1. Introduction
For integrable systems, the nonlinear Schrödinger equation plays an important role in various fields such as nonlinear optics [1, 2], water waves [3, 4], plasma [5], and Bose–Einstein condensates [6]. In optical fibers, the nonlinear Schrödinger equation can describe the propagation of a picosecond optical pulse [2, 7], but for high-bit-rate transmission systems, higher-order nonlinear and dispersive effects are taken into account, which yields the higher-order nonlinear Schrödinger equation involving the Hirota equation [8–12]. The exact localized wave solutions of the Hirota equation, such as multi-solitons, rogue waves, and breathers, have been extensively studied [13–22]. Furthermore, the explicit expressions of the asymptotic analyses of single-valley dark solitons (abbreviated as SVDS) and double-valley dark solitons (abbreviated as DVDS) have been given for the defocusing case, and a sufficient condition for elastic collisions has been obtained [21]. Notably, dark solitons with delayed nonlinear response and third-order dispersion, in contrast to those with only second-order dispersion and self-phase modulation, can admit single dark solitons with the same velocity under two different phase shifts identified as DVDSs [23]. Moreover, Hirota equations in different physical backgrounds have the characteristics of being multi-component and having variable coefficients [24]. And multi-component nonlinear systems are more widely used and possess more abundant dynamic phenomena than one-component systems [25–27]. In this work, we mainly study the dark soliton solutions of the defocusing couple Hirota equation, which is completely integrable and admits the following form [28–32]:1 ) is reduced to the coupled nonlinear Schrödinger equation.
$\begin{eqnarray}{\rm{i}}{{\boldsymbol{q}}}_{t}+\displaystyle \frac{1}{2}{{\boldsymbol{q}}}_{{xx}}-{\boldsymbol{q}}{{\boldsymbol{q}}}^{\dagger }{\boldsymbol{q}}+{\rm{i}}\alpha (3{{\boldsymbol{q}}}_{x}{{\boldsymbol{q}}}^{\dagger }{\boldsymbol{q}}+3{\boldsymbol{q}}{{\boldsymbol{q}}}^{\dagger }{{\boldsymbol{q}}}_{x}-{{\boldsymbol{q}}}_{{xxx}})={\bf{0}},\end{eqnarray}$
where α is the real parameter; ${\boldsymbol{q}}={({q}_{1},{q}_{2})}^{\top }$ is a two-dimensional complex vector; the superscripts ‘⊤' and ‘†' represent the transposition and conjugate transpose of the matrix, respectively. When α = 0, equation (In recent years, some exact solutions of the coupled Hirota equation, such as soliton solutions [13, 33], rogue wave solutions [26, 34], breather solutions [35], and traveling wave solutions [36] have also been derived. There are transition phenomena in the evolution process between solitons, breathers, and rogue waves in the focusing case [37–39]. Additionally, scholars pay attention to the dynamic behavior of the above exact solutions. For instance, elastic collisions are permitted in solutions such as SVDSs of the coupled Hirota equation [31, 40, 41]. Interestingly, in the coupled higher-order nonlinear Schrödinger equation, there exist the dark double-hump three-soliton solutions with higher order effects generated by the Hirota bilinear method, which admit elastic interactions among each other [41]. The soliton with a double-humped shape, or DVDS, has found extensive applications in power amplification processes owing to its wider pulse width and capacity to withstand higher power [42]. In fact, in the coupled Hirota equation, a single dark soliton can admit two types of intensity profiles: the dark soliton with a single valley and the dark soliton with double valleys. As far as our current state of knowledge allows us to ascertain that the question of whether there exists solely elastic interaction for DVDSs and SVDSs under the context of the coupled Hirota equation remains an open research field. The above problems in the coupled Hirota equation motivate us to further study the dynamic behaviors of its dark soliton solutions.
The paper is organized as follows: in section 2 , with the aid of the uniform Darboux transformation [43], we construct uniform expressions to represent the multi-dark soliton solutions consisting of SVDSs and DVDSs for the coupled Hirota equation. Meanwhile, we propose a sufficient condition for the existence of dark soliton solutions of the coupled Hirota equation by studying the corresponding characteristic equation. In section 3 , we explore the intriguing properties of these solutions through asymptotic analysis. It is revealed that the interaction among single dark soliton solutions can be divided into the following two cases: if the single dark soliton solution corresponds to an SVDS, it will inevitably result in an elastic collision. On the other hand, if the single dark soliton solution represents a DVDS, it is more likely to exhibit inelastic collision. The conclusions are given in section 4 .
2. The dark soliton solutions for the coupled Hirota equation
The coupled Hirota equation (1 ) admits the Lax pair1 ). 02 denotes the 2 × 2 null matrix. Utilizing the compatibility condition Φxt = Φtx of the Lax pair (2 ), we can obtain the zero curvature equation Ut − Vx + [U, V] = 0 with [U, V] = UV − VU, which results in the Hirota equation (1 ).
$\begin{eqnarray}\begin{array}{rcl}{{\rm{\Phi }}}_{x} & = & {\boldsymbol{U}}(\lambda ;{\boldsymbol{Q}}){\rm{\Phi }},\qquad {{\rm{\Phi }}}_{t}={\boldsymbol{V}}(\lambda ;{\boldsymbol{Q}}){\rm{\Phi }},\\ {\rm{\Phi }} & = & {\rm{\Phi }}(x,t;\lambda ),\end{array}\end{eqnarray}$
where $\begin{eqnarray*}\begin{array}{rcl}{\boldsymbol{U}} & \equiv & {\boldsymbol{U}}(\lambda ;{\boldsymbol{Q}})={\rm{i}}(\lambda {\sigma }_{3}+{\boldsymbol{Q}}),\qquad {\boldsymbol{Q}}=\left(\begin{array}{cc}0 & {{\boldsymbol{q}}}^{\dagger }\\ {\boldsymbol{q}} & {{\bf{0}}}_{2}\end{array}\right),\\ {\boldsymbol{V}} & \equiv & {\boldsymbol{V}}(\lambda ;{\boldsymbol{Q}})=(1-4\alpha \lambda )\left({\boldsymbol{U}}\lambda -\displaystyle \frac{{\rm{i}}{\sigma }_{3}}{2}({{\boldsymbol{Q}}}^{2}+{\rm{i}}{{\boldsymbol{Q}}}_{x})\right)\\ & & +{\rm{i}}\alpha ({{\boldsymbol{Q}}}_{{xx}}+2{{\boldsymbol{Q}}}^{3}-{\rm{i}}{{\boldsymbol{Q}}}_{x}{\boldsymbol{Q}}+{\rm{i}}{\boldsymbol{Q}}{{\boldsymbol{Q}}}_{x}),\end{array}\end{eqnarray*}$
σ3 = diag(1, −1, −1), $\lambda \in {\mathbb{C}}$ is a spectral parameter; q is defined in equation (To construct dark soliton solutions, we set the plane wave solution of the coupled Hirota equation (1 ) as ${q}_{i}={c}_{i}{{\rm{e}}}^{{\rm{i}}{\theta }_{i}}$, where θi = aix + bit, i = 1,2, ${b}_{i}=-\alpha \left({a}_{i}^{3}+{\sum }_{j=1}^{2}3{c}_{j}^{2}({a}_{i}^{2}+{a}_{j}^{2})\right)\,-\tfrac{1}{2}{a}_{i}^{2}-{\sum }_{j\,=\,1}^{2}{c}_{j}^{2}$, ai and ci are the wave number and amplitude of background, respectively. In order to solve the Lax pair (2 ) with the abovementioned plane wave solution, we first convert the variable coefficient differential equation into a constant coefficient differential equation utilizing the gauge transformation $\widetilde{{\rm{\Phi }}}={\boldsymbol{G}}{\rm{\Phi }}$, $\widetilde{{\rm{\Phi }}}=\widetilde{{\rm{\Phi }}}(x,t;\lambda )$, ${\boldsymbol{G}}=\mathrm{diag}(1,{{\rm{e}}}^{-{\rm{i}}{\theta }_{1}},{{\rm{e}}}^{-{\rm{i}}{\theta }_{2}})$. It should be pointed out that the spectral parameter λ must satisfy $\lambda \in {\mathbb{R}}$ to obtain the dark soliton solutions. The Lax pair (2 ) is then converted into4 ) and ${{\mathbb{I}}}_{3}$ denotes the 3 × 3 identity matrix. The coefficients of the algebraic expression (4 ) with respect to μ are real-valued if the spectral parameter λ is real, which guarantees that expression (4 ) possesses either real-valued roots or a set of complex conjugate roots. To get the dark soliton solutions of equation (1 ), it is necessary to possess a pair of conjugate complex roots μ and μ* of equation (4 ). It is straightforward to obtain the vector solution of equation (3 ) by this pair of complex roots, and then substituting the above solution into the transformation ${\rm{\Phi }}={{\boldsymbol{G}}}^{-1}\widetilde{{\rm{\Phi }}}$ yields2 ).
$\begin{eqnarray}{\widetilde{{\rm{\Phi }}}}_{x}={\rm{i}}{{\boldsymbol{U}}}_{1}\widetilde{{\rm{\Phi }}},\quad {\widetilde{{\rm{\Phi }}}}_{t}={\rm{i}}(2\alpha {{\boldsymbol{U}}}_{1}^{3}+{d}_{1}{{\boldsymbol{U}}}_{1}^{2}+{d}_{2}{{\boldsymbol{U}}}_{1}+{d}_{3})\widetilde{{\rm{\Phi }}},\end{eqnarray}$
where $\begin{eqnarray*}\begin{array}{rcl}{{\boldsymbol{U}}}_{1} & = & \left(\begin{array}{ccc}\lambda & -{c}_{1} & -{c}_{2}\\ {c}_{1} & -\lambda -{a}_{1} & 0\\ {c}_{2} & 0 & -\lambda -{a}_{2}\end{array}\right),\\ {d}_{1} & = & 3\alpha \left(\displaystyle \sum _{i=1}^{2}{a}_{i}\right)+\displaystyle \frac{1}{2},\end{array}\end{eqnarray*}$
$\begin{eqnarray*}\begin{array}{rcl}{d}_{2} & = & -6\alpha {\lambda }^{2}+\lambda +3\alpha {a}_{1}{a}_{2},\\ {d}_{3} & = & -{d}_{1}{\lambda }^{2}-3\alpha {a}_{1}{a}_{2}\lambda +\left({d}_{1}+\displaystyle \frac{1}{2}\right)\left(\displaystyle \sum _{i=1}^{2}{c}_{i}^{2}\right).\end{array}\end{eqnarray*}$
And the characteristic equation of matrix U1 is as follows: $\begin{eqnarray}\det ({{\boldsymbol{U}}}_{1}-\mu {{\mathbb{I}}}_{3})=0,\end{eqnarray}$
where μ is the eigenvalue of equation ( $\begin{eqnarray}\begin{array}{rcl}{\rm{\Phi }} & = & \left[\begin{array}{c}{{\rm{e}}}^{{\rm{i}}X}\\ \displaystyle \frac{{c}_{1}}{\chi +{a}_{1}}{{\rm{e}}}^{{\rm{i}}(X+{\theta }_{1})}\\ \displaystyle \frac{{c}_{2}}{\chi +{a}_{2}}{{\rm{e}}}^{{\rm{i}}(X+{\theta }_{2})}\end{array}\right],\\ X & = & \mu x+(2\alpha {\mu }^{3}+{d}_{1}{\mu }^{2}+{d}_{2}\mu +{d}_{3})t,\\ \chi & = & \lambda +\mu ,\end{array}\end{eqnarray}$
which is the vector solution of equation (We are going to employ the uniform Darboux transformation [43], which is widely used to generate solitonic solutions. Due to the limitations of the classical Darboux transformation, it is not feasible to directly derive the multi-dark soliton solutions of multi-component systems. Hence, we adopt the uniform Darboux transformation proposed in reference [43] to construct multi-dark soliton solutions of the coupled Hirota equation. According to equation (5 ), the uniform Darboux transformation can be constructed explicitly as1 ) can be expressed as7 ). According to solution (7 ), we could obtain the following straightforward features about SVDSs: the evolution direction of dark soliton solution (7 ) is along the trajectory x − v1t + γ1 = 0; the valley depths of ${\left|{q}_{i}^{[1]}\right|}^{2}$, i = 1, 2 are $\tfrac{{c}_{i}^{2}{\left({\mu }_{1I}\right)}^{2}}{{\left({\lambda }_{1}+{a}_{i}+{\mu }_{1R}\right)}^{2}+{\left({\mu }_{1I}\right)}^{2}}$, respectively.
$\begin{eqnarray}{{\boldsymbol{T}}}^{[1]}={{\mathbb{I}}}_{3}-\displaystyle \frac{({\mu }_{1}-{\mu }_{1}^{* }){{\rm{\Phi }}}_{1}{{\rm{\Phi }}}_{1}^{\dagger }{\sigma }_{3}}{2(\lambda -{\lambda }_{1})({{\rm{e}}}^{{\rm{i}}({X}_{1}-{X}_{1}^{* })}+{{\rm{e}}}^{2{\gamma }_{1}{\mu }_{1I}})},\end{eqnarray}$
where ${{\rm{\Phi }}}_{1}={\left.{\boldsymbol{G}}{\rm{\Phi }}\right|}_{\lambda ={\lambda }_{1},\mu ={\mu }_{1}}$, ${X}_{1}=X{| }_{\lambda ={\lambda }_{1},\mu ={\mu }_{1}}$, ${\mu }_{1I}={\mathfrak{I}}({\mu }_{1})$, ${\gamma }_{1}\in {\mathbb{R}}$ is an arbitrary constant. Furthermore, the SVDS corresponding to the spectral parameter λ1 for the coupled Hirota equation ( $\begin{eqnarray}\begin{array}{rcl}{{\boldsymbol{q}}}^{[1]}(x,t;{{\boldsymbol{c}}}_{1}^{(s)}) & = & {\left[{q}_{1}^{[1]}(x,t;{{\boldsymbol{c}}}_{1}^{(s)}),{q}_{2}^{[1]}(x,t;{{\boldsymbol{c}}}_{1}^{(s)})\right]}^{\top },\\ {q}_{i}^{[1]}(x,t;{{\boldsymbol{c}}}_{1}^{(s)}) & = & {c}_{i}\left(1-\displaystyle \frac{{B}_{i}}{2}+\displaystyle \frac{{B}_{i}}{2}\tanh ({Y}_{1})\right){{\rm{e}}}^{{\rm{i}}{\theta }_{i}},\end{array}\end{eqnarray}$
where ${{\boldsymbol{c}}}_{1}^{(s)}=\left({\lambda }_{1},{\gamma }_{1},{\mu }_{1}\right)$, ${\mu }_{1R}={\mathfrak{R}}({\mu }_{1})$, ${B}_{i}=\tfrac{{\mu }_{1}-{\mu }_{1}^{* }}{{\chi }_{1}+{a}_{i}}$, i = 1, 2, ${v}_{1}=-(2\alpha (3{\left({\mu }_{1R}\right)}^{2}-{\left({\mu }_{1I}\right)}^{2})+2{d}_{1}{\mu }_{1R}+{d}_{2})$, Y1 = μ1I(x − v1t + γ1) and v1 represents the velocity of SVDS (We select a set of parameters based on equation (7 ), allowing us to successfully present the density profile of the SVDS, as shown in figure 1. In particular, substituting the parameters a1 = 1, a2 = −0.4, c1 = 1, c2 = 1, and λ1 = 1 into the characteristic equation (4 ) to yield the complex root μ1 ≈ −0.2513 + 1.2203i and then substituting all parameters into the above results, we can obtain that: the velocity v1 of the dark soliton solution is approximately equal to 4.6289; the valley depths of ${\left|{q}_{1}^{[1]}\right|}^{2}$ and ${\left|{q}_{2}^{[1]}\right|}^{2}$ are approximately equal to 0.9601 and 0.5291, respectively; the evolution direction of dark soliton solution is along the trajectory x − v1t − 0.5 = 0, v1 ≈ 4.6289.
Figure 1. The density profiles of intensity square of the dark soliton solution with the parameters ${{\boldsymbol{c}}}_{1}^{(s)}\approx \left(-0.5,1,-0.2513+1.2203{\rm{i}}\right)$, a1 = 1, a2 = −0.4, c1 = 1, c2 = 1, and α = 0.65, which corresponds to an SVDS. (a) The density profile of $| {q}_{1}^{[1]}{| }^{2}$. (b) The density profile of $| {q}_{2}^{[1]}{| }^{2}$. |
Similarly, we are able to construct the expressions of the multi-dark soliton solutions corresponding to the spectral parameters λi, i = 1, 2, ⋯ , n for the coupled Hirota equation (1 ). Suppose that Lax pair (2 ) has n different solutions ${{\rm{\Phi }}}_{i}={\left.{\boldsymbol{G}}{\rm{\Phi }}\right|}_{\lambda ={\lambda }_{i},\mu ={\mu }_{i}}$, and ${{\rm{\Phi }}}_{j}^{\dagger }{\sigma }_{3}{{\rm{\Phi }}}_{j}=0$, if ${\lambda }_{j}\in {\mathbb{R}}$ , then the n-fold uniform Darboux transformation can be expressed as 5 ), ${\boldsymbol{c}}=({\lambda }_{1},{\lambda }_{2},\cdots ,{\lambda }_{n},{\gamma }_{1},{\gamma }_{2},\cdots ,{\gamma }_{n},{\mu }_{1},{\mu }_{2},\cdots ,{\mu }_{n})$. In fact, a multi-dark soliton ${{\boldsymbol{q}}}^{[n]}(x,t;{\boldsymbol{c}})$ can be seen as consisting of ns SVDSs and nd DVDSs. The SVDS corresponding to the spectral parameter ${\lambda }_{l}$ is defined as ${{\boldsymbol{q}}}^{[1]}(x,t;{{\boldsymbol{c}}}_{l}^{(s)})=[{q}_{1}^{[1]}(x,t;{{\boldsymbol{c}}}_{l}^{(s)}),{q}_{2}^{[1]}{(x,t;{{\boldsymbol{c}}}_{l}^{(s)})]}^{\top }$, and the DVDS with ${v}_{l-1}={v}_{l}$ corresponding to the spectral parameters ${\lambda }_{l-1}$ and ${\lambda }_{l}$ is denoted as ${{\boldsymbol{q}}}^{[2]}(x,t;{{\boldsymbol{c}}}_{l}^{(d)})={\left[{q}_{1}^{[2]}(x,t;{{\boldsymbol{c}}}_{l}^{(d)}),{q}_{2}^{[2]}(x,t;{{\boldsymbol{c}}}_{l}^{(d)})\right]}^{\top }$, where ${v}_{l}=-(2\alpha (3{\left({\mu }_{{lR}}\right)}^{2}-{\left({\mu }_{{lI}}\right)}^{2})+2{d}_{1}{\mu }_{{lR}}+{d}_{2})$, ${{\boldsymbol{c}}}_{l}^{(d)}=\left({\lambda }_{l-1},{\lambda }_{l},{\gamma }_{l-1},{\gamma }_{l},{\mu }_{l-1},{\mu }_{l}\right)$, $l=2,3,\cdots n$.
$\begin{eqnarray}{{\boldsymbol{T}}}^{[n]}={{\mathbb{I}}}_{3}-{{\rm{\Theta }}}_{n}{{\boldsymbol{R}}}_{n}^{-1}{\left(\lambda {{\mathbb{I}}}_{n}-{{\boldsymbol{D}}}_{n}\right)}^{-1}{{\rm{\Theta }}}_{n}^{\dagger }{\sigma }_{3},\end{eqnarray}$
where $\begin{eqnarray*}\begin{array}{rcl}{{\rm{\Theta }}}_{n} & = & \left[{{\rm{\Phi }}}_{1},{{\rm{\Phi }}}_{2},\cdots ,{{\rm{\Phi }}}_{n}\right],{{\boldsymbol{D}}}_{n}=\mathrm{diag}({\lambda }_{1},{\lambda }_{2},\ldots ,{\lambda }_{n}),{\nu }_{j}=\displaystyle \frac{1}{{\rm{i}}{\mu }_{{jI}}}{{\rm{e}}}^{2{\gamma }_{j}{\mu }_{{jI}}},j\,=\,1,2,\cdots ,n,\\ {{\boldsymbol{R}}}_{n} & = & \left[\begin{array}{cccc}\mathop{\mathrm{lim}}\limits_{\lambda \to {\lambda }_{1}}\displaystyle \frac{{{\rm{\Phi }}}_{1}^{\dagger }{\rm{\Lambda }}{{\rm{\Phi }}}_{1}(\lambda )}{\lambda -{\lambda }_{1}}+{\nu }_{1} & \displaystyle \frac{{{\rm{\Phi }}}_{1}^{\dagger }{\sigma }_{3}{{\rm{\Phi }}}_{2}}{{\lambda }_{2}-{\lambda }_{1}} & \cdots & \displaystyle \frac{{{\rm{\Phi }}}_{1}^{\dagger }{\sigma }_{3}{{\rm{\Phi }}}_{n}}{{\lambda }_{n}-{\lambda }_{1}}\\ \displaystyle \frac{{{\rm{\Phi }}}_{2}^{\dagger }{\sigma }_{3}{{\rm{\Phi }}}_{1}}{{\lambda }_{1}-{\lambda }_{2}} & \mathop{\mathrm{lim}}\limits_{\lambda \to {\lambda }_{2}}\displaystyle \frac{{{\rm{\Phi }}}_{2}^{\dagger }{\sigma }_{3}{{\rm{\Phi }}}_{2}(\lambda )}{\lambda -{\lambda }_{2}}+{\nu }_{2} & \cdots & \displaystyle \frac{{{\rm{\Phi }}}_{2}^{\dagger }{\sigma }_{3}{{\rm{\Phi }}}_{n}}{{\lambda }_{n}-{\lambda }_{2}}\\ \vdots & \vdots & \ddots & \vdots \\ \displaystyle \frac{{{\rm{\Phi }}}_{n}^{\dagger }{\sigma }_{3}{{\rm{\Phi }}}_{1}}{{\lambda }_{1}-{\lambda }_{n}} & \displaystyle \frac{{{\rm{\Phi }}}_{n}^{\dagger }{\sigma }_{3}{{\rm{\Phi }}}_{2}}{{\lambda }_{2}-{\lambda }_{n}} & \cdots & \mathop{\mathrm{lim}}\limits_{\lambda \to {\lambda }_{n}}\displaystyle \frac{{{\rm{\Phi }}}_{n}^{\dagger }{\sigma }_{3}{{\rm{\Phi }}}_{n}(\lambda )}{\lambda -{\lambda }_{n}}+{\nu }_{n}\end{array}\right].\end{array}\end{eqnarray*}$
The expressions for the multi-dark soliton solutions can be derived by the n-fold uniform Darboux transformation (
$\begin{eqnarray}\begin{array}{rcl}{{\boldsymbol{q}}}^{[n]}(x,t;{\boldsymbol{c}}) & = & {\left[{q}_{1}^{[n]}(x,t;{\boldsymbol{c}}),{q}_{2}^{[n]}(x,t;{\boldsymbol{c}})\right]}^{\top },\\ {q}_{i}^{[n]}(x,t;{\boldsymbol{c}}) & = & {c}_{i}\displaystyle \frac{\det ({{\boldsymbol{M}}}_{i})}{\det ({\boldsymbol{N}})}{{\rm{e}}}^{{\rm{i}}{\theta }_{i}},\end{array}\end{eqnarray}$
where $\begin{eqnarray}\begin{array}{rcl}{{\boldsymbol{M}}}_{i} & = & {\left(\displaystyle \frac{2\left({{\rm{e}}}^{{\rm{i}}({X}_{l}-{X}_{k}^{* })}+{\delta }_{k,l}\right)}{{\chi }_{l}-{\chi }_{k}^{* }}-2{\beta }_{l,i}{{\rm{e}}}^{{\rm{i}}({X}_{l}-{X}_{k}^{* })}\right)}_{1\leqslant k,l\leqslant n},\\ {\beta }_{l,i} & = & \displaystyle \frac{1}{{a}_{i}+{\chi }_{l}},\\ {\boldsymbol{N}} & = & {\left(\displaystyle \frac{2\left({{\rm{e}}}^{{\rm{i}}({X}_{l}-{X}_{k}^{* })}+{\delta }_{k,l}\right)}{{\chi }_{l}-{\chi }_{k}^{* }}\right)}_{1\leqslant k,l\leqslant n},\\ {\delta }_{k,l} & = & \left\{\begin{array}{ll}0, & k\ne l\\ {{\rm{e}}}^{2{\gamma }_{k}{\mu }_{{kI}}}, & k=l\end{array}\right.,\quad {\chi }_{l}={\lambda }_{l}+{\mu }_{l},\end{array}\end{eqnarray}$
Xl is defined in equation (We select two sets of parameters to construct two types of multi-dark soliton solutions respectively. The multi-dark soliton solution in figure 2 exhibits the dynamics of three SVDSs, whereas the multi-dark soliton solution in figure 3 displays the dynamics of a DVDS and an SVDS. Notably, in contrast to the scalar Hirota equation, the two valleys of the DVDS can remain relatively far away from each other.
Figure 2. The density profiles of intensity square of the dark soliton solution with the parameters ns = 3, nd = 0, a1 = 0.5, a2 = −0.4, c1 = 1, c2 = 1, α = 0.625, and c ≈ (1, 0.5, −1, 1, 1.2, 10, 0.0686 + 0.9824i, 0.0302 + 1.2606i, − 0.1154 + 1.0236i), which corresponds to a general multi-dark soliton solution. (a) The density profile of $| {q}_{1}^{[3]}{| }^{2}$. (b) The density profile of $| {q}_{2}^{[3]}{| }^{2}$. |
Figure 3. The density profiles of the intensity square of the dark soliton solution with the parameters ns = 1, nd = 1, a1 = − 0.2, a2 = − 0.4, c1 = 1, c2 = 1, α = 0.625, and ${\boldsymbol{c}}\approx \left(0.15,0.1,-1,1,1.2,10,0.15-1.4107{\rm{i}},0.1498+1.498{\rm{i}},0.1443+0.8251{\rm{i}}\right)$, which corresponds to a multi-dark soliton consisting of a symmetric DVDS and an SVDS. (a) The density profile of $| {q}_{1}^{[3]}{| }^{2}$. (b) The density profile of $| {q}_{2}^{[3]}{| }^{2}$. |
Whilst it is true that not all parameters selected can yield a dark soliton solution for the coupled Hirota equation, we shall endeavor to identify the underlying conditions that satisfy the existence of such solutions. Especially, we restrict our attention to the case of a1 > a2 and c1 = c2 in the subsequent proposition.
where ${{\rm{\Delta }}}_{1}={\left({a}_{1}-{a}_{2}\right)}^{4}\,+\,20{\left({a}_{1}-{a}_{2}\right)}^{2}{c}_{1}^{2}-8{c}_{1}^{4}$, ${{\rm{\Delta }}}_{2}\,=64{c}_{1}^{2}{\left({\left({a}_{1}-{a}_{2}\right)}^{2}+{c}_{1}^{2}\right)}^{3}$, ${{\rm{\Delta }}}_{3}={a}_{2}^{2}-{a}_{1}^{2}$, then the defocusing coupled Hirota equation admits the dark soliton solutions. 4 ) has three real roots; when ${\rm{\Delta }}\gt 0$, (4 ) has a real root and a pair of conjugate complex roots, which ensures the existence of the dark soliton solutions of equation (1 ). As previously discussed, we only need to consider ${\rm{\Delta }}\gt 0$. In this case, λ is located in $\left(\tfrac{{{\rm{\Delta }}}_{3}-\sqrt{{{\rm{\Delta }}}_{1}+\sqrt{{{\rm{\Delta }}}_{2}}}}{4\left({a}_{1}-{a}_{2}\right)},\tfrac{{{\rm{\Delta }}}_{3}-\sqrt{{{\rm{\Delta }}}_{1}-\sqrt{{{\rm{\Delta }}}_{2}}}}{4\left({a}_{1}-{a}_{2}\right)}\right)$ ∪$\left(\tfrac{{{\rm{\Delta }}}_{3}+\sqrt{{{\rm{\Delta }}}_{1}-\sqrt{{{\rm{\Delta }}}_{2}}}}{4\left({a}_{1}-{a}_{2}\right)},\tfrac{{{\rm{\Delta }}}_{3}+\sqrt{{{\rm{\Delta }}}_{1}+\sqrt{{{\rm{\Delta }}}_{2}}}}{4\left({a}_{1}-{a}_{2}\right)}\right)$, where ${\left({a}_{1}-{a}_{2}\right)}^{2}\geqslant 8{c}_{1}^{2}$ guarantees ${{\rm{\Delta }}}_{1}\geqslant {{\rm{\Delta }}}_{2}^{2}$ established, whereas without this condition, λ is located in $\left(\tfrac{{{\rm{\Delta }}}_{3}-\sqrt{{{\rm{\Delta }}}_{1}-\sqrt{{{\rm{\Delta }}}_{2}}}}{4\left({a}_{1}-{a}_{2}\right)},\tfrac{{{\rm{\Delta }}}_{3}+\sqrt{{{\rm{\Delta }}}_{1}-\sqrt{{{\rm{\Delta }}}_{2}}}}{4\left({a}_{1}-{a}_{2}\right)}\right)$.
If the following conditions $(1)$ or $(2)$ hold:
| 1. | (1)$\lambda \in \left(\tfrac{{{\rm{\Delta }}}_{3}-\sqrt{{{\rm{\Delta }}}_{1}+\sqrt{{{\rm{\Delta }}}_{2}}}}{4\left({a}_{1}-{a}_{2}\right)},\tfrac{{{\rm{\Delta }}}_{3}-\sqrt{{{\rm{\Delta }}}_{1}-\sqrt{{{\rm{\Delta }}}_{2}}}}{4\left({a}_{1}-{a}_{2}\right)}\right)$∪$\left(\tfrac{{{\rm{\Delta }}}_{3}+\sqrt{{{\rm{\Delta }}}_{1}-\sqrt{{{\rm{\Delta }}}_{2}}}}{4\left({a}_{1}-{a}_{2}\right)},\tfrac{{{\rm{\Delta }}}_{3}+\sqrt{{{\rm{\Delta }}}_{1}+\sqrt{{{\rm{\Delta }}}_{2}}}}{4\left({a}_{1}-{a}_{2}\right)}\right)$ with ${\left({a}_{1}-{a}_{2}\right)}^{2}\geqslant 8{c}_{1}^{2}$, |
| 2. | (2)$\lambda \in \left(\tfrac{{{\rm{\Delta }}}_{3}-\sqrt{{{\rm{\Delta }}}_{1}-\sqrt{{{\rm{\Delta }}}_{2}}}}{4\left({a}_{1}-{a}_{2}\right)},\tfrac{{{\rm{\Delta }}}_{3}+\sqrt{{{\rm{\Delta }}}_{1}-\sqrt{{{\rm{\Delta }}}_{2}}}}{4\left({a}_{1}-{a}_{2}\right)}\right)$ with ${\left({a}_{1}-{a}_{2}\right)}^{2}\lt 8{c}_{1}^{2}$, |
In order to construct dark soliton solutions by uniform Darboux transformation, equation (
$\begin{eqnarray*}{\rm{\Delta }}=-\displaystyle \frac{3\left({\left({\left({a}_{1}-{a}_{2}\right)}^{2}{\left(4\lambda +({a}_{1}+{a}_{2})\right)}^{2}-{{\rm{\Delta }}}_{1}\right)}^{2}-{{\rm{\Delta }}}_{2}\right)}{16{\left({a}_{1}-{a}_{2}\right)}^{2}}.\end{eqnarray*}$
According to the Cardan formula, when ${\rm{\Delta }}\leqslant 0$, (We can perform a similar analysis in the absence of the restrictions of a1 > a2 and c1 = c2, but we are unable to provide an explicit expression of the existence condition of the solution (9 ). In order to vividly demonstrate the relationship between parameters and the existence of dark soliton solutions, we plot figure 4. It is worth noting that the dark soliton solutions exist solely in the X-type region, with no such solutions being present in other regions. Moreover, the color bar in figure 4 indicates that the velocity of the dark soliton solution varies monotonically within some intervals. This figure agrees with the conditions of the existence of dark soliton solutions for the coupled Hirota equation (1 ).
Figure 4. The existence and velocity variation of the dark soliton solutions. The parameters are a2 = −0.4, c1 = c2 = 1, α = 0.625. The white square corresponds to the dark soliton solution in figure 1, the green triangle to the dark soliton solution in figure 2, and the pink pentagram to the special dark soliton solution in figure 3. The parameter selections of the solutions depicted in this figure are consistent with the requirement for the existence of solutions as stipulated in lemma |
3. The asymptotic analysis of the dark soliton solutions
In this section, we primarily employ asymptotic analysis to explore the evolution of the exact solutions for the coupled Hirota equation, which are composed of SVDSs and DVDSs.
For the multi-dark soliton solutions ${q}_{i}^{[n]}(x,t;{\boldsymbol{c}})$, $n={n}_{s}+2{n}_{d}$, i = 1, 2, composed of ns SVDSs and nd DVDSs, we define trajectories for SVDSs and DVDSs, respectively.
| 1. | (1)In the case of SVDSs, suppose ${v}_{1}^{(s)}\gt {v}_{2}^{(s)}\gt \cdots \gt {v}_{{n}_{s}}^{(s)}$, then we define the trajectories lj as $x-{v}_{j}^{(s)}t\,\equiv \mathrm{const}$, $j=1,2,\cdots ,{n}_{s}$. |
| 2. | (2)In the case of DVDSs, suppose ${v}_{1}^{(d)}\gt {v}_{2}^{(d)}\gt \cdots \gt {v}_{{n}_{d}}^{(d)}$, then we define the trajectories Lj as $x-{v}_{j}^{(d)}t\,\equiv \mathrm{const}$, $j=1,2,\cdots ,{n}_{d}$. |
We arrange all the velocities ${v}_{j}^{(s)}$, j = 1, 2, ⋯ , ns and ${v}_{j}^{(d)}$, j = 1, 2, ⋯ , nd in descending order as ${v}_{1}^{{\prime} }\gt {v}_{2}^{{\prime} }\gt \cdots \gt {v}_{{n}_{s}+{n}_{d}}^{{\prime} }$ resulting in a total of $\left(\displaystyle \genfrac{}{}{0em}{}{{n}_{s}+{n}_{d}}{{n}_{d}}\right)$ possible arrangements, and then we obtain the trajectories hj, j = 1, 2, ⋯ , ns + nd corresponding to the velocities ${v}_{j}^{{\prime} }$, j = 1, 2, ⋯ ,ns + nd, where hj ∈ Sl ∪ SL, ${{\boldsymbol{S}}}_{l}={\{{l}_{j}\}}_{j=1}^{{n}_{s}}$ and ${{\boldsymbol{S}}}_{L}={\{{L}_{j}\}}_{j=1}^{{n}_{d}}$. Note that ${q}_{i}^{[n]}(x,t;{\boldsymbol{c}})$ corresponds to spectral parameters λj, j = 1, 2, ⋯ n, where each DVDS corresponds to two distinct spectral parameters. We can also define the trajectories according to the number of the spectral parameters λj. Suppose that the velocities vj, j = 1, 2, ⋯ , n corresponding to λj satisfy v1 ≥ v2 ≥ ⋯ ≥ vn, where vj ∈ V(s) ∪ V(d), ${{\boldsymbol{V}}}^{(s)}=\{{v}_{j}^{(s)}\}{}_{j\,=\,1}^{{n}_{s}}$, ${{\boldsymbol{V}}}^{(d)}=\{{v}_{j}^{(d)}\}{}_{j\,=\,1}^{{n}_{d}}$, we define the trajectories gj as $x-{v}_{j}t\equiv \mathrm{const}$, j = 1, 2, ⋯ , ns + 2nd, where gj ∈ Sl ∪ SL. In order to distinguish whether the trajectory gj at the jth location corresponds to an SVDS or DVDS, we modify the subscripts of lj, j = 1, 2, ⋯ , ns and Lj, j = 1, 2, ⋯ , nd to their corresponding subscript of gj, j = 1, 2, ⋯ , n. Note that if gj−1 ∈ SL and gj ∈ SL, then gj−1 = gj, and we only take the subscript of L as j. For the sake of clarity, we provide an illustrative example. Suppose that the velocities vj, j = 1, 2, 3 corresponding to λj, j = 1, 2, 3 satisfy v1 > v2 ≥ v3. We define the trajectories gj, j = 1, 2, 3 where g1, g2 and g3 are l1, L1 and L1, respectively, and then we modify l1 to l1 and L1 to L3. 10 ). The determinants of matrices ${\boldsymbol{A}}$, ${\boldsymbol{B}}$, and ${\boldsymbol{C}}$ are $\det ({\boldsymbol{A}})\,={2}^{n}\tfrac{{\prod }_{1\leqslant k\lt l\leqslant n}(-{\chi }_{l}^{* }+{\chi }_{k}^{* })({\chi }_{l}-{\chi }_{k})}{{\prod }_{k\,=\,1}^{n}{\prod }_{l\,=\,1}^{n}({\chi }_{l}-{\chi }_{k}^{* })}$, $\det ({\boldsymbol{B}})={2}^{n}\tfrac{{{\rm{e}}}^{2{\sum }_{k\,=\,1}^{n}{\gamma }_{k}{\mu }_{{kI}}}}{{\prod }_{k\,=\,1}^{n}({\chi }_{k}-{\chi }_{k}^{* })}$, $\det ({\boldsymbol{C}})\,={\prod }_{k\,=\,1}^{n}\tfrac{{a}_{i}+{\chi }_{k}^{* }}{{a}_{i}+{\chi }_{k}}\det ({\boldsymbol{A}})$.
Set the matrices
$\begin{eqnarray}\begin{array}{rcl}{\boldsymbol{A}} & = & {\left(\displaystyle \frac{2}{{\chi }_{l}-{\chi }_{k}^{* }}\right)}_{1\leqslant k,l\leqslant n},\quad {\boldsymbol{B}}={\left(\displaystyle \frac{2{\delta }_{k,l}}{{\chi }_{l}-{\chi }_{k}^{* }}\right)}_{1\leqslant k,l\leqslant n},\\ {\boldsymbol{C}} & = & {\left(\displaystyle \frac{2}{{\chi }_{l}-{\chi }_{k}^{* }}-2{\beta }_{l,k}\right)}_{1\leqslant k,l\leqslant n},\end{array}\end{eqnarray}$
where ${\chi }_{l}$, ${\beta }_{l,k}$ and ${\delta }_{k,l}$ are defined in equation (Matrix ${\boldsymbol{A}}$ is a Cauchy matrix and matrix ${\boldsymbol{B}}$ is a diagonal matrix, thus we can directly identify the determinants of them. Additionally, matrix ${\boldsymbol{C}}$ can be rewritten as ${\boldsymbol{C}}={\left(\tfrac{{a}_{i}+{\chi }_{k}^{* }}{{a}_{i}+{\chi }_{l}}\tfrac{2}{{\chi }_{l}-{\chi }_{k}^{* }}\right)}_{1\leqslant k,l\leqslant n}$, enabling us to obtain the determinant of matrix ${\boldsymbol{C}}$.
For convenience, we introduce the following notations:7 ). Notably, since the parameter μj and λj are conjoined via the characteristic equation (4 ), the velocity vj is inherently linked to λj as well. With the aforementioned notational framework and results established, we are now poised to undertake an asymptotic analysis [44] of the dynamic behavior exhibited by both SVDSs and DVDSs.
$\begin{eqnarray}\begin{array}{rcl}{\xi }_{j}^{+} & = & \left\{\begin{array}{ll}0, & {v}_{j}={v}_{j+1}\\ \displaystyle \prod _{k=j+1}^{n}\displaystyle \frac{{a}_{i}+{\chi }_{k}^{* }}{{a}_{i}+{\chi }_{k}}, & {\rm{otherwise}}\end{array}\right.,\\ {\xi }_{j}^{-} & = & \left\{\begin{array}{ll}0, & {v}_{j}={v}_{j+1}\\ \displaystyle \prod _{k=1}^{j-2}\displaystyle \frac{{a}_{i}+{\chi }_{k}^{* }}{{a}_{i}+{\chi }_{k}}, & {v}_{j-1}={v}_{j}\\ \displaystyle \prod _{k=1}^{j-1}\displaystyle \frac{{a}_{i}+{\chi }_{k}^{* }}{{a}_{i}+{\chi }_{k}}, & {\rm{otherwise}}\end{array}\right.,\\ {\left({\zeta }_{j}^{[k]}\right)}^{-} & = & \left\{\begin{array}{ll}\displaystyle \prod _{l=1}^{j-2}{\left|{\chi }_{l}-{\chi }_{k}\right|}^{2}, & {v}_{j-1}={v}_{j}\\ \displaystyle \prod _{l=1}^{j-1}{\left|{\chi }_{l}-{\chi }_{k}\right|}^{2}, & {\rm{otherwise}}\end{array}\right.,\\ {\left({\zeta }_{j}^{[k]}\right)}^{+} & = & \displaystyle \prod _{l=j+1}^{n}{\left|{\chi }_{l}-{\chi }_{k}\right|}^{2},\\ {\left({\tilde{\zeta }}_{j}^{[k]}\right)}^{+} & = & \displaystyle \prod _{l=j+1}^{n}{\left|{\chi }_{l}^{* }-{\chi }_{k}\right|}^{2},\\ {\left({\tilde{\zeta }}_{j}^{[k]}\right)}^{-} & = & \left\{\begin{array}{ll}\displaystyle \prod _{l=1}^{j-2}{\left|{\chi }_{l}^{* }-{\chi }_{k}\right|}^{2}, & {v}_{j-1}={v}_{j}\\ \displaystyle \prod _{l=1}^{j-1}{\left|{\chi }_{l}^{* }-{\chi }_{k}\right|}^{2}, & {\rm{otherwise}}\end{array}\right.,\end{array}\end{eqnarray}$
where χj = λj + μj, the velocity vj is an expression related to λj. Indeed, we express the velocity vj in terms of the parameter μj as specified in equation (As $t\to \pm \infty $ , the multi-dark soliton solutions ${q}_{i}^{[n]}(x,t;{\boldsymbol{c}})$ in equation (
$\begin{eqnarray}{q}_{i}^{[n]}(x,t;{\boldsymbol{c}})=\displaystyle \sum _{j=1}^{n}{\xi }_{j}^{\pm }({q}_{i}^{\pm }(x,t;{{\boldsymbol{c}}}_{j}^{(p)})-{c}_{i}{{\rm{e}}}^{{\rm{i}}{\theta }_{i}})+{ \mathcal O }({{\rm{e}}}^{-\epsilon | t| }),\end{eqnarray}$
where i = 1, 2, p = s, d, $\epsilon ={\min }_{1\leqslant k\ne j\leqslant n}\left(| {v}_{k}-{v}_{j}| \right)$.| i | (i)As ${v}_{j-1}\ne {v}_{j}$ and ${v}_{j}\ne {v}_{j+1}$, ${q}_{i}^{\pm }(x,t;{{\boldsymbol{c}}}_{j}^{(s)})$ is the SVDS ${q}_{i}^{[1]}(x,t;{{\boldsymbol{c}}}_{j}^{(s)})$ with a shift ${x}_{0j}^{\pm }$, and its expression is $\begin{eqnarray}{q}_{i}^{\pm }(x,t;{{\boldsymbol{c}}}_{j}^{(s)})={c}_{i}\left(1-{B}_{j}+{B}_{j}\tanh ({Y}_{j}^{\pm })\right){{\rm{e}}}^{{\rm{i}}{\theta }_{i}},\end{eqnarray}$ where ${Y}_{j}^{\pm }=x-{v}_{j}t+{\gamma }_{j}+{x}_{0j}^{\pm }$, ${x}_{0j}^{\pm }=-\tfrac{1}{2}\mathrm{ln}(\tfrac{{\left({\zeta }_{j}^{[j]}\right)}^{\pm }}{{\left({\tilde{\zeta }}_{j}^{[j]}\right)}^{\pm }})$, ${\gamma }_{j}$ and Bj are defined in equations ( |
| ii | (ii)As ${v}_{j-1}={v}_{j}$, ${q}_{i}^{\pm }(x,t;{{\boldsymbol{c}}}_{j}^{(d)})$ represents a DVDS and its expression is $\begin{eqnarray}{q}_{i}^{\pm }(x,t;{{\boldsymbol{c}}}_{j}^{(d)})={c}_{i}\displaystyle \frac{\det ({\hat{{\boldsymbol{M}}}}_{i}^{\pm })}{\det ({\hat{{\boldsymbol{N}}}}^{\pm })}{{\rm{e}}}^{{\rm{i}}{\theta }_{i}},\end{eqnarray}$ where $\begin{eqnarray}\begin{array}{rcl}{\hat{{\boldsymbol{M}}}}_{i}^{\pm } & = & {\left(\displaystyle \frac{[{{\rm{e}}}^{{\rm{i}}({\hat{X}}_{l}^{\pm }-{\left({\widetilde{X}}_{k}^{\pm }\right)}^{* })}+{\delta }_{k,l}]}{{\chi }_{l}-{\chi }_{k}^{* }}-{\beta }_{l,i}{{\rm{e}}}^{{\rm{i}}({\hat{X}}_{l}^{\pm }-{\left({\widetilde{X}}_{k}^{\pm }\right)}^{* })}\right)}_{j-1\leqslant k,l\leqslant j},\\ {\hat{{\boldsymbol{N}}}}^{\pm } & = & {\left(\displaystyle \frac{[{{\rm{e}}}^{{\rm{i}}({\hat{X}}_{l}^{\pm }-{\left({\widetilde{X}}_{k}^{\pm }\right)}^{* })}+{\delta }_{k,l}]}{{\chi }_{l}-{\chi }_{k}^{* }}\right)}_{j-1\leqslant k,l\leqslant j},\end{array}\end{eqnarray}$ ${\hat{X}}_{l}^{\pm }={X}_{l}+\mathrm{ln}({\left({\zeta }_{j}^{[l]}\right)}^{\pm })$, ${\widetilde{X}}_{l}^{\pm }={X}_{l}+\mathrm{ln}({\left({\tilde{\zeta }}_{j}^{[l]}\right)}^{\pm })$, ${\hat{{\boldsymbol{M}}}}_{i}^{\pm }$ and ${\hat{{\boldsymbol{N}}}}^{\pm }$ are both 2 × 2 matrices; ${\beta }_{l,i}$ and ${\delta }_{k,l}$ are defined in equation ( |
The proof of theorem
$\begin{eqnarray}{q}_{i}^{[n]}(x,t;{\boldsymbol{c}})={c}_{i}\displaystyle \frac{\det ({{\boldsymbol{E}}}^{\dagger }{\boldsymbol{C}}{\boldsymbol{E}}+{\boldsymbol{B}})}{\det ({{\boldsymbol{E}}}^{\dagger }{\boldsymbol{A}}{\boldsymbol{E}}+{\boldsymbol{B}})}{{\rm{e}}}^{{\rm{i}}{\theta }_{i}},\end{eqnarray}$
where matrices ${\boldsymbol{A}}$, ${\boldsymbol{B}}$, ${\boldsymbol{C}}$ are defined in lemma $\begin{eqnarray}{\boldsymbol{E}}=\mathrm{diag}\left({{\rm{e}}}^{{\rm{i}}{X}_{1}},{{\rm{e}}}^{{\rm{i}}{X}_{2}},\cdots ,{{\rm{e}}}^{{\rm{i}}{X}_{n}}\right).\end{eqnarray}$
Moreover, the multi-dark soliton solutions can be expressed as $\begin{eqnarray}\begin{array}{l}{q}_{i}^{[n]}(x,t;{\boldsymbol{c}})\\ \quad ={c}_{i}\displaystyle \frac{\det ({\left({{\boldsymbol{F}}}_{j}^{\pm }\right)}^{\dagger }{{\boldsymbol{E}}}^{\dagger }{\boldsymbol{C}}{\boldsymbol{E}}{{\boldsymbol{F}}}_{j}^{\pm }+{\left({{\boldsymbol{F}}}_{j}^{\pm }\right)}^{\dagger }{\boldsymbol{B}}{{\boldsymbol{F}}}_{j}^{\pm })}{\det ({\left({{\boldsymbol{F}}}_{j}^{\pm }\right)}^{\dagger }{{\boldsymbol{E}}}^{\dagger }{\boldsymbol{A}}{\boldsymbol{E}}{{\boldsymbol{F}}}_{j}^{\pm }+{\left({{\boldsymbol{F}}}_{j}^{\pm }\right)}^{\dagger }{\boldsymbol{B}}{{\boldsymbol{F}}}_{j}^{\pm })}{{\rm{e}}}^{{\rm{i}}{\theta }_{i}},\end{array}\end{eqnarray}$
where ${{\boldsymbol{F}}}_{j}^{+}=\mathrm{diag}\left({\overbrace{1,\cdots ,1}}^{j},{{\rm{e}}}^{-{\rm{i}}{X}_{j+1}},\cdots ,{{\rm{e}}}^{-{\rm{i}}{X}_{n}}\right)$, ${{\boldsymbol{F}}}_{j}^{-}=\mathrm{diag}({{\rm{e}}}^{-{\rm{i}}{X}_{1}},\cdots ,{{\rm{e}}}^{-{\rm{i}}{X}_{j-1}},{\overbrace{1,\cdots ,1}}^{n-j+1},)$. The problem of identifying the expressions for the multi-dark soliton solutions in equation ( $\begin{eqnarray}\begin{array}{l}\det ({\left({{\boldsymbol{F}}}_{j}^{+}\right)}^{\dagger }{{\boldsymbol{E}}}^{\dagger }{\boldsymbol{A}}{\boldsymbol{E}}{{\boldsymbol{F}}}_{j}^{+}+{\left({{\boldsymbol{F}}}_{j}^{+}\right)}^{\dagger }{\boldsymbol{B}}{{\boldsymbol{F}}}_{j}^{+})\\ \quad =\det ({{\boldsymbol{B}}}_{j-1})\det ({{\boldsymbol{A}}}_{n-j+1}){{\rm{e}}}^{({\rm{i}}{X}_{j}-{\rm{i}}{X}_{j}^{* })}\\ \quad +\det ({{\boldsymbol{B}}}_{j})\det ({{\boldsymbol{A}}}_{n-j})+{ \mathcal O }({{\rm{e}}}^{-\epsilon | t| }),\end{array}\end{eqnarray}$
where ε is defined in equation( $\begin{eqnarray}\begin{array}{l}\det ({\left({{\boldsymbol{F}}}_{j}^{-}\right)}^{\dagger }{{\boldsymbol{E}}}^{\dagger }{\boldsymbol{A}}{\boldsymbol{E}}{{\boldsymbol{F}}}_{j}^{-}+{\left({{\boldsymbol{F}}}_{j}^{-}\right)}^{\dagger }{\boldsymbol{B}}{{\boldsymbol{F}}}_{j}^{-})\\ \quad =\det ({{\boldsymbol{A}}}_{j})\det ({{\boldsymbol{B}}}_{n-j}){{\rm{e}}}^{({\rm{i}}{X}_{j}-{\rm{i}}{X}_{j}^{* })}\\ \quad +\det ({{\boldsymbol{A}}}_{j-1})\det ({{\boldsymbol{B}}}_{n-j+1})+{ \mathcal O }({{\rm{e}}}^{-\epsilon | t| }),\end{array}\end{eqnarray}$
where matrix ${{\boldsymbol{A}}}_{j}$ is a submatrix of the matrix ${\boldsymbol{A}}$ that contains the first j rows and columns extracted from matrix ${\boldsymbol{A}};$ matrix ${{\boldsymbol{B}}}_{n-j}$ is composed of the last n − j rows and columns of matrix ${\boldsymbol{B}}$. Similarly, we can identify the determinant of the numerator $\det ({\left({{\boldsymbol{F}}}_{j}^{-}\right)}^{\dagger }{{\boldsymbol{E}}}^{\dagger }{\boldsymbol{C}}{\boldsymbol{E}}{{\boldsymbol{F}}}_{j}^{-}+{\left({{\boldsymbol{F}}}_{j}^{-}\right)}^{\dagger }{\boldsymbol{B}}{{\boldsymbol{F}}}_{j}^{-})$. In this way, it is easy to calculate the determinant of the numerator and denominator in equation ( $\begin{eqnarray}{q}_{i}^{[n]}(x,t;{\boldsymbol{c}},{l}_{j}^{\pm })={c}_{i}{\xi }_{j}^{\pm }\displaystyle \frac{{\left({{\boldsymbol{g}}}_{j}^{\pm }\right)}^{\top }{\boldsymbol{p}}}{{\left({{\boldsymbol{h}}}_{j}^{\pm }\right)}^{\top }{\boldsymbol{p}}}{{\rm{e}}}^{{\rm{i}}{\theta }_{i}}+{ \mathcal O }({{\rm{e}}}^{-\epsilon | t| }),\end{eqnarray}$
where $\begin{eqnarray}\begin{array}{rcl}{\boldsymbol{p}} & = & {\left(\mathrm{1,1}\right)}^{\top },{{\boldsymbol{h}}}_{j}^{\pm }={\left({{\rm{e}}}^{2{\gamma }_{j}{\mu }_{{jI}}},\displaystyle \frac{{\left({\zeta }_{j}^{[j]}\right)}^{\pm }}{{\left({\tilde{\zeta }}_{j}^{[j]}\right)}^{\pm }}{{\rm{e}}}^{({\rm{i}}{X}_{j}-{\rm{i}}{X}_{j}^{* })}\right)}^{\top },\\ {{\boldsymbol{g}}}_{j}^{\pm } & = & {\left({{\rm{e}}}^{2{\gamma }_{j}{\mu }_{{jI}}},\displaystyle \frac{\left({a}_{i}+{\chi }_{j}^{* }\right){\left({\zeta }_{j}^{[j]}\right)}^{\pm }}{({a}_{i}+{\chi }_{j}){\left({\tilde{\zeta }}_{j}^{[j]}\right)}^{\pm }}{{\rm{e}}}^{\left({\rm{i}}{X}_{j}-{\rm{i}}{X}_{j}^{* }\right)}\right)}^{\top },\end{array}\end{eqnarray}$
${\xi }_{j}^{\pm }$ and ${\left({\zeta }_{j}^{[j]}\right)}^{\pm }$ are defined in equation ( $\begin{eqnarray}{q}_{i}^{[n]}(x,t;{\boldsymbol{c}},{l}_{j}^{\pm })={\xi }_{j}^{\pm }{q}_{i}^{\pm }(x,t;{{\boldsymbol{c}}}_{j}^{(s)})+{ \mathcal O }({{\rm{e}}}^{-\epsilon | t| }),\end{eqnarray}$
where ${q}_{i}^{\pm }(x,t;{{\boldsymbol{c}}}_{j})$ is corresponded to equation (Then we conduct the asymptotic analysis of the multi-dark soliton solutions along the trajectory Lj. The multi-dark soliton solutions can be further expressed as
$\begin{eqnarray}\begin{array}{l}{q}_{i}^{[n]}(x,t;{\boldsymbol{c}})\\ \quad ={c}_{i}\displaystyle \frac{\det ({\left({{\boldsymbol{F}}}_{j-1}^{\pm }\right)}^{\dagger }{{\boldsymbol{E}}}^{\dagger }{\boldsymbol{C}}{\boldsymbol{E}}{{\boldsymbol{F}}}_{j-1}^{\pm }+{\left({{\boldsymbol{F}}}_{j-1}^{\pm }\right)}^{\dagger }{\boldsymbol{B}}{{\boldsymbol{F}}}_{j-1}^{\pm })}{\det ({\left({{\boldsymbol{F}}}_{j-1}^{\pm }\right)}^{\dagger }{{\boldsymbol{E}}}^{\dagger }{\boldsymbol{A}}{\boldsymbol{E}}{{\boldsymbol{F}}}_{j-1}^{\pm }+{\left({{\boldsymbol{F}}}_{j-1}^{\pm }\right)}^{\dagger }{\boldsymbol{B}}{{\boldsymbol{F}}}_{j-1}^{\pm })}{{\rm{e}}}^{{\rm{i}}{\theta }_{i}},\end{array}\end{eqnarray}$
where ${{\boldsymbol{F}}}_{j-1}^{+}=\mathrm{diag}\left(\mathop{\overbrace{1,\cdots ,1}}\limits^{j},{{\rm{e}}}^{-{\rm{i}}{X}_{j+1}},\cdots ,{{\rm{e}}}^{-{\rm{i}}{X}_{n}}\right)$, ${{\boldsymbol{F}}}_{j-1}^{-}=\mathrm{diag}\left({{\rm{e}}}^{-{\rm{i}}{X}_{1}},\cdots ,{{\rm{e}}}^{-{\rm{i}}{X}_{j-2}},\mathop{\overbrace{1,\cdots ,1}}\limits^{n-j+2}\right)$, matrix ${\boldsymbol{E}}$ is defined in equation ( $\begin{eqnarray}\begin{array}{l}\det ({\left({{\boldsymbol{F}}}_{j-1}^{+}\right)}^{\dagger }{{\boldsymbol{E}}}^{\dagger }{\boldsymbol{A}}{\boldsymbol{E}}{{\boldsymbol{F}}}_{j-1}^{+}+{\left({{\boldsymbol{F}}}_{j-1}^{+}\right)}^{\dagger }{\boldsymbol{B}}{{\boldsymbol{F}}}_{j-1}^{+})\\ \quad =\det ({{\boldsymbol{A}}}_{n-j+2}^{[j]})\det ({{\boldsymbol{B}}}_{j}^{[j-1]}){{\rm{e}}}^{({\rm{i}}{X}_{j-1}-{\rm{i}}{X}_{j-1}^{* })}\\ \quad +\det ({{\boldsymbol{A}}}_{n-j+1})\det ({{\boldsymbol{B}}}_{j-1}){{\rm{e}}}^{({\rm{i}}{X}_{j}-{\rm{i}}{X}_{j}^{* })}\\ \quad +\det ({{\boldsymbol{A}}}_{n-j+2})\det ({{\boldsymbol{B}}}_{j-2}){{\rm{e}}}^{({\rm{i}}{X}_{j-1}-{\rm{i}}{X}_{j-1}^{* })}{{\rm{e}}}^{({\rm{i}}{X}_{j}-{\rm{i}}{X}_{j}^{* })}\\ \quad +\det ({{\boldsymbol{A}}}_{n-j})\det ({{\boldsymbol{B}}}_{j})+{ \mathcal O }({{\rm{e}}}^{-\epsilon | t| })\end{array}\end{eqnarray}$
as $t\to +\infty $, and $\begin{eqnarray}\begin{array}{l}\det ({\left({{\boldsymbol{F}}}_{j-1}^{-}\right)}^{\dagger }{{\boldsymbol{E}}}^{\dagger }{\boldsymbol{A}}{\boldsymbol{E}}{{\boldsymbol{F}}}_{j-1}^{-}+{\left({{\boldsymbol{F}}}_{j-1}^{-}\right)}^{\dagger }{\boldsymbol{B}}{{\boldsymbol{F}}}_{j-1}^{-})\\ \quad =\det ({{\boldsymbol{A}}}_{j-1})\det ({{\boldsymbol{B}}}_{n-j+1}){{\rm{e}}}^{({\rm{i}}{X}_{j-1}-{\rm{i}}{X}_{j-1}^{* })}\\ \quad +\det ({{\boldsymbol{A}}}_{j}^{[j-1]})\det ({{\boldsymbol{B}}}_{n-j+2}^{[j]}){{\rm{e}}}^{({\rm{i}}{X}_{j}-{\rm{i}}{X}_{j}^{* })}\\ \quad +\det ({{\boldsymbol{A}}}_{j})\det ({{\boldsymbol{B}}}_{n-j}){{\rm{e}}}^{({\rm{i}}{X}_{j-1}-{\rm{i}}{X}_{j-1}^{* })}{{\rm{e}}}^{({\rm{i}}{X}_{j}-{\rm{i}}{X}_{j}^{* })}\\ \quad +\det ({{\boldsymbol{A}}}_{j-2})\det ({{\boldsymbol{B}}}_{n-j+2})+{ \mathcal O }({{\rm{e}}}^{-\epsilon | t| })\end{array}\end{eqnarray}$
as $t\to -\infty $, where $\epsilon ={\min }_{1\leqslant k\ne j-1,j\leqslant n}\left(| {v}_{k}-{v}_{j}| \right);$ matrix ${{\boldsymbol{A}}}_{j}^{[j-1]}$ and matrix ${{\boldsymbol{A}}}_{n-j+2}^{[j]}$ are submatrices of matrix ${\boldsymbol{A}};$ the matrix ${{\boldsymbol{A}}}_{j}^{[j-1]}$ includes the first j rows and columns extracted from matrix ${\boldsymbol{A}}$ except the $j-1$ th row and column, and the matrix ${{\boldsymbol{A}}}_{n-j+2}^{[j]}$ contains the last $n-j+2$ rows and columns extracted from matrix ${\boldsymbol{A}}$ except the jth row and column; this is also how the matrices ${{\boldsymbol{B}}}_{j}^{[j-1]}$ and ${{\boldsymbol{B}}}_{n-j+2}^{[j]}$, which are submatrices of matrix ${\boldsymbol{B}}$, are defined. For the determinants of the numerator $\det ({\left({{\boldsymbol{F}}}_{j-1}^{+}\right)}^{\dagger }{{\boldsymbol{E}}}^{\dagger }{\boldsymbol{C}}{\boldsymbol{E}}{{\boldsymbol{F}}}_{j-1}^{+}+{({{\boldsymbol{F}}}_{j-1}^{+})}^{\dagger }{\boldsymbol{B}}{{\boldsymbol{F}}}_{j-1}^{+})$ and $\det ({\left({{\boldsymbol{F}}}_{j-1}^{-}\right)}^{\dagger }{{\boldsymbol{E}}}^{\dagger }{\boldsymbol{C}}{\boldsymbol{E}}{{\boldsymbol{F}}}_{j-1}^{-}+{({{\boldsymbol{F}}}_{j-1}^{-})}^{\dagger }{\boldsymbol{B}}{{\boldsymbol{F}}}_{j-1}^{-})$, we can obtain similar results through replacing matrix ${\boldsymbol{A}}$ in equation ( $\begin{eqnarray}{q}_{i}^{[n]}(x,t;{\boldsymbol{c}},{L}_{j}^{\pm })={c}_{i}{\xi }_{j}^{\pm }\displaystyle \frac{{\left({{\boldsymbol{g}}}_{j-1}^{\pm }\right)}^{\top }{\boldsymbol{Q}}{{\boldsymbol{g}}}_{j}^{\pm }}{{\left({{\boldsymbol{h}}}_{j-1}^{\pm }\right)}^{\top }{\boldsymbol{Q}}{{\boldsymbol{h}}}_{j}^{\pm }}{{\rm{e}}}^{{\rm{i}}{\theta }_{i}}+{ \mathcal O }({{\rm{e}}}^{-\epsilon | t| }),\end{eqnarray}$
where $\begin{eqnarray}{\boldsymbol{Q}}=\left[\begin{array}{cc}1 & 1\\ 1 & {\left|\displaystyle \frac{{\chi }_{j}-{\chi }_{j-1}}{{\chi }_{j}^{* }-{\chi }_{j-1}}\right|}^{2}\end{array}\right],\end{eqnarray}$
${{\boldsymbol{g}}}_{j}^{\pm }$ and ${{\boldsymbol{h}}}_{j}^{\pm }$ are defined in equation ( $\begin{eqnarray}{q}_{i}^{[n]}(x,t;{\boldsymbol{c}},{L}_{j}^{\pm })={\xi }_{j}^{\pm }{q}_{i}^{\pm }(x,t;{{\boldsymbol{c}}}_{j}^{(d)})+{ \mathcal O }({{\rm{e}}}^{-\epsilon | t| }),\end{eqnarray}$
where ${q}_{i}^{\pm }(x,t;{{\boldsymbol{c}}}_{j}^{(d)})$ is defined in equation (Evidently, it is worth noting that as t → ±∞ , the SVDS satisfies14 ). Undoubtedly, the interactions for SVDSs are always elastic. Figure 5 depicts an example of observing changes following the collision of two SVDSs. The shapes of the SVDSs do not change after the collision, indicating that the SVDSs admit elastic collisions. Next, we would like to look into the interaction between an SVDS and a DVDS. Following the collision with a DVDS, the SVDS retains its original form, as shown in figure 6, which implies that the collision for the SVDS is still elastic. However, after colliding with the SVDS, the shape of the DVDS changes significantly, implying that the DVDS admits an inelastic collision.
$\begin{eqnarray*}{\left|{q}_{i}^{+}(x-{x}_{0j}^{+},t;{{\boldsymbol{c}}}_{j}^{(s)})\right|}^{2}={\left|{q}_{i}^{-}(x-{x}_{0j}^{-},t;{{\boldsymbol{c}}}_{j}^{(s)})\right|}^{2}\end{eqnarray*}$
implying that the SVDSs keep their shape following a collision with a phase shift where i = 1, 2, j = 1, 2, ⋯ ,n, and ${x}_{0j}^{\pm }$ is defined in equation (
Figure 5. The collision dynamics of two SVDSs. Left panels: dynamical evolution of dark soliton solution $| {q}_{1}^{[2]}{| }^{2}$ before (t = −2, (a)) and after (t = 10, (c)) the collision. Right panels: dynamical evolution of dark soliton solution $| {q}_{2}^{[2]}{| }^{2}$ before (t = −2, (b)) and after (t = 10, (d)) the collision. The solid red line describes the evolution of the dark soliton solution ( |
Figure 6. The collision dynamics of an SVDS and a DVDS. Left panels: dynamical evolution of dark soliton solution $| {q}_{1}^{[3]}{| }^{2}$ before (t = − 3, (a)) and after (t = 5, (c)) the collision. Right panels: dynamical evolution of dark soliton solution $| {q}_{2}^{[3]}{| }^{2}$ before (t = −3, (b)) and after (t = 5, (d)) the collision. The solid red line describes the evolution of the dark soliton solution ( |
In fact, a plethora of experimental evidence has demonstrated that inelastic collisions occur in most cases for DVDSs, which is consistent with the outcomes we discussed in theorem 2 . For example, the two DVDSs in figure 7 do not keep their pre-collision shape after the collision implying that they both exhibit inelastic collisions.
Figure 7. The collision dynamics of two DVDSs. Left panels: dynamical evolution of multi-dark soliton solution $| {q}_{1}^{[4]}{| }^{2}$ before (t = −50, (a)) and after (t = 50, (c)) the collision. Right panels: dynamical evolution of dark soliton solution $| {q}_{2}^{[4]}{| }^{2}$ before (t = −50, (b)) and after (t = 50, (d)) the collision. The solid red line describes the evolution of the dark soliton solution ( |
In light of this, we proceed to ascertain the conditions that give rise to elastic behavior in collisions for DVDSs. It should be highlighted that the asymptotic expression of a DVDS before and after the collision differs primarily in terms of the phase shift. Thus, the collision is elastic if the phase differences of the two valleys of the DVDS are equal before and after the collision; otherwise, it is inelastic. From the asymptotic expressions (30 ) we can also derive the elastic condition for the DVDSs as follows:10 ). Different from figure 6, the two valleys of the DVDS in figure 8 are separated by a relatively wide distance (the displacement difference between the two valleys before and after the collision of the DVDS is much smaller than the initial distance of the two valleys). The interaction between the two valleys is extremely weak in this case, so even if the DVDS in figure 8 has an inelastic collision, the shape change after the collision is easily ignored.
$\begin{eqnarray}\left\{\begin{array}{l}{\prod }_{l=1}^{j-2}{\left|\displaystyle \frac{{\chi }_{l}-{\chi }_{j}}{{\chi }_{l}-{\chi }_{j-1}}\right|}^{2}=\displaystyle \prod _{l=j+1}^{n}{\left|\displaystyle \frac{{\chi }_{l}-{\chi }_{j}}{{\chi }_{l}-{\chi }_{j-1}}\right|}^{2},\\ {\prod }_{l=1}^{j-2}{\left|\displaystyle \frac{{\chi }_{l}^{* }-{\chi }_{j}}{{\chi }_{l}^{* }-{\chi }_{j-1}}\right|}^{2}=\displaystyle \prod _{l=j+1}^{n}{\left|\displaystyle \frac{{\chi }_{l}^{* }-{\chi }_{j}}{{\chi }_{l}^{* }-{\chi }_{j-1}}\right|}^{2},\end{array}\right.\end{eqnarray}$
where χl is defined in equation (
Figure 8. The collision dynamics of an SVDS and a DVDS. Left panels: Dynamical evolution of a multi-dark soliton solution $| {q}_{1}^{[3]}{| }^{2}$ before (t = −1, (a)) and after (t = 6, (c)) the collision. Right panels: dynamical evolution of a multi-dark soliton solution $| {q}_{2}^{[3]}{| }^{2}$ before (t = −1, (b)) and after (t = 6, (d)) the collision. The solid red line describes the evolution of the dark soliton solution ( |
4. Conclusions
In summary, we provide a sufficient condition for the existence of dark soliton solutions and proceed to derive the uniform expressions of such solutions including both SVDSs and DVDSs by means of the uniform Darboux transformation. The analysis indicates that while elastic collisions are a common feature of SVDSs, inelastic collisions are prevalent in most instances for DVDSs. Notably, we also propose a condition that guarantees elastic collisions for DVDSs. The dark soliton solutions derived from the defocusing coupled Hirota equation possess the potential for applications in physical fields such as signal transmission and modulation in the realm of fiber optic communication [32, 45]. Furthermore, our results also shed new light on the fundamental properties of dark solitons, and may provide a promising avenue for future research in the fields of nonlinear optics and photonics [46, 47].