Bilinear Forms and Dark-Dark Solitons for the Coupled Cubic-Quintic Nonlinear SchrÕdinger Equations with Variable Coefficients in a Twin-Core Optical Fiber or Non-Kerr Medium*

Chu Mei-Xia; Tian Bo; Yuan Yu-Qiang; Zhang Ze; He-Yuan Tian and

doi:10.1088/0253-6102/71/12/1393

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Communications in Theoretical Physics ›› 2019, Vol. 71 ›› Issue (12) : 1393-1398. DOI: 10.1088/0253-6102/71/12/1393

Bilinear Forms and Dark-Dark Solitons for the Coupled Cubic-Quintic Nonlinear SchrÕdinger Equations with Variable Coefficients in a Twin-Core Optical Fiber or Non-Kerr Medium*

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Abstract

Twin-core optical fibers are applied in such fields as the optical sensing and optical communication, and propagation of the pulses, Gauss beams and laser beams in the non-Kerr media is reported.Studied in this paper are the coupled cubic-quintic nonlinear SchrÕdinger equations with variable coefficients, which describe the effects of quintic nonlinearity for the ultrashort optical pulse propagation in a twin-core optical fiber or non-Kerr medium. Based on the integrable conditions, bilinear forms are derived, and dark-dark soliton solutions can be constructed in terms of the Gramian via the Kadomtsev-Petviashvili hierarchy reduction. Propagation and interaction of the dark-dark solitons are presented and discussed through the graphic analysis. With different values of the delayed nonlinear response effect $b (z)$ , where $z$ represents direction of the propagation, the linear- and parabolic-shaped one dark-dark soltions can be derived. Interactions between the parabolic- and periodic-shaped two dark-dark solitons are presented with $b (z)$ as the linear and periodic functions, respectively. Directions of velocities of the two dark-dark solitons vary with $z$ and the amplitudes of the solitons remain unchanged can be observed. Interactions between the two dark-dark solitons of different types are displayed, and we observe that the velocity of one soliton is zero and direction of the velocity of the other soliton vary with $z$ . We find that those interactions are elastic.

Key words

optical fiber / coupled cubic-quintic nonlinear SchrÕ / dinger equations / dark-dark solitons / Kadomtsev-Petviashvili hierarchy reduction

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Chu Mei-Xia, Tian Bo, Yuan Yu-Qiang, et al. Bilinear Forms and Dark-Dark Solitons for the Coupled Cubic-Quintic Nonlinear SchrÕdinger Equations with Variable Coefficients in a Twin-Core Optical Fiber or Non-Kerr Medium*[J]. Communications in Theoretical Physics, 2019, 71(12): 1393-1398 https://doi.org/10.1088/0253-6102/71/12/1393

1 Introduction

Twin-core optical fibers have been applied in such fields as the optical sensing^[1] and optical communication,^[2] which have embodied in the optical fiber sensors,^[1] optical interferometers^[2] and optical switches.^[3] Propagation of the pulses,^[4] Gauss beams^[5] and laser beams^[6-8] in the non-Kerr media has attracted the researchers' attention.

Applications of the nonlinear SchrÕdinger (NLS)-type equations in nonlinear optics,^{[6, 9-10]} plasma physics,^[11-13] hydrodynamics^[14-15] and Bose-Einstein condensation^{[11, 16]} have attracted the researchers' attention. In nonlinear optical fibers, optical solitons, which are produced by the balance between the nonlinear effect and group velocity dispersion, have been described by the NLS-type equations.^{[6, 17-20]} With the consideration of the two solitons simultaneously propagate in the two-mode fiber or birefringent, the coupled NLS equations have been investigated.^[21-24]

When the intensity of the incident light field becomes stronger and non-Kerr nonlinearity plays a role, the NLS-type equations with higher-order nonlinearity terms have been used for describing the pulses in optical fibers.^[25-29] Combination of the cubic and quintic nonlinearity terms has allowed stable multidimensional structure exist of the cubic-quintic NLS equation.^[30-31] So as to describe the effects of quintic nonlinearity for the ultrashort optical pulse propagation in a twin-core optical fiber or non-Kerr medium, people have studied the following coupled cubic-quintic NLS equations with variable coefficients,^[32-35]

i q_{1, z} + k (z) q_{1, t t} + φ (z) (| q_{1} |^{2} + | q_{2} |^{2}) q_{1} + ς (z) (| q_{1} |^{2} + | q_{2} |^{2})^{2} q_{1} - i l (z) [(| q_{1} |^{2} + | q_{2} |^{2}) q_{1}]_{t} + i h (z) (q_{1}^{*} q_{1, t} + q_{2}^{*} q_{2, t}) q_{1} = 0, i q_{2, z} + m (z) q_{2, t t} + π (z) (| q_{1} |^{2} + | q_{2} |^{2}) q_{2} + ω (z) (| q_{1} |^{2} + | q_{2} |^{2})^{2} q_{2} - i n (z) [(| q_{1} |^{2} + | q_{2} |^{2}) q_{2}]_{t} + i b (z) (q_{1}^{*} q_{1, t} + q_{2}^{*} q_{2, t}) q_{2} = 0,

(1)

where

q_{1}

and

q_{2}

represent the two components of the electromagnetic fields, the subscripts

z

and

t

denote the partial derivatives with respect to the scaled distance along the direction of the propagation and retarded time, respectively,

" * "

represents the complex conjugate,

i = \sqrt{- 1}

k (z)

and

m (z)

depict the group velocity dispersions,

φ (z)

and

π (z)

denote the nonlinearity parameters,

ς (z)

and

ω (z)

are the saturations of the nonlinear refractive indexes,

l (z)

and

n (z)

represent the self-steepening effects,

h (z)

and

b (z)

are the delayed nonlinear response effects. To our knowledge, the conservation laws, Lax pair, Darboux transformation (DT),^[32] rogue wave solutions,^[33] mixed-type vector solitons,^[34] bright and dark solitons and B\"{a}cklund transformations^[35] for Eqs. (1) have been studied. Refs. ^{[32, 35]} have presented the integrable conditions for Eqs. (1), which can be rewritten as

n (z) = h (z) = l (z) = 4 R k (z) = 4 R m (z) = \frac{ς (z)}{R} = \frac{ω (z)}{R} = \frac{4 R φ (z)}{v} = \frac{4 R π (z)}{v} = b (z),

(2)

where

R

and

v

are the real constants.

Special cases of Eqs. (1) have been given as

(i) When

k (z) = m (z) = 1

φ (z) = π (z) = 2,

ς (z) = ω (z) = ρ_{1}^{2},

l (z) = n (z) = h (z) = b (z) = 2 ρ_{1},

Eqs. (1) have been reduced to the coupled cubic-quintic NLS equations,^[36]

i q_{1, z} + q_{1, t t} + 2 (| q_{1} |^{2} + | q_{2} |^{2}) q_{1} + ρ_{1}^{2} (| q_{1} |^{2} + | q_{2} |^{2})^{2} q_{1} - 2 i ρ_{1} [(| q_{1} |^{2} + | q_{2} |^{2}) q_{1}]_{t} + 2 i ρ_{1} (q_{1}^{*} q_{1, t} + q_{2}^{*} q_{2, t}) q_{1} = 0, i q_{2, z} + q_{2, t t} + 2 (| q_{1} |^{2} + | q_{2} |^{2}) q_{2} + ρ_{1}^{2} (| q_{1} |^{2} + | q_{2} |^{2})^{2} q_{2} - 2 i ρ_{1} [(| q_{1} |^{2} + | q_{2} |^{2}) q_{2}]_{t} + 2 i ρ_{1} (q_{1}^{*} q_{1, t} + q_{2}^{*} q_{2, t}) q_{2} = 0 .

(3)

Soliton solutions and DT for this case have been investigateded.^[36]

(ii) When

q_{1} = q

and

q_{2} = 0

, Eqs. (1) have been reduced to the variable coefficients Kundu-Eckhaus equation,^[37]

i q_{z} + k (z) q_{t t} + φ (z) | q |^{2} q + ς (z) | q |^{4} q - i b (z) (| q |^{2})_{t} q = 0, (4)

which depicts the ultra-short femtosecond pulses propagate in optical fibers, where

q (z, t)

are the electromagnetic wave,

b (z)

is the nonlinear dispersion. Soliton interactions for this case have been studied via the Hirota method.^[37]

However, dark-dark soliton solutions for Eqs. (1) have not been studied. In Sec. 2 of this paper, under Integrable Conditions (2), bilinear forms will be constructed, and dark-dark soliton solutions for Eqs. (1) will be derived through the Kadomtsev-Petviashvili (KP) hierarchy reduction. In Sec. 3, we will graphically analyze the interaction and propagation of dark solitons. Conclusions will be written in Sec. 4.

2 Bilinear Forms and Dark Soliton Solutions for Eqs. (1)

In this section, we will consider the following variable transformations,

q_{1} = μ_{1} \frac{g}{f} e^{i c_{1} t + i F_{1} (z) + 2 i R θ}, q_{2} = μ_{2} \frac{h}{f} e^{i c_{2} t + i F_{2} (z) + 2 i R θ},

(5)

where

F_{j} (z) = \frac{1}{4 R} [v (| μ_{1} |^{2} + | μ_{2} |^{2}) - c_{j}^{2}] \int b (z) d z, (j = 1, 2), θ = \int (| μ_{1} |^{2} \frac{| g |^{2}}{f^{2}} + | μ_{2} |^{2} \frac{| h |^{2}}{f^{2}}) d t + S (z), S (z) = \int [- \int (| μ_{1} |^{2} + | μ_{2} |^{2})_{z} d t + \frac{i b (z)}{4 R} \times (u_{1, t} u_{1}^{*} - u_{1} u_{1, t}^{*} + u_{2, t} u_{2}^{*} - u_{2} u_{2, t}^{*})] d z, u_{j} = μ_{j} \frac{g}{f} e^{i c_{j} t} + i F_{j} (z),

(6)

g

and

h

represent the complex functions of

z

and

t

f

denotes a real function,

c_{j}

's and

μ_{j}

's are the real and complex constants, respectively. Substituting Variable Transformations (5) into Eqs. (1), we can derive the bilinear forms for Eqs. (1) with Integrable Conditions (2) as

Bracket argument to \\ must be a dimension

where the Hirota bilinear operators

D_{z}, D_{t}

are defined as^[38]

D_{z}^{l_{1}} D_{t}^{l_{2}} (F \cdot H) = (\frac{\partial}{\partial z} - \frac{\partial}{\partial z^{'}})^{l_{1}} (\frac{\partial}{\partial t} - \frac{\partial}{\partial t^{'}})^{l_{2}} F (z, t) H (z^{'}, t^{'}) |_{z^{'} = z, t^{'} = t},

F (z, t)

denotes an analytic function of

z

and

t

H (z^{'}, t^{'})

represents an analytic function of the formal variables

z^{'}

and

t^{'}

l_{1}

and

l_{2}

are the non-negative integers.

Via Integrable Conditions (2), dark-dark soliton solutions in terms of the Gramian for Eqs. (1) can be constructed as

q_{1} = μ_{1} \frac{g}{f} e^{i c_{1} t + i F_{1} (z) + 2 i R θ}, q_{2} = μ_{2} \frac{h}{f} e^{i c_{2} t + i F_{2} (z) + 2 i R θ}

(7)

where

f = | I + M |, g = | I + V |, h = | I + W |,

(8)

I

represents the

N \times N

identity matrix,

N

is an non-negative integer,

M, V

and

W

are the

N \times N

matrices, of which elements are

m_{k l}, v_{k l}

and

w_{k l}

, respectively, defined as

m_{k l} = \frac{1}{p_{k} + p_{l}^{*}} e^{ξ_{k} + ξ_{l}^{*}}, v_{k l} = - \frac{p_{k} - i c_{1}}{p_{l}^{*} + i c_{1}} m_{k l}, w_{k l} = - \frac{p_{k} - i c_{2}}{p_{l}^{*} + i c_{2}} m_{k l},

(9)

ξ_{k} = p_{k} t + (i / 4 R) p_{k}^{2} \int b (z) d z + ξ_{k 0}

(k, l = 1, 2, \dots, N)

p_{k}

's and

ξ_{k 0}

's are the complex constants, where

p_{k}

satisfies the constraint

\frac{| μ_{1} |^{2}}{(p_{k} - i c_{1}) (p_{k}^{*} + i c_{1})} + \frac{| μ_{2} |^{2}}{(p_{k} - i c_{2}) (p_{k}^{*} + i c_{2})} = - \frac{2}{v} .

(10)

It is worth noting that the proof of the above process is similar to that in Ref. ^[39]. It is observed that

v

should be negative according to Constraint (10).

When

N = 1

, one dark-dark soliton solutions via Solutions (7) can be derived as

q_{1} = \frac{μ_{1}}{2} e^{i c_{1} t + i F_{1} (z) + 2 i R θ} \times [1 + ϱ_{1} + (ϱ_{1} - 1) \tanh (\frac{ξ_{1} + ξ_{1}^{*} + s_{1}}{2})], q_{2} = \frac{μ_{2}}{2} e^{i c_{2} t + i F_{2} (z) + 2 i R θ} \times [1 + ι_{1} + (ι_{1} - 1) \tanh (\frac{ξ_{1} + ξ_{1}^{*} + s_{1}}{2})],

(11)

where

ϱ_{1} = - \frac{p_{1} - i c_{1}}{p_{1}^{*} + i c_{1}}, ι_{1} = - \frac{p_{1} - i c_{2}}{p_{1}^{*} + i c_{2}}, s_{1} = \ln (\frac{p_{1} + p_{1}^{*}}{2}),

ξ_{1} = p_{1} t + \frac{i}{4 R} p_{1}^{2} \int b (z) d z + ξ_{k 0}

v

c_{1}

c_{2}

μ_{1}

μ_{2}

, and

p_{1}

satisfy

\frac{| μ_{1} |^{2}}{(p_{1} - i c_{1}) (p_{1}^{*} + i c_{1})} + \frac{| μ_{2} |^{2}}{(p_{1} - i c_{2}) (p_{1}^{*} + i c_{2})} = - \frac{2}{v} .

(12)

Based on Solutions (11), we require that

s_{1} > 0

, i.e.,

p_{1} + p_{1}^{*} > 0

, to guarantee Solutions (11) against the singularity.

When

N = 2

, two dark-dark solutions via Solutions (7) can be presented as

q_{1} = μ_{1} \frac{g}{f} e^{i c_{1} t + i F_{1} (z) + 2 i R θ}, q_{2} = μ_{2} \frac{h}{f} e^{i c_{2} t + i F_{2} (z) + 2 i R θ},

(13)

Where

\begin{matrix} f = 1 + e^{ξ_{1} + ξ_{1}^{*} + s_{1}} + e^{ξ_{2} + ξ_{2}^{*} + \ln s_{2}} + e^{ξ_{1} + ξ_{1}^{*} + ξ_{2} + ξ_{2}^{*} + \ln s_{3}}, \\ g = 1 + ϱ_{1} e^{ξ_{1} + ξ_{1}^{*} + s_{1}} + ϱ_{2} e^{ξ_{2} + ξ_{2}^{*} + \ln s_{2}} + ϱ_{1} ϱ_{2} e^{ξ_{1} + ξ_{1}^{*} + ξ_{2} + ξ_{2}^{*} + \ln s_{3}}, \\ h = 1 + i o t a_{1} e^{ξ_{1} + ξ_{1}^{*} + s_{1}} + i o t a_{2} e^{ξ_{2} + ξ_{2}^{*} + \ln s_{2}} + i o t a_{1} i o t a_{2} e^{ξ_{1} + ξ_{1}^{*} + ξ_{2} + ξ_{2}^{*} + \ln s_{3}}, \\ ϱ_{2} = - \frac{p_{2} - i c_{1}}{p_{2}^{*} + i c_{1}}, i o t a_{2} = - \frac{p_{2} - i c_{2}}{p_{2}^{*} + i c_{2}}, \\ s_{2} = \frac{1}{p_{2} + p_{2}^{*}}, s_{3} = \frac{| p_{1} - p_{2} |^{2}}{(p_{1} + p_{1}^{*}) (p_{2} + p_{2}^{*}) | p_{1} + p_{2}^{*} |^{2}}, \\ ξ_{k} = p_{k} t + \frac{i}{4 R} p_{k}^{2} i n t b (z) d z + ξ_{k 0}, (k = 1, 2), \end{matrix}

(14)

v

c_{1}

c_{2}

μ_{1}

μ_{2}

p_{1}

, and

p_{2}

satisfy Constraint (10). Similarly, we require that

s_{1} > 0

and

s_{2} > 0

, i.e.,

p_{1} + p_{1}^{*} > 0

and

p_{2} + p_{2}^{*} > 0

, to guarantee Solutions (11) against the singularity.

3 Discussions

Based on Solutions (11) and (13), dark-dark solitons for Eqs. (1) will be displayed graphically. Via Solutions (11), amplitude

A_{k}

and velocity

\vec{v}

q_{1}

and

q_{2}

can be given as

A_{k} = \frac{| μ_{1} |}{2} [2 - (2 - \frac{p_{1} - i c_{k}}{p_{1}^{*} + i c_{k}} - \frac{p_{1}^{*} + i c_{k}}{p_{1} - i c_{k}})^{1 / 2}], v = | - \frac{i}{4 R} (p_{1} - p_{1}^{*}) b (z) |,

(15)

which is the magnitude of

\vec{v}

Direction of

\vec{v}

: When

- (i / 4 R) (p_{1} - p_{1}^{*}) b (z) > 0

, the soliton propagates along the positive direction of

t

axis; When

- (i / 4 R) (p_{1} - p_{1}^{*}) b (z) < 0

, the soliton propagates along the negative direction of

t

axis.

We can see that the amplitude is irrelevant to

b (z)

, the velocity of

q_{k}

is relevant to

b (z)

. When

(p_{1} - i c_{k}) / (p_{1}^{*} + i c_{k}) = 1

, amplitude of

q_{k}

reaches the maximum, and minimum intensity drops to zero at the same time. The

q_{k}

component is black with

p_{1} - p_{1}^{*} = 2 i c_{k}

, or else it is grey.

Hereafter we mainly discuss the case of

c_{1} \neq c_{2}

for Eqs. (1). Based on Solutions (11),

q_{1} = χ q_{2}

(where

χ

is a constant) when

c_{1} = c_{2}

, and the soliton is black in both the

q_{1}

and

q_{2}

components; when

c_{1} \neq c_{2}

q_{1} \neq χ q_{2}

More on the solitonic issues can be seen, e.g., in Refs. ^[40].

In Fig. 1, we choose

c_{1} \neq c_{2}

, the

q_{2}

component is black when

p_{1} - p_{1}^{*} = 2 i c_{2}

. When the variable coefficient

b (z)

is a constant, velocity of the one dark-dark soliton remains unchanged when the soliton propagates, as shown in Fig. 1.

When

b (z) = 2 z

, Fig. 2 depicts the parabolic-shaped dark-dark soliton, of which the direction of the velocity changes with

z

Fig. 1 (Color online) Linear-shaped dark-dark soliton via Solutions (11) with the parameters as $p_{1} = 1 / 2 + i / 2,$ $c_{1} = 0, c_{2} = 1 / 2,$ $μ_{1} = 1,$ $μ_{2} = 3 / 2,$ $v = - 2 / 11$ , $R = 1$ , $ξ_{1}^{(0)} = 0$ and $b (z) = 1 / 7$ .

Full size|PPT slide

Fig. 2 (Color online) Parabolic-shaped dark-dark soliton via Solutions (11) with the same parameters as those in Fig. 1 except that $b (z) = 2 z$ .

Full size|PPT slide

When Im

(p_{1})

(p_{2}) < 0

, Figs. 3--4 show the head-on interactions between the two solitons of the same type with Im

(p_{k})

being the imaginary part of

p_{k}

. The soliton

S_{1}

is black and

S_{2}

is grey in the

q_{1}

component when

p_{1} - p_{1}^{*} = 2 i c_{1}

, and just the opposite, in the

q_{2}

component. When

b (z) = 8 / 5 + z

, Fig. 3 depicts interaction of the parabolic-shaped two dark-dark solitons, of which the directions of the velocities vary with

z

The amplitudes in both the

q_{1}

and

q_{2}

components remain unchanged during the propagation. When

b (z) = 5 / 2 + (5 / 2) \sin [(11 / 10) z]

, interaction of the periodic-shaped two dark-dark solitons is displayed in Fig. 4. Amplitudes of the two solitons keep invariant during those interactions except that some phase shifts vary, implying that those interactions are elastic.

Fig. 3 (Color online) Head-on interaction between the parabolic-shaped dark-dark solitons via Solutions (13), with $p_{1} = 1 + i / 2,$ $p_{2} = 1 - i / 2$ , $c_{1} = 1 / 2,$ $c_{2} = - 1 / 2,$ $μ_{1} = μ_{2} = (2 \sqrt{3}) / 3,$ $v = - 1$ , $R = 2$ , $ξ_{1}^{(0)} = ξ_{2}^{(0)} = 0$ and $b (z) = 3 + 2 z$ .

Full size|PPT slide

Fig. 4 (Color online) Head-on interaction between the periodic-shaped dark-dark solitons via Solutions (13) with the same parameters as those in Fig. 3 except that $b (z) = 5 / 2 + 5 / 2 \sin [(11 / 10) z]$ .

Full size|PPT slide

Fig. 5 (Color online) Overtaking interaction between the parabolic-shaped two dark-dark soltions via Solutions (13), with $p_{1} = 1 + i / 3$ , $p_{2} = 1 + i$ , $c_{1} = 1$ , $c_{2} = - 1$ , $μ_{1} = (\sqrt{13} / 3)$ , $μ_{2} = 5 / 3$ , $v = - 1$ , $R = 3 / 10$ , $ξ_{1}^{(0)} = 2$ , $ξ_{2}^{(0)} = 0$ , and $b (z) = 1 / 7 + (7 / 10) z$ .

Full size|PPT slide

When the two dark-dark solitons have the same velocity, which means that Im

(p_{1})

= Im

(p_{2})

, we derive the bound-state dark-dark solitons based on Solutions (13). Since the coefficients of the same nonlinear terms for Eqs. (1) are the same, similar to the analysis in Ref. ^[41], we cannot obtain the bound-state dark-dark solitons. When Im

(p_{1})

(p_{2}) > 0

, Figs. 5--6 illustrate the overtaking interactions of the two solitons with the same type. In Fig. 5, parabolic-shaped two dark-dark solitons are displayed. Directions of the velocities of the two solitons change with

z

simultaneously. Fig. 6 displays the periodic-shaped two dark-dark solitons, of which the directions of the velocities vary periodically. As we can see, the amplitudes of the above two dark-dark solitons are unaffected by those interactions, namely, those interactions are elastic.

Fig. 6 (Color online) Overtaking interaction between the periodic-shaped two dark-dark soltions via Solutions (13) with the same parameters as those in Fig. 5 except that $b (z) = 1 + \sin (z / 2)$ .

Full size|PPT slide

Figures 7--8 show the interactions between the two solitons of different types when Im

(q_{1}) = 0

. Velocities of the linear-shaped dark soliton are zero. Fig. 7 illustrates the interaction between the linear-shaped dark soliton

S_{1}

and parabolic-shaped dark soliton

S_{2}

. Interaction between the linear-shaped dark soliton and periodic-shaped dark soliton is displayed in Fig. 8. Similar to the above, those interactions are elastic.

Fig. 7 (Color online) Interaction between the two dark-dark soltions via Solutions (13), with $p_{1} = 1,$ $p_{2} = 3 / 4 + 1 / 2 i$ , $c_{1} = 0$ , $c_{2} = - 1$ , $μ_{1} = (13 \sqrt{38}) / 76$ , $μ_{2} = (3 \sqrt{285}) / 38$ , $v = - 1$ , $R = 1 / 2$ , $ξ_{1}^{(0)} = ξ_{2}^{(0)} = 0$ , and $b (z) = z$ .

Full size|PPT slide

Fig. 8 (Color online) The same as Fig. 7 except that $b (z) = - 2 + 2 \sin (3 z / 2)$ .

Full size|PPT slide

4 Conclusions

Twin-core optical fibers have been applied in such fields as the optical sensing and optical communication, and propagation of the pulses, Gauss beams and laser beams in the non-Kerr media has been reported. The coupled cubic-quintic NLS equations with variable coefficients have been investigated, i.e., Eqs. (1), which describe the effects of quintic nonlinearity for the ultrashort optical pulse propagation in a twin-core optical fiber or non-Kerr medium. Through the KP hierarchy reduction, Dark-dark Soliton Solutions (7) have been constructed in terms of the Gramian and bilinear forms have been derived with Integrable Conditions (2). When the delayed nonlinear response effect

b (z)

is the constant and linear functions, the linear- and parabolic-shaped disproportional one dark-dark soltions have been displayed in Figs. 1

-

2, respectively. With

b (z)

as the linear and periodic functions and Im

(p_{1}) \neq 0

, head-on interactions between the two dark-dark solitons of the parabolic- and periodic-shaped have been presented in Figs. 3--4, respectively, overtaking interactions between the two dark-dark solitons of the same type have been presented in Figs. 5--6, while the velocities of the two dark-dark solitons have altered with

z

changing and the amplitudes of the two dark-dark solitons have been seen to remain unchanged. Besides, when Im

(p_{1}) = 0

, interactions between the two dark-dark solitons of the different types have been displayed in Figs. 7

-

8, and we have seen that the velocities of the one soliton is zero and direction of the velocity of the other soliton vary with

z

. Graphically analyzing the interactions between the two solitons in Figs. 3

-

8 has indicated that those interactions are elastic.

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Min

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Tao

, Phys. Rev. E 91 ( 2015) 033202.

https://doi.org/10.1103/PhysRevE.91.033202

https://www.ncbi.nlm.nih.gov/pubmed/25871233

Abstract

Via the Nth Darboux transformation, a chain of nonsingular localized-wave solutions is derived for a nonlocal nonlinear Schrödinger equation with the self-induced parity-time (PT) -symmetric potential. It is found that the Nth iterated solution in general exhibits a variety of elastic interactions among 2N solitons on a continuous-wave background and each interacting soliton could be the dark or antidark type. The interactions with an arbitrary odd number of solitons can also be obtained under different degenerate conditions. With N=1 and 2, the two-soliton and four-soliton interactions and their various degenerate cases are discussed in the asymptotic analysis. Numerical simulations are performed to support the analytical results, and the stability analysis indicates that the PT-symmetry breaking can also destroy the stability of the soliton interactions.

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W. J.

Liu

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Yang

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Liu

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https://doi.org/10.1103/PhysRevE.96.042201

https://www.ncbi.nlm.nih.gov/pubmed/29347524

Cited in this article [1] Abstract

The interactions of multiple solitons show different properties with two-soliton interactions. For the difficulty of deriving multiple soliton solutions, it is rare to study multiple soliton interactions analytically. In this paper, three-soliton interactions in inhomogeneous optical fibers, which are described by the variable coefficient Hirota equation, are investigated. Via the Hirota bilinear method and symbolic computation, analytic three-soliton solutions are obtained. According to the obtained solutions, properties and features of three-soliton interactions are discussed by changing the third-order dispersion (TOD) and other relevant coefficients, and some plentiful structure of three-soliton interactions are presented for the first time. The influences of TOD on the intensity and propagation distance of solitons are described, which can be used to realize the soliton control. Besides, the method that can achieve the phase reverse of solitons is suggested, and bound states of three solitons are observed, which have potential applications in the mode-locked fiber lasers. Furthermore, comparing to two-soliton interactions, a novel phenomenon of three-soliton interactions with a strong phase shift at x=0 is revealed, which is potentially useful for optical logic switches.

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Funding

*Supported by the National Natural Science Foundation of China under Grant(Nos. 11772017)

Supported by the National Natural Science Foundation of China under Grant(Nos. 11805020)

Supported by the National Natural Science Foundation of China under Grant(Nos. 11272023)

Supported by the National Natural Science Foundation of China under Grant(Nos. 11471050)

the Fund of State Key Laboratory of Information Photonics and Optical Communications Beijing University of Posts and Telecommunications, China IPOC(2017ZZ05)

the Fundamental Research Funds for the Central Universities of China under Grant(No. 2011BUPTYB02)

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Abstract
Key words
Cite this article
1 Introduction
2 Bilinear Forms and Dark Soliton Solutions for Eqs. (1)
3 Discussions
Fig. 1 (Color online) Linear-shaped dark-dark soliton via Solutions (11) with the parameters as $p_{1} = 1 / 2 + i / 2,$ $c_{1} = 0, c_{2} = 1 / 2,$ $μ_{1} = 1,$ $μ_{2} = 3 / 2,$ $v = - 2 / 11$ , $R = 1$ , $ξ_{1}^{(0)} = 0$ and $b (z) = 1 / 7$ .
Fig. 2 (Color online) Parabolic-shaped dark-dark soliton via Solutions (11) with the same parameters as those in Fig. 1 except that $b (z) = 2 z$ .
Fig. 3 (Color online) Head-on interaction between the parabolic-shaped dark-dark solitons via Solutions (13), with $p_{1} = 1 + i / 2,$ $p_{2} = 1 - i / 2$ , $c_{1} = 1 / 2,$ $c_{2} = - 1 / 2,$ $μ_{1} = μ_{2} = (2 \sqrt{3}) / 3,$ $v = - 1$ , $R = 2$ , $ξ_{1}^{(0)} = ξ_{2}^{(0)} = 0$ and $b (z) = 3 + 2 z$ .
Fig. 4 (Color online) Head-on interaction between the periodic-shaped dark-dark solitons via Solutions (13) with the same parameters as those in Fig. 3 except that $b (z) = 5 / 2 + 5 / 2 \sin [(11 / 10) z]$ .
Fig. 5 (Color online) Overtaking interaction between the parabolic-shaped two dark-dark soltions via Solutions (13), with $p_{1} = 1 + i / 3$ , $p_{2} = 1 + i$ , $c_{1} = 1$ , $c_{2} = - 1$ , $μ_{1} = (\sqrt{13} / 3)$ , $μ_{2} = 5 / 3$ , $v = - 1$ , $R = 3 / 10$ , $ξ_{1}^{(0)} = 2$ , $ξ_{2}^{(0)} = 0$ , and $b (z) = 1 / 7 + (7 / 10) z$ .
Fig. 6 (Color online) Overtaking interaction between the periodic-shaped two dark-dark soltions via Solutions (13) with the same parameters as those in Fig. 5 except that $b (z) = 1 + \sin (z / 2)$ .
Fig. 7 (Color online) Interaction between the two dark-dark soltions via Solutions (13), with $p_{1} = 1,$ $p_{2} = 3 / 4 + 1 / 2 i$ , $c_{1} = 0$ , $c_{2} = - 1$ , $μ_{1} = (13 \sqrt{38}) / 76$ , $μ_{2} = (3 \sqrt{285}) / 38$ , $v = - 1$ , $R = 1 / 2$ , $ξ_{1}^{(0)} = ξ_{2}^{(0)} = 0$ , and $b (z) = z$ .
Fig. 8 (Color online) The same as Fig. 7 except that $b (z) = - 2 + 2 \sin (3 z / 2)$ .
4 Conclusions
References
Funding
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Received	Published
2019-08-15	2019-12-01
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2019-12-05

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Abstract

Key words

Cite this article

1 Introduction

2 Bilinear Forms and Dark Soliton Solutions for Eqs. (1)

3 Discussions

Fig. 1 (Color online) Linear-shaped dark-dark soliton via Solutions (11) with the parameters as $p_{1} = 1 / 2 + i / 2,$ $c_{1} = 0, c_{2} = 1 / 2,$ $μ_{1} = 1,$ $μ_{2} = 3 / 2,$ $v = - 2 / 11$ , $R = 1$ , $ξ_{1}^{(0)} = 0$ and $b (z) = 1 / 7$ .

Fig. 2 (Color online) Parabolic-shaped dark-dark soliton via Solutions (11) with the same parameters as those in Fig. 1 except that $b (z) = 2 z$ .

Fig. 4 (Color online) Head-on interaction between the periodic-shaped dark-dark solitons via Solutions (13) with the same parameters as those in Fig. 3 except that $b (z) = 5 / 2 + 5 / 2 \sin [(11 / 10) z]$ .

Fig. 6 (Color online) Overtaking interaction between the periodic-shaped two dark-dark soltions via Solutions (13) with the same parameters as those in Fig. 5 except that $b (z) = 1 + \sin (z / 2)$ .

Fig. 8 (Color online) The same as Fig. 7 except that $b (z) = - 2 + 2 \sin (3 z / 2)$ .

4 Conclusions

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References

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Footnotes

Funding

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模态框（Modal）标题

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Abstract

Key words

Cite this article

1 Introduction

2 Bilinear Forms and Dark Soliton Solutions for Eqs. (1)

3 Discussions

Fig. 1 (Color online) Linear-shaped dark-dark soliton via Solutions (11) with the parameters as p1=1/2+i/2, c1=0,c2=1/2, μ1=1, μ2=3/2, v=−2/11, R=1, ξ1(0)=0 and b(z)=1/7.

Fig. 2 (Color online) Parabolic-shaped dark-dark soliton via Solutions (11) with the same parameters as those in Fig. 1 except that b(z)=2z.

Fig. 3 (Color online) Head-on interaction between the parabolic-shaped dark-dark solitons via Solutions (13), with p1=1+i/2, p2=1−i/2, c1=1/2, c2=−1/2, μ1=μ2=(23)/3, v=−1, R=2, ξ1(0)=ξ2(0)=0 and b(z)=3+2z.

Fig. 4 (Color online) Head-on interaction between the periodic-shaped dark-dark solitons via Solutions (13) with the same parameters as those in Fig. 3 except that b(z)=5/2+5/2sin⁡[(11/10)z].

Fig. 5 (Color online) Overtaking interaction between the parabolic-shaped two dark-dark soltions via Solutions (13), with p1=1+i/3, p2=1+i, c1=1, c2=−1, μ1=(13/3), μ2=5/3, v=−1, R=3/10, ξ1(0)=2, ξ2(0)=0, and b(z)=1/7+(7/10)z.

Fig. 6 (Color online) Overtaking interaction between the periodic-shaped two dark-dark soltions via Solutions (13) with the same parameters as those in Fig. 5 except that b(z)=1+sin⁡(z/2).

Fig. 7 (Color online) Interaction between the two dark-dark soltions via Solutions (13), with p1=1, p2=3/4+1/2i, c1=0, c2=−1, μ1=(1338)/76, μ2=(3285)/38, v=−1, R=1/2, ξ1(0)=ξ2(0)=0, and b(z)=z.

Fig. 8 (Color online) The same as Fig. 7 except that b(z)=−2+2sin⁡(3z/2).

4 Conclusions

{{custom_sec.title}}

{{custom_sec.title}}

References

{{custom_fnGroup.title_en}}

Footnotes

Funding

RIGHTS & PERMISSIONS

Fig. 1 (Color online) Linear-shaped dark-dark soliton via Solutions (11) with the parameters as $p_{1} = 1 / 2 + i / 2,$ $c_{1} = 0, c_{2} = 1 / 2,$ $μ_{1} = 1,$ $μ_{2} = 3 / 2,$ $v = - 2 / 11$ , $R = 1$ , $ξ_{1}^{(0)} = 0$ and $b (z) = 1 / 7$ .

Fig. 2 (Color online) Parabolic-shaped dark-dark soliton via Solutions (11) with the same parameters as those in Fig. 1 except that $b (z) = 2 z$ .

Fig. 4 (Color online) Head-on interaction between the periodic-shaped dark-dark solitons via Solutions (13) with the same parameters as those in Fig. 3 except that $b (z) = 5 / 2 + 5 / 2 \sin [(11 / 10) z]$ .

Fig. 6 (Color online) Overtaking interaction between the periodic-shaped two dark-dark soltions via Solutions (13) with the same parameters as those in Fig. 5 except that $b (z) = 1 + \sin (z / 2)$ .

Fig. 8 (Color online) The same as Fig. 7 except that $b (z) = - 2 + 2 \sin (3 z / 2)$ .