1. Introduction
With the balance between the nonlinearity and group velocity dispersion (GVD) effects, solitons arise in an optical fiber [1, 2]. Since there are potential applications in ultrafast signal routing systems or telecommunications, optical solitons have been widely studied theoretically and experimentally [3-8].
The nonlinear Schrödinger (NLS) equation can be used to describe the pulse evolution in the picosecond regime [9-11], and it has two different types of localized solutions, i.e., bright and dark soliton solutions, which exist in the anomalous and normal dispersion regimes, respectively [12, 13]. In an infinitely extended constant background, a dark soliton appears as an intensity dip, and its existence was first predicted in [1, 14] and experimentally confirmed in [15]. Appearing in the positive group velocity dispersion regime, the dark soliton possesses some advantages among all types of solitons [16]. For example, compared with the bright solitons, the dark ones are found to have better stability against various perturbations such as fiber loss, the Raman effect and so on [17-20]. Recently, the dark soliton has been observed in such fields as fiber optics, plasmas and Bose-Einstein condensates [21-23], and it has been found to have many applications, such as optical logic devices [24] and waveguide optics as dynamic switches and junctions [25]. Some methods can be applied for deriving the solitons, such as the similarity transformation and Darboux transformation (DT): (a) For the similarity transformation, nonlinear equations can be converted into a set of the integrable (1+1)-dimensional coupled NLS equations, and bright/dark soliton solutions for the original equations are subsequently constructed [26-30]; (b) DT is a powerful method for constructing N-soliton solutions, which is expressed in terms of determinants such as the Wronskian or Grammian [31].
In practice, the inhomogeneity of the optical fiber should be considered; the variable-coefficient NLS equations are thus considered to be more realistic than their constant-coefficient counterparts [32]. In this paper, we pay attention to a variable-coefficient higher-order NLS equation with gain or loss term [33, 34],1 ) have been investigated [33]. One and two dark soliton solutions for equation (1 ) have been derived via the Hirota method [34].
$\begin{eqnarray}\begin{array}{l}{\rm{i}}{q}_{x}+a(x){q}_{{tt}}+b(x)| q{| }^{2}q+{\rm{i}}[c(x){q}_{{ttt}}+d(x)| q{| }^{2}{q}_{t}]\\ +\,f(x)[{q}_{{tttt}}+{\gamma }_{1}(x){q}_{t}^{2}{q}^{* }+{\gamma }_{2}(x)q| {q}_{t}{| }^{2}\\ +\,{\gamma }_{3}(x){q}_{{tt}}| q{| }^{2}+{\gamma }_{4}(x){q}_{{tt}}^{* }{q}^{2}+{\gamma }_{5}(x)q| q{| }^{4}]+{\rm{i}}g(x)q=0,\end{array}\end{eqnarray}$
which can be applied to describe the propagation of an ultrashort pulse in the inhomogeneous optical fiber, where a(x) is the group-velocity dispersion, b(x) represents the self-phase modulation, c(x) and f(x) are respectively the third- and fourth-order dispersions, d(x) denotes the time-delay correction, g(x) stands for the gain [g(x) < 0] or loss [g(x) > 0] coefficient, while γj(x) (j = 1, 2, 3, 4, 5) are all the real functions. Lax pair, conservation laws, nonautonomous breathers and rogue waves for equation (To our knowledge, multi-dark soliton solutions for equation (1 ) have not been reported in the existing literature. Since collisions among multi-dark solitons are meaningful and useful in the inhomogeneous optical fiber, derivation of the dark soliton solutions expressed in terms of the determinants is necessary. This paper is organized as follows: in section 2 , n(1, 2, …)th-iterated binary DT is established based on the Lax pair, and multi-dark soliton solutions in the determinant form for equation (1 ) are thus obtained in section 3 . The discussion about effects of some coefficients on the properties of solitons and summary will be presented in section 4 .
2. The binary DT for equation (1 )
First, we rewrite the Lax pair associated with equation (1 ) as1 ) under the following coefficient constraints:
$\begin{eqnarray}{{\rm{\Psi }}}_{t}=U{\rm{\Psi }},\end{eqnarray}$
$\begin{eqnarray}{{\rm{\Psi }}}_{x}=V{\rm{\Psi }},\end{eqnarray}$
where ${\rm{\Psi }}={({{\rm{\Psi }}}_{1},{{\rm{\Psi }}}_{2})}^{{\rm{T}}}$, with &PSgr;1 and &PSgr;2 as the functions of x and t, T denotes the transpose of a vector, and the matrices U and V are written as: $\begin{eqnarray}U={\rm{i}}\lambda J+{\rm{i}}Q,\end{eqnarray}$
$\begin{eqnarray}\begin{array}{rcl}V & = & 8{\rm{i}}f(x){\lambda }^{4}+[4{\rm{i}}c(x)J+8{\rm{i}}f(x)Q]{\lambda }^{3}\\ & & +[-2a(x){\rm{i}}J+4{\rm{i}}c(x)Q+4f(x){{JQ}}_{t}\\ & & -4{\rm{i}}f(x){{JQ}}^{2}]{\lambda }^{2}+[-2f(x)({Q}_{t}Q-{{QQ}}_{t})\\ & & -2{\rm{i}}c(x){{JQ}}^{2}-2{\rm{i}}a(x)Q-4{\rm{i}}f(x){Q}^{3}\\ & & +2c(x){{JQ}}_{t}-2{\rm{i}}f(x){Q}_{{tt}}]\lambda -{\rm{i}}f(x){{JQ}}_{t}^{2}\\ & & -a(x){{JQ}}_{t}-6f(x){{JQ}}^{2}{Q}_{t}\\ & & -{\rm{i}}c(x){Q}_{{tt}}-f(x){{JQ}}_{{ttt}}-2{\rm{i}}c(x){Q}^{3},\end{array}\end{eqnarray}$
with $\begin{eqnarray}J=\left(\begin{array}{ll}1 & \ 0\\ 0 & -1\end{array}\right),\ \ \ \ \ Q=\left(\begin{array}{ll}\ 0 & q\\ -{q}^{* } & \ 0\end{array}\right),\end{eqnarray}$
while λ is a spectral parameter. It can be verified that the compatibility condition ${U}_{x}-{V}_{t}+{UV}-{VU}=0$ gives rise to equation ( $\begin{eqnarray}\begin{array}{rcl}b(x) & = & -2a(x),\ d(x)=-6c(x),{\gamma }_{1}(x)=-6,\\ {\gamma }_{2}(x) & = & -4,{\gamma }_{3}(x)=-8,\\ {\gamma }_{4}(x) & = & -2,{\gamma }_{5}(x)=6,g(x)=0.\end{array}\end{eqnarray}$
Next, n-fold binary DT for equation (1 ) will be constructed based on the Lax pair (2 ), in order to obtain the dark soliton solutions [35].
The total differential can be derived as
$\begin{eqnarray}\begin{array}{rcl}{\rm{d}}{\rm{\Omega }}({{\rm{\Psi }}}_{1},{\rm{\Psi }}) & = & {{\rm{\Psi }}}_{1}^{\dagger }{\rm{\Psi }}{\rm{d}}t+{{\rm{\Psi }}}_{1}^{\dagger }J\{8f(x)({\lambda }^{3}+{\lambda }^{2}{\lambda }_{1}^{\ast }\\ & & +\lambda {\lambda }_{1}^{{\ast }^{2}}+{\lambda }_{1}^{{\ast }^{3}})J+[4c(x)J+8f(x)Q]\\ & & \times ({\lambda }^{2}+\lambda {\lambda }_{1}^{\ast }+{\lambda }_{1}^{{\ast }^{2}})+[-2a(x)J\\ & & +4c(x)Q-4{\rm{i}}f(x){{JQ}}_{t}-4f(x){{JQ}}^{2}]\\ & & \times (\lambda +{\lambda }_{1}^{\ast })+[2{\rm{i}}f(x)({Q}_{t}Q-{{QQ}}_{t})\\ & & -2c(x){{JQ}}^{2}-2a(x)Q-4f(x){Q}^{3}\\ & & +2{\rm{i}}c(x){{JQ}}_{t}-2f(x){Q}_{{tt}}]\}{\rm{\Psi }}{\rm{d}}x,\end{array}\end{eqnarray}$
with $\begin{eqnarray}{\rm{\Omega }}({{\rm{\Psi }}}_{1},{\rm{\Psi }})=\displaystyle \frac{{{\rm{\Psi }}}_{1}^{\dagger }J{\rm{\Psi }}}{{\rm{i}}(\lambda -{\lambda }_{1}^{* })}+C,\end{eqnarray}$
$\begin{eqnarray}{\rm{\Omega }}({{\rm{\Psi }}}_{1},{{\rm{\Psi }}}_{1})=\displaystyle \frac{{{\rm{\Psi }}}_{1}^{\dagger }J{{\rm{\Psi }}}_{1}}{{\rm{i}}({\lambda }_{1}-{\lambda }_{1}^{* })}+C,\end{eqnarray}$
when ${\lambda }_{1}$ is complex, and $\begin{eqnarray}{\rm{\Omega }}({{\rm{\Psi }}}_{1},{{\rm{\Psi }}}_{1})=\mathop{\mathrm{lim}}\limits_{\lambda \to {\lambda }_{1}}\displaystyle \frac{{{\rm{\Psi }}}_{1}^{\dagger }J{{\rm{\Psi }}}_{1}}{{\rm{i}}(\lambda -{\lambda }_{1})}+C,\end{eqnarray}$
when ${\lambda }_{1}$ is real, supposing ${\rm{\Psi }}$ and ${{\rm{\Psi }}}_{1}$ are respectively the solutions of the Lax pair (For convenience, we set C = 0. The one-fold binary DT matrix for equation (1 ) has the form of4 ).
$\begin{eqnarray}T[1]=I-\displaystyle \frac{{{\rm{\Psi }}}_{1}{{\rm{\Psi }}}_{1}^{\dagger }J}{{\rm{i}}(\lambda -{\lambda }_{1}^{* }){\rm{\Omega }}({{\rm{\Psi }}}_{1},{{\rm{\Psi }}}_{1})},\end{eqnarray}$
we then have $\begin{eqnarray}{\rm{\Psi }}[1]={\rm{\Psi }}-\displaystyle \frac{{{\rm{\Psi }}}_{1}{\rm{\Omega }}({{\rm{\Psi }}}_{1},{\rm{\Psi }})}{{\rm{\Omega }}({{\rm{\Psi }}}_{1},{{\rm{\Psi }}}_{1})},\end{eqnarray}$
$\begin{eqnarray}Q[1]=Q+{\rm{i}}\left[J,\displaystyle \frac{{{\rm{\Psi }}}_{1}{{\rm{\Psi }}}_{1}^{\dagger }}{{\rm{\Omega }}({{\rm{\Psi }}}_{1},{{\rm{\Psi }}}_{1})}\right],\end{eqnarray}$
where [j] (1, 2, …) signifies the j-th iteration, I is an identity matrix, &PSgr;[1] is a new eigenfunction, which has the same form as that of &PSgr; with Q replaced by Q[1] in expressions (Continue such a process n times, the n-th binary DT for equation (1 ) have the following forms:
Supposing &PSgr;j's (j = 1, 2, …, n) are the solutions of the Lax pair (2 ) with λ = λj, respectively, then we have the n-fold binary DT for equation (1 ) as
$\begin{eqnarray}{\rm{\Psi }}[n]={\rm{\Psi }}-{\rm{\Delta }}{P}^{-1}{\rm{\Gamma }},\end{eqnarray}$
$\begin{eqnarray}Q[n]=Q-{\rm{i}}[J,{\rm{\Delta }}{P}^{-1}{{\rm{\Delta }}}^{\dagger }],\end{eqnarray}$
with $\begin{eqnarray}\begin{array}{rcl}{\rm{\Delta }} & = & [{{\rm{\Psi }}}_{1},{{\rm{\Psi }}}_{2},\ldots ,\,{{\rm{\Psi }}}_{n}],\\ P & = & \left[\begin{array}{ll}{\rm{\Omega }}({{\rm{\Psi }}}_{1},{{\rm{\Psi }}}_{1}) & \ {\rm{\Omega }}({{\rm{\Psi }}}_{1},{{\rm{\Psi }}}_{2})\ \ldots \ {\rm{\Omega }}({{\rm{\Psi }}}_{1},{{\rm{\Psi }}}_{n})\\ {\rm{\Omega }}({{\rm{\Psi }}}_{2},{{\rm{\Psi }}}_{1}) & \ {\rm{\Omega }}({{\rm{\Psi }}}_{2},{{\rm{\Psi }}}_{2})\ \ldots \ {\rm{\Omega }}({{\rm{\Psi }}}_{2},{{\rm{\Psi }}}_{n})\\ \ \ \ \ \vdots & \ \ \ \ \ \ \vdots \ \ \ \ \ \ \ \ \ldots \ \ \ \ \ \ \vdots \\ {\rm{\Omega }}({{\rm{\Psi }}}_{n},{{\rm{\Psi }}}_{1}) & \ {\rm{\Omega }}({{\rm{\Psi }}}_{n},{{\rm{\Psi }}}_{2})\ \ldots \ {\rm{\Omega }}({{\rm{\Psi }}}_{n},{{\rm{\Psi }}}_{n})\end{array}\right],\\ {\rm{\Gamma }} & = & \left[\begin{array}{c}{\rm{\Omega }}({{\rm{\Psi }}}_{1},{\rm{\Psi }})\\ {\rm{\Omega }}({{\rm{\Psi }}}_{2},{\rm{\Psi }})\\ \vdots \\ {\rm{\Omega }}({{\rm{\Psi }}}_{n},{\rm{\Psi }})\end{array}\right].\end{array}\end{eqnarray}$
To obtain the dark-soliton solutions for equation (1 ), we then consider the limit form of the binary DT in the case of ${\lambda }_{1}={\lambda }_{1}^{* }$:
Setting &PSgr;1 and Φ1 are two different solutions for the Lax pair (2 ) with λ = λ1, and satisfy ${{\rm{\Psi }}}_{1}^{\dagger }J{{\rm{\Phi }}}_{1}={C}_{1}=\mathrm{const}\ne 0$, ${{\rm{\Phi }}}_{1}^{\dagger }J{{\rm{\Phi }}}_{1}=0$ and ${{\rm{\Psi }}}_{1}^{\dagger }J{{\rm{\Psi }}}_{1}=0$. Assuming ${{\rm{\Theta }}}_{1}(\nu )={{\rm{\Psi }}}_{1}(\nu )\,+\tfrac{\beta (\nu -{\lambda }_{1})}{{C}_{1}}{{\rm{\Phi }}}_{1}({\lambda }_{1})$, we have12 ) can be written as17 ).
$\begin{eqnarray}\begin{array}{l}\mathop{\mathrm{lim}}\limits_{\nu \to {\lambda }_{1}}{\left[\displaystyle \frac{{{\rm{\Psi }}}_{1}^{\dagger }J{{\rm{\Theta }}}_{1}(\nu )}{{\rm{i}}(\nu -{\lambda }_{1})}\right]}_{t}={{\rm{\Psi }}}_{1}^{\dagger }{{\rm{\Psi }}}_{1},\\ \mathop{\mathrm{lim}}\limits_{\nu \to {\lambda }_{1}}{\left[\displaystyle \frac{{{\rm{\Psi }}}_{1}^{\dagger }J{{\rm{\Theta }}}_{1}(\nu )}{{\rm{i}}(\nu -{\lambda }_{1})}\right]}_{x}={{\rm{\Psi }}}_{1}^{\dagger }J\{32f(x){\lambda }^{3}J+3{\lambda }^{2}[4c(x)J\\ \quad +\,8f(x)Q]+2\lambda [-2a(x)J+4c(x)Q\\ \quad -\,4{\rm{i}}f(x){{JQ}}_{t}-4f(x){{JQ}}^{2}]\\ \quad +\,[2{\rm{i}}f(x)({Q}_{t}Q-{{QQ}}_{t})-2c(x){{JQ}}^{2}\\ \quad -\,2a(x)Q-4f(x){Q}^{3}+2{\rm{i}}c(x){{JQ}}_{t}-2f(x){Q}_{{tt}}]\}{{\rm{\Psi }}}_{1},\end{array}\end{eqnarray}$
with $\begin{eqnarray}{\rm{\Omega }}({{\rm{\Psi }}}_{1},{{\rm{\Psi }}}_{1})=\mathop{\mathrm{lim}}\limits_{\nu \to {\lambda }_{1}}\displaystyle \frac{{{\rm{\Psi }}}_{1}^{\dagger }J{{\rm{\Theta }}}_{1}(\nu )}{{\rm{i}}(\nu -{\lambda }_{1})}.\end{eqnarray}$
Then, transformations ( $\begin{eqnarray}{\rm{\Psi }}[1]=\mathop{\mathrm{lim}}\limits_{\nu \to {\lambda }_{1}}\left(I+\displaystyle \frac{{\lambda }_{1}-\nu }{\lambda -{\lambda }_{1}}\displaystyle \frac{{{\rm{\Psi }}}_{1}{{\rm{\Psi }}}_{1}^{\dagger }J}{{{\rm{\Psi }}}_{1}^{\dagger }J{{\rm{\Theta }}}_{1}(\nu )}\right){\rm{\Psi }},\end{eqnarray}$
$\begin{eqnarray}Q[1]=Q+\mathop{\mathrm{lim}}\limits_{\nu \to {\lambda }_{1}}\left[J,\displaystyle \frac{(\nu -{\lambda }_{1}){{\rm{\Psi }}}_{1}{{\rm{\Psi }}}_{1}^{\dagger }J}{{{\rm{\Psi }}}_{1}^{\dagger }J{{\rm{\Theta }}}_{1}(\nu )}\right],\end{eqnarray}$
and the constraint condition $\beta +{\mathrm{lim}}_{\nu \to {\lambda }_{1}}\tfrac{{{\rm{\Psi }}}_{1}^{\dagger }{\sigma }_{3}{{\rm{\Psi }}}_{1}(\nu )}{\nu -{\lambda }_{1}}\ne 0$ can avoid the singularity in expressions (3. Dark soliton solutions for equation (1 )
In this section, the one- and multi-dark soliton solutions will be derived based on the binary DT in the previous section.
3.1. One dark soliton solutions
We aim to derive the one dark soliton solutions for equation (1 ) with the seed solutions
$\begin{eqnarray}q=\kappa {{\rm{e}}}^{{\rm{i}}[\alpha (x)+\eta t]},\end{eqnarray}$
while $\begin{eqnarray}\begin{array}{rcl}\alpha (x) & = & -\displaystyle \int a(x)({\eta }^{2}+2{\kappa }^{2})+c(x)\eta ({\eta }^{2}+6{\kappa }^{2})\\ & & +f(x)({\eta }^{4}+12{\eta }^{2}{\kappa }^{2}+6{\kappa }^{4}){\rm{d}}x,\end{array}\end{eqnarray}$
where κ and η are both the real parameters.Substituting the following gauge transformation2 ), then we have
$\begin{eqnarray}{\rm{\Psi }}=\left(\begin{array}{ll}1 & \ \ \ 0\\ 0 & \ {{\rm{e}}}^{-{\rm{i}}[\alpha (x)+\eta t]}\end{array}\right){{\rm{\Psi }}}_{0},\end{eqnarray}$
into Lax pair ( $\begin{eqnarray}{{\rm{\Psi }}}_{0,t}={\rm{i}}\left(\begin{array}{ll}\lambda & \ \ \kappa \\ -\kappa & \ \eta -\lambda \end{array}\right){{\rm{\Psi }}}_{0},\end{eqnarray}$
$\begin{eqnarray}{{\rm{\Psi }}}_{0,x}={\rm{i}}\left(\begin{array}{ll}{V}_{1} & {V}_{2}\\ -{V}_{2} & {V}_{3}\end{array}\right){{\rm{\Psi }}}_{0},\end{eqnarray}$
with $\begin{eqnarray}\begin{array}{rcl}{V}_{1} & = & -a(x)({\kappa }^{2}+2{\lambda }^{2})+2c(x)(\eta {\kappa }^{2}+\lambda {\kappa }^{2}+2{\lambda }^{3})\\ & & +f(x)[3{\kappa }^{2}({\kappa }^{2}+{\eta }^{2})+4\eta \lambda {\kappa }^{2}\\ & & +4{\kappa }^{2}{\lambda }^{2}+8{\lambda }^{4}],\\ {V}_{2} & = & \kappa [-a(x)(\eta +2\lambda )+c(x)({\eta }^{2}+2{\kappa }^{2}+2\eta \lambda +4{\lambda }^{2})\\ & & +f(x)({\eta }^{3}+6\eta {\kappa }^{2}+2\lambda {\eta }^{2}\\ & & +4\lambda {\kappa }^{2}+4\eta {\lambda }^{2}+8{\lambda }^{3})],\\ {V}_{3} & = & -a(x)({\eta }^{2}+{\kappa }^{2}-2{\lambda }^{2})+c(x)({\eta }^{3}+4\eta {\kappa }^{2}\\ & & -2\lambda {\kappa }^{2}-4{\lambda }^{3})+f(x)({\eta }^{4}+9{\eta }^{2}{\kappa }^{2}\\ & & +3{\kappa }^{4}-4\lambda \eta {\kappa }^{2}-4{\lambda }^{2}{\kappa }^{2}-8{\lambda }^{4}),\end{array}\end{eqnarray}$
where &PSgr;0 is a new 3 × 1 vector function.Setting
$\begin{eqnarray}{U}_{0}=\left(\begin{array}{ll}\lambda & \ \ \kappa \\ -\kappa & \ \eta -\lambda \end{array}\right),\end{eqnarray}$
we can derive the characteristic equation of U0: $\begin{eqnarray}\det (\rho -{U}_{0})=0,\end{eqnarray}$
with ρ as a complex parameter.If μj and ${\mu }_{i}^{* }$ are respectively the real and complex root of equation (24 ) with λ = λj and λ = λi, then
$\begin{eqnarray}{\rho }_{j}-{\lambda }_{j}+\displaystyle \frac{{\kappa }^{2}}{{\rho }_{j}+{\lambda }_{j}-\eta }=0,\end{eqnarray}$
$\begin{eqnarray}{\rho }_{i}^{* }-{\lambda }_{i}+\displaystyle \frac{{\kappa }^{2}}{{\rho }_{i}^{* }+{\lambda }_{i}-\eta }=0.\end{eqnarray}$
Subtracting equation (26 ) from (25 ), we obtain
$\begin{eqnarray}\displaystyle \frac{{\rho }_{j}-{\lambda }_{j}-{\rho }_{i}^{* }+{\lambda }_{i}}{{\rho }_{i}^{* }-{\lambda }_{j}-{\rho }_{j}+{\lambda }_{i}}+\displaystyle \frac{{\kappa }^{2}}{({\rho }_{j}+{\lambda }_{j}-\eta )({\rho }_{i}^{* }+{\lambda }_{i}-\eta )}=0.\end{eqnarray}$
With the expressions derived above, the solution for Lax pair (21 ) with λ = λl (l = 1, 2, …) can be written as
$\begin{eqnarray}{{\rm{\Psi }}}_{0l}=\left[\begin{array}{l}{{\rm{e}}}^{{\rm{i}}{X}_{l}}\\ \displaystyle \frac{\kappa }{\eta -{\lambda }_{l}-{\rho }_{l}}{{\rm{e}}}^{{\rm{i}}{X}_{l}}\end{array}\right],\end{eqnarray}$
with $\begin{eqnarray}\begin{array}{rcl}{X}_{l} & = & {\rho }_{l}t+\displaystyle \frac{1}{{\left({\lambda }_{l}-{\rho }_{l}\right)}^{2}}\left\{\displaystyle \int {\left({\lambda }_{l}-{\rho }_{l}\right)}^{2}(-2{\kappa }^{2}+{\lambda }_{l}^{2}\right.\\ & & -2{\lambda }_{l}{\rho }_{l}-{\rho }_{l}^{2})a(x)-({\lambda }_{l}-{\rho }_{l})[3{\left({\kappa }^{2}-{\lambda }_{l}^{2}\right)}^{2}\\ & & -6{\rho }_{l}{\lambda }_{l}^{3}+6{\kappa }^{2}{\rho }_{l}^{2}+2{\lambda }_{l}{\rho }_{l}^{3}+{\rho }_{l}^{4}]c(x)+4{\kappa }^{6}\\ & & +12{\kappa }^{2}{\left({\lambda }_{l}-{\rho }_{l}\right)}^{2}{\left({\lambda }_{l}+{\rho }_{l}\right)}^{2}+6{\kappa }^{4}({\lambda }_{l}-{\rho }_{l})\\ & & \times ({\lambda }_{l}+3{\rho }_{l})-{\left({\lambda }_{l}-{\rho }_{l}\right)}^{2}(7{\lambda }_{l}^{4}-4{\rho }_{l}{\lambda }_{l}^{3}\\ & & \left.-6{\lambda }_{l}^{2}{\rho }_{l}^{2}-4{\lambda }_{l}{\rho }_{l}^{3}-{\rho }_{l}^{4}){\rm{d}}x\right\}.\end{array}\end{eqnarray}$
With expression (27 ), we have17 ), we have1 ) has the form as
$\begin{eqnarray}\mathop{\mathrm{lim}}\limits_{\nu \to {\lambda }_{1}}\displaystyle \frac{{{\rm{\Psi }}}_{1}^{\dagger }J{{\rm{\Theta }}}_{1}(\nu )}{\nu -{\lambda }_{1}}=\displaystyle \frac{2[{{\rm{e}}}^{2\varsigma {\rho }_{1I}}+{{\rm{e}}}^{{\rm{i}}({X}_{1}-{X}_{1}^{* })}]}{{\rho }_{1}-{\rho }_{1}^{* }},\end{eqnarray}$
while setting $\beta =\tfrac{2{{\rm{e}}}^{2\varsigma {\rho }_{1I}}}{{\rho }_{1}-{\rho }_{1}^{* }}$, where ς is a real parameter. Then from expressions ( $\begin{eqnarray}Q[1]=Q+\left[J,\displaystyle \frac{({\rho }_{1}-{\rho }_{1}^{* }){{\rm{\Psi }}}_{1}{{\rm{\Psi }}}_{1}^{\dagger }J}{2[{{\rm{e}}}^{2\varsigma {\rho }_{1I}}+{{\rm{e}}}^{{\rm{i}}({X}_{1}-{X}_{1}^{* })}]}\right].\end{eqnarray}$
Then, the one dark soliton solutions for equation ( $\begin{eqnarray}q=\kappa \left(1-\displaystyle \frac{B}{2}+\displaystyle \frac{B}{2}\tanh Y\right){{\rm{e}}}^{{\rm{i}}[\alpha (x)+\eta t]},\end{eqnarray}$
with $\begin{eqnarray}B=\displaystyle \frac{{\rho }_{1}-{\rho }_{1}^{* }}{\eta -{\lambda }_{1}-{\rho }_{1}},\ Y=\displaystyle \frac{{\rm{i}}}{2}[({X}_{1}-{X}_{1}^{* })+2{\rm{i}}\varsigma {\rho }_{1I}].\end{eqnarray}$
3.2. Multi-dark soliton solutions
Following the previous steps, the expressions for the n-dark soliton solutions for equation (24 ) are given as
$\begin{eqnarray}q[n]=2{{\rm{e}}}^{{\rm{i}}[\alpha (x)+\eta t]}\displaystyle \frac{| {{\rm{\Lambda }}}_{1}| }{| {\rm{\Delta }}| },\end{eqnarray}$
with $\begin{eqnarray*}\begin{array}{rcl}{{\rm{\Lambda }}}_{1} & = & \left[\begin{array}{ccccc}-{\rm{i}}\tfrac{2[{{\rm{e}}}^{2\varsigma {\mu }_{1I}}+{{\rm{e}}}^{{\rm{i}}({X}_{1}-{X}_{1}^{* })}]}{{\rho }_{1}-{\rho }_{1}^{* }} & -{\rm{i}}\tfrac{2{{\rm{e}}}^{{\rm{i}}({X}_{2}-{X}_{1}^{* })}}{{\rho }_{2}-{\lambda }_{2}-{\rho }_{1}^{* }+{\lambda }_{1}} & \ldots & -{\rm{i}}\tfrac{2{{\rm{e}}}^{{\rm{i}}({X}_{n}-{X}_{1}^{* })}}{{\rho }_{n}-{\lambda }_{n}-{\rho }_{1}^{* }+{\lambda }_{1}} & \tfrac{{{\rm{e}}}^{-{\rm{i}}{X}_{1}^{* }}}{\eta -{\lambda }_{1}-{\rho }_{1}^{* }}\\ -{\rm{i}}\tfrac{2{{\rm{e}}}^{{\rm{i}}({X}_{1}-{X}_{2}^{* })}}{{\rho }_{1}-{\lambda }_{1}-{\rho }_{2}^{* }+{\lambda }_{2}} & -{\rm{i}}\tfrac{2[{{\rm{e}}}^{2\varsigma {\rho }_{2I}}+{{\rm{e}}}^{{\rm{i}}({X}_{2}-{X}_{2}^{* })}]}{{\rho }_{2}-{\rho }_{2}^{* }} & \ldots & -{\rm{i}}\tfrac{2{{\rm{e}}}^{{\rm{i}}({X}_{n}-{X}_{2}^{* })}}{{\rho }_{n}-{\lambda }_{n}-{\rho }_{2}^{* }+{\lambda }_{2}} & \tfrac{{{\rm{e}}}^{-{\rm{i}}{X}_{2}^{* }}}{\eta -{\lambda }_{2}-{\rho }_{2}^{* }}\\ \vdots & \vdots & \ldots & \vdots & \vdots \\ -{\rm{i}}\tfrac{2{{\rm{e}}}^{{\rm{i}}({X}_{1}-{X}_{n}^{* })}}{{\rho }_{1}-{\lambda }_{1}-{\rho }_{n}^{* }+{\lambda }_{n}} & -{\rm{i}}\tfrac{2{{\rm{e}}}^{{\rm{i}}({X}_{2}-{X}_{n}^{* })}}{{\rho }_{2}-{\lambda }_{2}-{\rho }_{n}^{* }+{\lambda }_{n}} & \ldots & -{\rm{i}}\tfrac{2[{{\rm{e}}}^{2\varsigma {\rho }_{{nI}}}+{{\rm{e}}}^{{\rm{i}}({X}_{n}-{X}_{n}^{* })}]}{{\rho }_{n}-{\rho }_{n}^{* }} & \tfrac{{{\rm{e}}}^{-{\rm{i}}{X}_{n}^{* }}}{\eta -{\lambda }_{n}-{\rho }_{n}^{* }}\\ {{\rm{e}}}^{{\rm{i}}{X}_{1}} & {{\rm{e}}}^{{\rm{i}}{X}_{2}} & \ldots & {{\rm{e}}}^{{\rm{i}}{X}_{n}} & \tfrac{\kappa }{2}\end{array}\right],\\ {\rm{\Delta }} & = & \left[\begin{array}{cccc}-{\rm{i}}\tfrac{2[{{\rm{e}}}^{2\varsigma {\rho }_{1I}}+{{\rm{e}}}^{{\rm{i}}({X}_{1}-{X}_{1}^{* })}]}{{\rho }_{1}-{\rho }_{1}^{* }} & -{\rm{i}}\tfrac{2{{\rm{e}}}^{{\rm{i}}({X}_{2}-{X}_{1}^{* })}}{{\rho }_{2}-{\lambda }_{2}-{\rho }_{1}^{* }+{\lambda }_{1}} & \ldots & -{\rm{i}}\tfrac{2{{\rm{e}}}^{{\rm{i}}({X}_{n}-{X}_{1}^{* })}}{{\rho }_{n}-{\lambda }_{n}-{\rho }_{1}^{* }+{\lambda }_{1}}\\ -{\rm{i}}\tfrac{2{{\rm{e}}}^{{\rm{i}}({X}_{1}-{X}_{2}^{* })}}{{\rho }_{1}-{\lambda }_{1}-{\rho }_{2}^{* }+{\lambda }_{2}} & -{\rm{i}}\tfrac{2[{{\rm{e}}}^{2\varsigma {\rho }_{2I}}+{{\rm{e}}}^{{\rm{i}}({X}_{2}-{X}_{2}^{* })}]}{{\rho }_{2}-{\rho }_{2}^{* }} & \ldots & -{\rm{i}}\tfrac{2{{\rm{e}}}^{{\rm{i}}({X}_{n}-{X}_{2}^{* })}}{{\rho }_{n}-{\lambda }_{n}-{\rho }_{2}^{* }+{\lambda }_{2}}\\ \vdots & \vdots & \ldots & \vdots \\ -{\rm{i}}\tfrac{2{{\rm{e}}}^{{\rm{i}}({X}_{1}-{X}_{n}^{* })}}{{\rho }_{1}-{\lambda }_{1}-{\rho }_{n}^{* }+{\lambda }_{n}} & -{\rm{i}}\tfrac{2{{\rm{e}}}^{{\rm{i}}({X}_{2}-{X}_{n}^{* })}}{{\rho }_{2}-{\lambda }_{2}-{\rho }_{n}^{* }+{\lambda }_{n}} & \ldots & -{\rm{i}}\tfrac{2[{{\rm{e}}}^{2\varsigma {\rho }_{{nI}}}+{{\rm{e}}}^{{\rm{i}}({X}_{n}-{X}_{n}^{* })}]}{{\rho }_{n}-{\rho }_{n}^{* }}\end{array}\right].\end{array}\end{eqnarray*}$
4. Discussions and conclusions
With the expressions of the one-soliton solutions [i.e., (32 )], two-soliton solutions [i.e., (34 ) with n = 2] and three-soliton solutions [i.e., (34 ) with n = 3], our purpose in this section is discussing the properties of the dark solitons with a(x), c(x) and f(x), which are respectively the group-velocity dispersion, third- and fourth-order dispersions.
Figure 1(a) shows that the one dark soliton propagates with its velocity and amplitude stable, while figure 1(b) reveals that the dark soliton keeps its shape but the velocity during the propagation changes. It is shown in figure 2(a) that c(x) can influence the soliton's amplitude without affecting its traveling velocity. In figure 2(b), we observe that the amplitude for the soliton does not alter during the propagation, but the velocity for the soliton changes around x = 0.
Figure 1. Dark-one solitons via solutions ( |
Figure 2. Parameters are the same as figure 1(a) except for: (a) $c(x)=2.2x;$ (b) $f(x)=0.8x$. |
Figure 3(a) illustrates the elastic collision between the two dark solitons, which implies that the shapes of the solitons remain unvarying before and after collision except for certain phase shifts. It can be found in figure 3(b) that a(x) has no effects on the two solitons' amplitudes, but influences their traveling directions. In figures 4(a) and (b), c(x) and f(x) are respectively chosen as the functions of x, we can find that shapes of the two dark solitons do not change, but the velocities do. In conclusion, only the velocities of the solitons can be influenced by a(x), c(x) and f(x), leaving their elasticity unaltered.
Figure 3. Dark-two solitons via solutions ( |
Figure 4. Parameters are the same as figure 3(a) except for: (a) $c(x)=0.2x;$ (b) $f(x)=0.2x$. |
Figure 5(a) reveals the elastic collision among the three dark solitons, and we find that the group-velocity dispersion a(x) cannot influence the solitons' amplitudes except their velocities in figure 5(b). In figures 6(a) and (b), c(x) and f(x) are also observed to have no effect on the shapes of the three dark solitons while influencing their velocities and directions, respectively.
Figure 5. Dark-three solitons via solutions ( |
Figure 6. Parameters are the same as figure 5(a) except for: (a) $c(x)=2.2x;$ (b) f(x) = 0.8x. |
In summary, the binary DT has been constructed in this manuscript and expressions of the dark-soliton solutions for a variable-coefficient higher-order NLS equation with gain or loss term, i.e., equation (1 ), have been illustrated. Compared with the results derived before, dark soliton solutions expressed in terms of the determinant, i.e., equation (34 ), have been obtained, then we can study the properties of the multi-dark solitons. Properties of solitons, i.e., propagations of the one solitons, collisions between the two solitons and collisions among the three solitons, have been graphically simulated. Furthermore, influences of the coefficients a(x), c(x) and f(x) on the dark solitons have been discussed in figures. We have concluded that a(x), c(x) and f(x) can influence the velocities of the dark solitons, but have no effect on the solitons' shapes and elasticity. Generally, we have shown that the dark solitons can be controlled, and it has no influence on the mutual collisions. Our results have certain applications in producing the dark solitons, and may be meaningful to manage the propagation and collisions of the solitons in optical communication systems.