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]
$$ i{}q_{1,z}\!+\!k(z)q_{1,tt}\!+\!\varphi(z)(|q_1|^2\!+\!|q_2|^2)q_1\!+\!\varsigma(z)(|q_1|^2\!+\!|q_2|^2)^2q_1 \ \ -i{}l(z)[(|q_1|^2\!+\!|q_2|^2)q_1]_t\!+\! i h(z)(q_1^*q_{1,t}\!+\!q_2^*q_{2,t})q_1\!=\!0\,, i{}q_{2,z}\!+\!m(z)q_{2,tt}\!+\!\pi(z)(|q_1|^2\!+\!|q_2|^2)q_2\!+\!\omega(z)(|q_1|^2\!+\!|q_2|^2)^2q_2\! \ \ -i{}n(z)[(|q_1|^2\!+\!|q_2|^2)q_2]_t\!+\!i b(z)(q_1^*q_{1,t}\!+\!q_2^*q_{2,t})q_2\!=\!0\,, $$
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, $\varphi(z)$ and $\pi(z)$ denote the nonlinearity parameters, $\varsigma(z)$ and $\omega(z)$ 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
$$ n(z)=h(z)=l(z)=4Rk(z)=4Rm(z)=\frac{\varsigma(z)}{R}\\ =\frac{\omega(z)}{R}=\frac{4R\varphi(z)}{v}=\frac{4R\pi(z)}{v}=b(z)\,, $$
where $R$ and $v$ are the real constants.
Special cases of Eqs. (1) have been given as
(i) When $k(z)=m(z)=1$, $\varphi(z)=\pi(z)=2,$ $\varsigma(z)=\omega(z)=\rho_1^2,$ $l(z)=n(z)=h(z)=b(z)=2\rho_1,$ Eqs. (1) have been reduced to the coupled cubic-quintic NLS equations,[36]
$$ i{}q_{1,z}\!+\!q_{1,tt}\!+\!2(|q_1|^2+|q_2|^2)q_1\!+\!\rho_1^2(|q_1|^2\!+\!|q_2|^2)^2q_1\\ \ \ -2i\rho_1[(|q_1|^2\!+\!|q_2|^2)q_1]_t\!+\!2i\rho_1(q_1^*q_{1,t}\!+\!q_2^*q_{2,t})q_1\!=\!0\,,\\ i{}q_{2,z}\!+\!q_{2,tt}\!+\!2(|q_1|^2\!+\!|q_2|^2)q_2\!+\!\rho_1^2(|q_1|^2\!+\!|q_2|^2)^2q_2\\ \ \ -2i\rho_1[(|q_1|^2\!+\!|q_2|^2)q_2]_t\!+\!2i\rho_1(q_1^*q_{1,t}\!+\!q_2^*q_{2,t})q_2\!=\!0\,. $$
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_{tt}\!+\!\varphi(z)|q|^2q\!+\!\varsigma(z)|q|^4q\!-\!i{}b(z)(|q|^2)_tq\!=\!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,
$$ q_{1}=\mu_1\frac{g}{f}e^{i{}c_1t+i{}F_1(z)+2i{}R\theta}\,,\\q_{2}=\mu_2\frac{h}{f}e^{i{}c_2t+i{}F_2(z)+2i{}R\theta}\,, $$
where
$ F_j(z)=\frac{1}{4R}[v(|\mu_1|^2\!+|\mu_2|^2)-c_j^2]\int b(z)\textrm{d}z\,,\ \ (j=1,2)\,,\\ \theta=\int \Big(|\mu_1|^2\frac{|g|^2}{f^2}+|\mu_2|^2\frac{|h|^2}{f^2}\Big)\textrm{d}t + S(z)\,, \\ \\ S(z)=\int\Big[-\int(|\mu_1|^2+|\mu_2|^2)_z d t+\frac{i b(z)}{4R} \\ \hphantom{S(z)=} \times (u_{1,t}u^*_1-u_{1}u^*_{1,t}+u_{2,t}u^*_2-u_{2}u^*_{2,t})\Big]d z\,, \\ u_j=\mu_j \frac{g}{f}e^{i c_j t}+i F_j(z)\,, $
$g$ and $h$ represent the complex functions of $z$ and $t$, $f$ denotes a real function, $c_j$'s and $\mu_j$'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
$ \Big[i{}D_z+\frac{b(z)}{4R}D_t^2+i{}c_1\frac{b(z)}{2R}D_t\Big]g\cdot f=0\,,\\ \Big[i{}D_z+\frac{b(z)}{4R}D_t^2+i{}c_2\frac{b(z)}{2R}D_t\Big]h\cdot f=0\,,\\ [D_t^2\!+\!v(|\mu_1|^2\!+\!|\mu_2|^2)]f\cdot f\!=\!v(|\mu_1|^2|g|^2\!+\!|\mu_2|^2|h|^2), $
where the Hirota bilinear operators $D_z, D_t$ are defined as[38]
$ D_{z}^{l_1}D_{t}^{l_2}(F\cdot H) \\ \;\;=\Big(\frac{\partial}{\partial z}-\frac{\partial}{\partial z^{\prime}}\Big)^{l_1}\Big(\frac{\partial}{\partial t}-\frac{\partial}{\partial t^{\prime}}\Big)^{l_2} F(z,t)H(z^{\prime},t^{\prime})| _{z^{\prime}=z, t^{\prime}=t}\,,$
$F(z, t)$ denotes an analytic function of $z$ and $t$, $H(z^{\prime}, t^{\prime})$ represents an analytic function of the formal variables $z^{\prime}$ and $t^{\prime}$, $l_1$ 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=\mu_1\frac{g}{f}{e}^{i c_1t+i F_1(z)+2i R\theta},\\ q_2=\mu_2\frac{h}{f}{e}^{i c_2t+i F_2(z)+2i R\theta}$
where
$$ f=|I+M|,\quad g=|I+V|,\quad h=|I+W|,$$
$I$ 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 $m_{kl}, v_{kl}$ and $w_{kl}$, respectively, defined as
$$ m_{kl}=\frac{1}{p_k+p_l^*}e^{\xi_k+\xi_l^*},\quad v_{kl}=-\frac{p_k-i{}c_1}{p_l^*+i{}c_1}m_{kl},\\ w_{kl}=-\frac{p_k-i{}c_2}{p_l^*+i{}c_2}m_{kl},$$
$\xi_k=p_kt+({i}/{4R})p_k^2\int b(z)\textrm{d}z+\xi_{k0}$, $(k, l=1,2,\ldots,N)$, $p_k$'s and $\xi_{k0}$'s are the complex constants, where $p_k$ satisfies the constraint
$$ \frac{|\mu_1|^2}{(p_k-i{}c_1)(p_k^*+i{}c_1)}+\frac{|\mu_2|^2}{(p_k-i{}c_2)(p_k^*+i{}c_2)}=-\frac{2}{v}. $$
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{\mu_1}{2}e^{i{}c_1t+i{}F_1(z)+2i{}R\theta} \hphantom{q_1=} \times \Big[1+\varrho_1+(\varrho_1-1)\tanh\Big(\frac{\xi_1+\xi_1^*+ s_1}{2}\Big)\Big], \hphantom{q_1=} q_2=\frac{\mu_2}{2}e^{i{}c_2t+i{}F_2(z)+2i{}R\theta}\hphantom{q_2=}\times \Big[1+\iota_1+(\iota_1-1)\tanh\Big(\frac{\xi_1+\xi_1^*+ s_1}{2}\Big)\Big], $
where
$ \varrho_1\!=\!-\frac{p_1-i{}c_1}{p_1^*+i{}c_1},\ \ \iota_1=-\frac{p_1-i{}c_2}{p_1^*+i{}c_2},s_1=\ln \Big(\frac{p_1+p_1^*}{2}\Big), $
$ \xi_1=p_1t+\frac{i}{4R}p_1^2\int b(z)\textrm{d}z+\xi_{k0}$,
$v$, $c_1$, $c_2$, $\mu_1$, $\mu_2$, and $p_1$ satisfy
$$ \frac{|\mu_1|^2}{(p_1-i{}c_1)(p_1^*+i{}c_1)}+\frac{|\mu_2|^2}{(p_1-i{}c_2)(p_1^*+i{}c_2)}=-\frac{2}{v}. $$
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
$$ q_1=\mu_1\frac{g}{f}e^{i c_1t+i F_1(z)+2i R\theta}\,,\\\ \\ q_2=\mu_2\frac{h}{f}e^{i c_2t+i F_2(z)+2i R\theta}\,, $$
Where
$$ \displaylines{f\!\!=\!\!1\!+\!e^{\xi_1\!+\!\xi_1^*\!+\!s_1}\!+\!e^{\xi_2\!+\!\xi_2^*\!+\!\ln s_2}\!+\!e^{\xi_1\!+\!\xi_1^*\!+\!\xi_2\!+\!\xi_2^*\!+\!\ln s_3},\hfill{}\cr g\!\!=\!\!1\!+\!\varrho_1e^{\xi_1\!+\!\xi_1^*\!+\!s_1}\!+\!\varrho_2e^{\xi_2\!+\!\xi_2^*\!+\!\ln s_2}\!+\!\varrho_1\varrho_2e^{\xi_1\!+\!\xi_1^*\!+\!\xi_2\!+\!\xi_2^*\!+\!\ln s_3}\,,\hfill{}\cr h\!\!=\!\!1\!+\!iota_1e^{\xi_1\!+\!\xi_1^*\!+\!s_1}\!+\!iota_2e^{\xi_2\!+\!\xi_2^*\!+\!\ln s_2}\!+\!iota_1iota_2e^{\xi_1\!+\!\xi_1^*\!+\!\xi_2\!+\!\xi_2^*\!+\!\ln s_3}\,,\hfill{}\cr \varrho_2=-\frac{p_2-i{}c_1}{p_2^*+i{}c_1},\quad iota_2=-\frac{p_2-i{}c_2}{p_2^*+i{}c_2},\hfill{}\cr s_2=\frac{1}{p_2+p_2^*},\quad s_3=\frac{|p_1-p_2|^2}{(p_1+p_1^*)(p_2+p_2^*)|p_1+p_2^*|^2},\hfill{}\cr \xi_k=p_kt+\frac{i}{4R}p_k^2int b(z)\textrm{d}z+\xi_{k0}\,, \quad (k=1, 2)\,,\hfill{}\cr}$$
$v$, $c_1$, $c_2$, $\mu_1$, $\mu_2$, $p_1$, 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 $\vec{v}$ of $q_1$ and $q_2$ can be given as
$$ A_k=\frac{|\mu_1|}{2}\Big[2-\Big(2-\frac{p_1-i{}c_k}{p_1^*+i{}c_k}-\frac{p_1^*+i{}c_k}{p_1-i{}c_k}\Big)^{{1}/{2}}\Big],\\ v=\Big|-\frac{i}{4R}(p_1-p_1^*)b(z)\Big|\,, $$
which is the magnitude of $\vec{v}$.
Direction of $\vec{v}$: When $-({i}/{4R})(p_1-p_1^*)b(z) > 0$, the soliton propagates along the positive direction of $t$ axis; When $-({i}/{4R})(p_1-p_1^*)b(z) < 0$, 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 $p_1-p_1^*=2i{}c_k$, or else it is grey.
Hereafter we mainly discuss the case of $c_1\neq c_2$ for Eqs. (1). Based on Solutions (11), $q_1=\chi q_2$ (where $\chi$ is a constant) when $c_1=c_2$, and the soliton is black in both the $q_1$ and $q_2$ components; when $c_1\neq c_2$, $q_1\neq\chi q_2$.
More on the solitonic issues can be seen, e.g., in Refs. [40].
When $b(z)=2z$, 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 $p_1={1}/{2}+{i}/{2},$ $c_1=0, c_2={1}/{2},$ $\mu_1=1,$ $\mu_2={3}/{2},$ $v=-{2}/{11}$, $R=1$, $\xi_1^{(0)}=0$ 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 $b(z)=2z$. |
When Im$(p_1)$Im$(p_2)<0$, 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 $p_1-p_1^*=2i{}c_1$, 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 $p_1=1+{i}/{2},$ $p_2=1-{i}/{2}$, $c_1={1}/{2},$ $c_2=-{1}/{2},$ $\mu_1=\mu_2=({2\sqrt{3}})/{3},$ $v=-1$, $R=2$, $\xi_1^{(0)}=\xi_2^{(0)}=0$ 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 $b(z)={5}/{2}+{5}/{2}\sin[({11}/{10})z]$. |
Fig. 5 (Color online) Overtaking interaction between the parabolic-shaped two dark-dark soltions via Solutions (13), with $p_1=1+{i}/{3}$, $p_2=1+i$, $c_1=1$, $c_2=-1$, $\mu_1=({\sqrt{13}}/{3})$, $\mu_2={5}/{3}$, $v=-1$, $R={3}/{10}$, $\xi_1^{(0)}=2$, $\xi_2^{(0)}=0$, and $b(z)={1}/{7}+({7}/{10})z$. |
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$(p_1)$Im$(p_2)>0$, 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 $b(z)=1+\sin({z}/{2})$. |
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 $p_1=1,$ $p_2={3}/{4}+{1}/{2}i$, $c_1=0$, $c_2=-1$, $\mu_1=({13\sqrt{38}})/{76}$, $\mu_2=({3\sqrt{285}})/{38}$, $v=-1$, $R={1}/{2}$, $\xi_1^{(0)}=\xi_2^{(0)}=0$, and $b(z)=z$. |
Fig. 8 (Color online) The same as Fig. 7 except that $b(z)=-2+2\sin({3z}/{2})$. |
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$(p_1)\neq0$, 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.