results in the dispersion ω =
√
ρ0 k 1 − β 2 k 2 /ρ0 . This dispersion is negative (i.e., the wave velocity decreases as the wave vector k
√
increases) and is unstable for short waves with k > kcr , where kcr = ρ0 /β. But this instability results
from truncating the dispersion relation
ω 2 = ρ0
k2
1 + β 2 k 2 /ρ0
(19)
for system (4)–(6) and does not correspond to any actual physical effect for shallow water waves [19].
Although this system is linearly stable for all wave numbers k, it is not known to be integrable. Therefore,
it is not as suitable for studying wave interactions as the RNLS, which is completely integrable and admits
a rich variety of exact solutions.
5. Steady-state flow and solitons
We now consider system (10), (11) for the steady-state flow. It describes motion with the fixed velocity
u(x, t) = u0 = const for which continuity equation (10) implies that ∂ρ/∂t + u0 ∂ρ/∂x = 0 or that the fluid
density has the traveling-wave form ρ = ρ(x − u0 t), where x = βx and t = βt. Equation (11) in this case
gives
2 1 ∂ 2 ρ 1 1 ∂ρ
ρ+
−
= a = const .
(20)
ρ ∂x2
2 ρ ∂x
This has a simple physical interpretation. Motion with a fixed velocity implies that the sum of all forces
acting on the system is zero. In our case, Eq. (20) describes compensation of the nonlinearity by the
quantum potential. With (16) taken into account, this equation gives
√
1 ∂2 ρ
= a.
ρ + 2√
ρ ∂x2
In terms of y(z) =
(21)
√
ρ, z = x − u0 t, we then obtain the nonlinear equation
d2 y a
1
− y + y3 = 0
2
dz
2
2
(22)
a
1
− y 2 + y 4 = 2b = const .
2
4
(23)
or, multiplying by y and integrating once,
dy
dz
2
This equation has the solution
y = 2p dn[p(x − u0 t), κ],
(24)
where dn is the Jacobi elliptic function with the modulus κ and p is an arbitrary constant. It is related to
the integration constants a > 0 and b < 0 by the equations
a
= 1 + κ2 ,
2p2
994
b
= −κ2 ,
2p4
(25)
√
where κ = 1 − κ2 is the complimentary modulus of the Jacobi elliptic function. These equations fix the
relation between the integration constants
b
a = 2p 1 − 4
2p
2
and the modulus κ =
(26)
1 + b/(2p4 ). For the density ρ, we then have the traveling-wave solution
ρ(x, t) = 4p2 dn2 [p(x − u0 t), κ].
(27)
With fixed potential (21) from Eq. (13) for u0 = −(2/β)∂S/∂x, we have the Hamilton–Jacobi equation
−
1 ∂S
1
+ 2
β ∂t
β
which has a solution in the form
∂S
∂x
u0
S = β S0 − x +
2
2
+
a
= 0,
2
u20
a
+
t
4
2
(28)
(29)
or, with (26) taken into account,
2
u0
u0
+ p2 (2 − κ2 ) t .
S = β S0 − x +
2
4
(30)
For the degenerate case κ = 1, which corresponds to b = 0 and hence a = 2p2 , elliptic solution (27), (30)
becomes the soliton
ρ(x, t) = 4p2 sech2 [p(x − u0 t)],
2
(31)
u0
u0
+ p2 t .
S = β S0 − x +
2
4
A more general form of the traveling wave appears if we consider a solution of system (10), (11) in the
form u(x, t) = u(x−u0 t), ρ(x, t) = ρ(x−u0 t). Then the first equation in (10) implies that d[(u−u0 )ρ]/dz = 0,
where z = x − u0 t and
C
u = u0 +
(32)
ρ
with C = const. Substituting (32) in Eq. (11) and integrating once, we obtain
2 1 d2 ρ 1 1 dρ
1 C2
+ρ+
−
= A = const .
2 ρ2
ρ dz 2
2 ρ dz
(33)
Multiplying by ρ2 and differentiating once, we obtain (the traveling-wave form of the KdV equation)
d3 ρ
dρ
dρ
− 2A
ρ
+ 3ρ
3
dz
dz
dz
= 0,
(34)
d2 ρ 3 2
+ ρ − 2Aρ = B = const .
dz 2
2
(35)
which, after one integration, implies that
This gives the equation
dρ
dz
2
= −ρ3 + 2Aρ2 + 2Bρ + C 2
(36)
995
and the solution
ρ(x − u0 t) = α1 + (α3 − α1 ) dn
2
1
(α3 − α1 )1/2 (x − u0 t), κ
2
(37)
with the modulus κ2 = (α3 − α2 )/(α3 − α1 ) of the elliptic function, the constants α1 , α2 , and α3 , and
u(x − u0 t) = u0 −
(α1 α2 α3 )1/2
.
ρ
(38)
Solution (37) was first reported in [18]. In particular cases, we have the following reductions:
1. If α1 = α2 = 0 and hence κ2 = 1, then (37) reduces to (27) with the parameter values α3 = 4k 2
and b = −α2 k 2 /2 and the velocity u = u0 .
2. If α1 = α2 = 0, then again κ2 = 1, but the solution
ρ(x − u0 t) = α1 + (α3 − α1 ) sech
2
1
1/2
(α3 − α1 ) (x − u0 t)
2
(39)
has a nontrivial asymptotic behavior. The physical value relevant for plasma physics is α1 = 1,
which leads to lim|x|→∞ ρ = 1. Setting α3 ≡ σ 2 , we can write the solution in the form
ρ(x − u0 t) = 1 +
(σ 2 − 1)
√
.
cosh2 [ σ 2 − 1(x − u0 t)/2]
(40)
Using these results, we can now construct solutions of RNLS (15). Substituting Eqs. (27) and (30) in
Eq. (14) and changing the parameter k ≡ p/β, we obtain the quasiperiodic solution
2
u0
u0 2 2
2
+ β k (2 − κ ) t ,
ψ(x , t ) = 2βk dn[k(x − u0 t ), κ] exp −i φ0 − x +
2
4
(41)
where x = βx and t = βt. In the limit κ = 1, it gives the envelope soliton solution
ψ(x , t ) = 2βk
2
u0
1
u0 2 2 exp
−i
φ
x
+
β
−
+
k
t
.
0
cosh k(x − u0 t )
2
4
(42)
For RD system (18), we correspondingly have the dissipative periodic solution
e
(±)
2
v
v 2
2
+ k (2 − κ ) τ ,
(x , τ ) = ±2βk dn[k(x − vτ ), κ] exp ± φ0 − x +
2
4
(43)
where the velocity v ≡ u0 /β, k ≡ p/β, and the dissipative analogue of the envelope soliton
e(±) (x , τ ) = ±2βk
2
1
v v
2
τ
−
+
exp
±
φ
x
+
k
0
cosh k(x − vτ )
2
4
is the so-called dissipaton solution [1].
996
(44)
6. Bilinear form and solitons
6.1. Trivial boundary conditions. The key for constructing multisoliton solutions is RD representation (15) of RNLS (18). Because the RD system is algebraically similar to the NLS equation, it is easy to
write the bilinear representation for it. Representing the two real functions e(+) and e(−) in terms of three
real functions,
G±
e(±) = 2β
,
(45)
F
we obtain the bilinear system of equations
(±Dτ − Dx2 )(G(±) · F ) = 0,
Dx2 (F · F ) = −2G(+) G(−) .
(46)
2
Dx2 (F · F )
2 ∂ log F
=
4β
.
F2
∂x2
(47)
The corresponding solution of RNLS (15) is
|ψ(x, t)|2 = ρ = −e(+) e(−) = 2β 2
±
+
−
The one-dissipaton solution of system (46) is given by G± = ±eη1 , F = 1 + eη1 +η1 +φ1,1 , and eφ1,1 =
±(0)
±(0)
and k1± and η1
are constants. In terms of the redefined
(k1+ + k1− )−2 , where η1± ≡ k1± x ± (k1± )2 τ + η1
parameters k ≡ (k1+ + k1− )/2 and v ≡ −(k1+ − k1− ), it takes form (44). In the parameter space (v, k), there
exists the critical value vcrit = 2k such that for v < vcrit , we have e± → 0 at infinity, and the boundary
conditions hence vanish for the dissipaton. At the critical value, the solution is a kink steady state in
the moving frame e± = ±ke±kξ0 (1 ∓ tanh kξ) with the constant asymptotic behavior e± → ±2ke±kξ0 for
x → ∓∞ and e± → ±0 for x → ±∞. In the supercritical case v > vcrit , e± → ±∞ for x → ∓∞ and
e± → ±0 for x → ±∞.
For the two-dissipaton solution, we have
±
G =± e
η1±
+
+e
−
η2±
+
+
±±
k̆12
±∓ +−
k21
k11
−
2
e
+
η1+ +η1− +η2±
−
+
+
−
eη1 +η2
eη2 +η1
eη2 +η2
eη1 +η1
F = 1 + +− 2 + +− 2 + +− 2 + +− 2 +
(k11 )
(k12 )
(k21 )
(k22 )
±±
k̆12
±∓ +−
k12
k22
2
e
η2+ +η2− +η1±
++ −−
k̆12
k̆12
+− +− +− +−
k12 k21 k11 k22
2
,
+
(48)
−
+
−
eη1 +η1 +η2 +η2 ,
(49)
ab
ab
where kij
≡ kia + kjb , k̆ij
≡ kia − kjb , and ηi± ≡ ki± x ± (ki± )2 τ + η ±(0) . This solution shows the resonance
character of the dissipaton interaction [1].
6.2. Nontrivial boundary conditions. As Hirota first noted, the bilinear form of the equations
should be modified for the NLS equation of defocusing type with nonvanishing boundary conditions [20].
After substituting representation (45) in system (18), we choose the decoupling system in the form
(±Dτ − Dx2 + λ)(G± · F ) = 0,
(Dx2 − λ)(F · F ) = −2G+ G− ,
(50)
where we introduce the constant λ to be determined. Equation (45) and the second equation in (50) imply
that −e(+) e(−) = −4β 2 [λ/2 − (log F )xx ]. Expanding G± and F in Hirota’s power series,
G± = ±g0± (1 + g1± + 2 g2± + . . . ),
F = 1 + f1 + 2 f2 + . . . ,
(51)
and requiring that lim|x|→∞ (log F )xx = 0, we obtain the boundary condition
α1 = lim [−e
|x|→∞
(+) (−)
e
] = lim
|x|→∞
−4β
2
λ
− (log F )xx
2
= −2β 2 λ,
(52)
997
which fixes the constant λ = −α1 /(2β 2 ). In the zeroth-order approximation, we have the system
(±Dτ − Dx2 + λ)(g0± · 1) = 0,
(Dx2 − λ)(1 · 1) = 2g0+ g0− .
(53)
±
It has a solution of the form g0± = β ± eθ , where β02 ≡ β + β − = −λ/2 = α1 /(4β 2 ), θ± = ±kx ± (k 2 − λ)t,
and β ± = β0 e±γ0 . Using the properties of Hirota’s derivatives
Dx (f g · h) =
Dx2 (f g
∂f
gh + f Dx (g · h),
∂x
∂2f
∂f
· h) =
gh + 2 Dx (g · h) + f Dx2 (g · h),
2
∂x
∂x
(54)
we rewrite the bilinear system as
(∓Dτ ± 2kDx + Dx2 )((1 + g1± + 2 g2± + · · · ) · (1 + f1 + 2 f2 + · · · )) = 0,
(Dx2 + 2β02 )((1 + f1 + · · · ) · (1 + f1 + · · · )) = 2β02 (1 + g1+ + · · · )(1 + g1− + . . . ).
(55)
6.2.1. One-soliton solution. In the first order, we have the system
(∓∂τ ± 2k∂x + ∂x2 )g1± + (±∂τ ∓ 2k∂x + ∂x2 )f1 = 0,
(56)
(∂x2 + 2β02 )f1 = β02 (g1+ + g1− ).
η1
Considering a solution of the form g1± = a±
and f1 = b1 eη1 , where η1 = k1 x + ω1 τ + η10 , we obtain
1e
−
2
2
a+
1 = γ1 b1 , a1 = b1 /γ1 , and γ1 = (ω1 − 2kk1 + k1 )/(ω1 − 2kk1 − k1 ), where dispersion formula is
ω1± = k1 2k ± k12 + 4β02 .
(57)
We note that in contrast to the defocusing NLS equation, no restrictions on the values of k1 appear in our
case.
Truncating Hirota’s expansion at this level gives one dissipative soliton
e(±) = ±2ββ ± e±[kx+(k
2
+2β02 )τ ] 1
+ γ1±1 eη̃1
,
1 + eη̃1
(58)
where we absorb the constant b1 in the exponential form η̃1 = k1 x + ω1 τ + η10 + log b1 . We then have the
one-soliton density
(1 + γ1 eη̃1 )(1 + γ1−1 eη̃1 )
ρ = −e(+) e(−) = α1
.
(59)
(1 + eη̃1 )2
This solution can be represented in the form
±1
√
η̃
+ 1 γ ±1 − 1
±1 ±[kx+(k2 +2β02 )τ ] γ
+
tanh
= ± α1 µ e
2
2
2
(60)
2
2
√ µ
η̃
e(+) = + α1 e+[kx+(k +2β0 )τ ] γ + 1 + (γ − 1) tanh
,
2
2
√
1
η̃
1 −[kx+(k2 +2β02 )τ ] 1
(−)
e
+1+
− 1 tanh
= − α1
e
,
2µ
γ
γ
2
(61)
e
or
998
(±)
and the product is
ρ = −e
(+) (−)
e
(γ − 1)2
= α1 1 +
4γ cosh2 (η̃/2)
with the asymptotic behavior lim|x|→∞ ρ → α1 . Explicitly, it is
k1 k2
ρ = −e(+) e(−) = α1 1 + 12 sech2
x + 2k ± k12 + 4β02 τ + x0
,
4β0
2
(62)
(63)
where β02 = 1/(4β 2 α1 ). For the velocity field, we have
(−)
u=
(+)
ex
ex
(γ 2 − 1)k1
−
=
−2k
−
.
e(−)
e(+)
(γ − 1)2 + 4γ cosh2 (η̃/2)
We consider the particular solution for k = 0. It then follows that
ω1 = ±k1 k12 + 4β02
(64)
(65)
is the Bogoliubov dispersion from the superfluidity theory of a weakly nonideal Bose gas. For k1 2β0 , it
has the nonrelativistic free-particle form ω1 ≈ k12 , while for k1 2β0 , it has the relativistic collective form
ω1 ≈ 2β0 k1 . The solution for the plus sign of the dispersion has the form
√
α1 µ
v 2 − 4β02
(+)
2β02 τ
2
2
(x + vτ + x0 ) ,
=
e
v + v − 4β0 tanh
e
2
v − v 2 − 4β02
(66)
√
2 − 4β 2
α1 /µ
v
(−)
−2β02 τ
0
2
(x + vτ + x0 ) ,
=−
e
v − v 2 − 4β0 tanh
e
2
v + v 2 − 4β02
and the density is
ρ = −e
(+) (−)
e
= α1
v 2 − 4β02
sech2
1+
4β02
2
v − 4β02
(x + vτ + x0 ) .
2
(67)
This shows that soliton velocity is bounded below by the modulus |v| > 2β0 and the soliton hence has the
“tachyonic” character. These results show that in contrast to the defocusing NLS equation with the soliton
velocity bounded above (subsonic type), the RNLS soliton has a velocity bounded below (supersonic type).
Another difference is that the soliton of the defocusing (repulsive) NLS equation is a hole-like (bubble)
excitation with |ψ|2 = ρ < 1, while we have a wall-like form ρ > 1 for the RNLS soliton.
6.2.2. Two-soliton solution. To construct a two-soliton solution following Hirota [20], we consider
(±)
g1
η1
η2
= a±
+ a±
1e
2e ,
f1 = eη1 + eη2 .
(68)
Substituting them in the bilinear equations
η1
η2
η1
(∓Dτ ± 2kDx + Dx2 )(a±
+ a±
+ eη2 ) = 0,
1e
2 e ) · 1 + 1 · (e
η1
η2
η1
η2
2(Dx2 + 2β02 )(1 · (eη1 + eη2 )) = 2β02 (a+
+ a+
+ a−
+ a−
1e
2e
1e
2 e ),
(69)
we obtain the system
2 η1
a±
+ (±∂τ ∓ 2k∂x + ∂x2 )eη1 +
1 (∓∂τ ± 2k∂x + ∂x )e
2 η2
+ (±∂τ ∓ 2k∂x + ∂x2 )eη2 = 0,
+ a±
2 (∓∂τ ± 2k∂x + ∂x )e
(70)
− η1
− η2
(k12 + 2β02 )(eη1 + eη2 ) = β02 [(a+
+ (a+
1 + a1 )e
2 + a2 )e ].
999
Using the dispersion relations
we obtain
a+
i =
ωi± = ki 2k ± ki2 + 4β02 ,
(ωi − 2kki ) + ki2
≡ e φi ,
(ωi − 2kki ) − ki2
a−
i =
i = 1, 2,
(ωi − 2kki ) − ki2
1
= + e−φi .
2
(ωi − 2kki ) + ki
ai
(71)
(72)
The last relations imply that
ωi − 2kki = ki2 coth
and
ki = 2β0 sinh
φi
2
φi
,
2
(73)
(74)
and hence
ωi − 2kki = 2β02 sinh φi .
(75)
We note that the two signs in dispersion relations (71) correspond to the simple replacement φi → −φi in
the above formulas. First, we restrict our consideration to the same sign for both frequencies. In the next
order, we have the system
(∓Dτ ± 2kDx + Dx2 )(g2± · 1 + g1± · f1 + 1 · f2 ) = 0,
(Dx2 + 2β02 )(2 · f2 + f1 · f1 ) = 2β02 (g2+ + g2− + g1+ g1− ).
(76)
We rewrite the first equation as
(∓∂τ ± 2k∂x + ∂x2 )g2± + (±∂τ ∓ 2k∂x + ∂x2 )f2 +
2
+ {a±
1 [∓(ω1 − ω2 ) ± 2k(k1 − k2 ) + (k1 − k2 ) ] +
2
η1 +η2
+ a±
= 0.
2 [∓(ω2 − ω1 ) ± 2k(k2 − k1 ) + (k1 − k2 ) ]}e
(77)
This implies a solution of the form
η1 +η2
g2± = a±
,
12 e
f2 = b12 eη1 +η2 .
(78)
Then the second equation in the system implies the relations
+ +
φ1 +φ2
a+
b12 ,
12 = a1 a2 b12 = e
hence
b12 =
− −
−(φ1 +φ2 )
a−
b12 ,
12 = a1 a2 b12 = e
sinh2 ((φ1 − φ2 )/4)
,
sinh2 ((φ1 + φ2 )/4)
(79)
(80)
and the first equation in (76) is satisfied automatically. As a result, we have the solution
g0± (1 + g1± + g2± )
1 + f1 + f2
(81)
g0± (1 + eη1 ±φ1 + eη2 ±φ2 + b12 eη1 +η2 ±(φ1 +φ2 ) )
.
1 + eη1 + eη2 + b12 eη1 +η2
(82)
e(±) = ±2β
or
e(±) = ±2β
1000
For the particular parameterization β0 = 1/2 (which implies that α1 = 1), we have the two-soliton solution
for the density
ρ=
where
2
[sinh ((φ1 + φ2 )/4)(1 +
eη1
A+ A−
,
+ eη2 ) + sinh2 ((φ1 − φ2 )/4)eη1 +η2 ]2
φ1 + φ2
φ1 − φ2 η1 +η2 ±(φ1 +φ2 )
(1 + eη1 ±φ1 + eη2 ±φ2 ) + sinh2
e
,
4
4
φi
1
φi
(0)
+ sinh φi τ + ηi , i = 1, 2.
ηi = sinh x + 2k sinh
2
2
2
(83)
A± = sinh2
(84)
If one of the parameters φi vanishes or if φ1 = φ2 , then the solution reduces to the one-soliton form. For
example, if φ2 = 0, then
sinh2 (φ1 /2)
ρ=1+
.
(85)
cosh2 (η1 /2)
Analyzing the two-soliton solution in the soliton moving frames, we can see that it describes a collision
of two solitons of type (85) moving in the same direction with the initial position shifts
∆xi = (−1)i−1
sinh((φ1 − φ2 )/4)
2
log
,
sinh(φi /2)
sinh((φ1 + φ2 )/4)
i = 1, 2,
(86)
and hence sinh(φ1 /2)∆x1 + sinh(φ2 /2)∆x2 = 0.
We obtain a different form of the two-soliton solution if we choose opposite signs for frequencies (71),
and hence
φ1
+
ω1 = 2β0 2k sinh
+ β0 sinh φ1 ,
2
(87)
φ1
−
− β0 sinh φ1 .
ω2 = 2β0 2k sinh
2
±φ1
∓φ2
, a±
,
Then a±
1 =e
2 = e
+ +
φ1 −φ2
a+
b12 ,
12 = a1 a2 b12 = e
and
b12 =
− −
−φ1 +φ2
a−
b12 ,
12 = a1 a2 b12 = e
cosh2 ((φ1 + φ2 )/4)
.
cosh2 ((φ1 − φ2 )/4)
(88)
(89)
For β0 = 1/2, the two-soliton solution is
ρ=
where
[cosh ((φ1 − φ2 )/4)(1 +
2
eη1
B+ B−
,
+ eη2 ) + cosh2 ((φ1 + φ2 )/4)eη1 +η2 ]2
φ1 − φ2
φ1 + φ2 η1 +η2 ±φ1 ∓φ2
(1 + eη1 ±φ1 + eη2 ∓φ2 ) + cosh2
e
,
4
4
1
φ1
φ1
+ sinh φ1 τ,
η1 = sinh (x − x1 ) + 2k sinh
2
2
2
1
φ2
φ2
− sinh φ2 τ.
η2 = sinh (x − x2 ) + [2k sinh
2
2
2
(90)
B± = cosh2
(91)
1001
Fig. 1
It describes a collision of two solitons of form (85) moving in opposite directions with the initial position
shifts
cosh((φ1 + φ2 )/4)
2
∆xi = (−1)i−1
log
, i = 1, 2.
(92)
sinh(φi /2)
cosh((φ1 − φ2 )/4)
In Fig. 1, we show a 3D plot of this solution.
For the velocity field, we have
(−)
u=
(+)
ex
ex
− (+) =
(−)
e
e
= − 2k + +
−
where
cosh2 ((φ1 − φ2 )/4)a− + cosh2 ((φ1 − φ2 )/4)b−
−
cosh2 ((φ1 − φ2 )/4)(1 + eη1 −φ1 + eη2 +φ2 ) + cosh2 ((φ1 + φ2 )/4)eη1 +η2 −φ1 +φ2
cosh2 ((φ1 − φ2 )/4)a+ + cosh2 ((φ1 − φ2 )/4)b+
,
cosh2 ((φ1 − φ2 )/4)(1 + eη1 +φ1 + eη2 −φ2 ) + cosh2 ((φ1 + φ2 )/4)eη1 +η2 +φ1 −φ2
φ1
φ2
a∓ = sinh eη1 ∓φ1 + sinh eη2 ±φ2 ,
2
2
φ2 η1 +η2 ∓φ1 ±φ2
φ1
+ sinh
b∓ = sinh
.
e
2
2
(93)
(94)
It has the same phase shift (92) and describes a collision of two solitons of the form
u = −2k −
ki sinh φi
,
2 cosh((ηi + φi )/2) cosh((ηi − φi )/2)
i = 1, 2.
(95)
Acknowledgments. This work was supported in part by the Institute of Mathematics, Academia
Sinica, Taipei, Taiwan, and the Izmir Institute of Technology, Izmir, Turkey (BAP Project No. 2005 IYTE
08).
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