⇦Chinese Physics Letters, 2019, Vol. 36, No. 11, Article code 110201 Construction of Multi-soliton Solutions of the $N$-Coupled Hirota Equations in an Optical Fiber * Zhou-Zheng Kang (康周正)1,2, Tie-Cheng Xia (夏铁成)1** Affiliations 1Department of Mathematics, Shanghai University, Shanghai 200444 2College of Mathematics, Inner Mongolia University for Nationalities, Tongliao 028043 Received 22 July 2019, online 21 October 2019 *Supported by the National Natural Science Foundation of China under Grant Nos 11975145, 11271008 and 61072147.
**Corresponding author. Email: xiatc@shu.edu.cn Citation Text: Kang Z Z and Xia T C 2019 Chin. Phys. Lett. 36 110201    Abstract This work aims to study the $N$-coupled Hirota equations in an optical fiber under the zero boundary condition at infinity. By analyzing the spectral problem, a matrix Riemann–Hilbert problem on the real axis is strictly established. Then, by solving the presented matrix Riemann–Hilbert problem under the constraint of no reflection, the bright multi-soliton solutions to the $N$-coupled Hirota equations are explicitly gained. DOI:10.1088/0256-307X/36/11/110201 PACS:02.30.Jr, 02.30.Ik, 05.45.Yv © 2019 Chinese Physics Society Article Text Considerable efforts have been devoted to investigating soliton solutions of nonlinear evolution equations (NLEEs) for a profound understanding of various complex nonlinear phenomena occurring in fluid dynamics, plasma physics, oceanography, optics, condensed matter physics, and so on. Currently, a variety of efficient approaches[1–11] have been available for searching for soliton solutions of NLEEs. One of the most famous methods is the inverse scattering transform (IST) method,[12,13] which plays a key role in investigating exact solutions and long-time asymptotics of integrable NLEEs with particular initial data. As a modern version of the IST method, the Riemann–Hilbert method for integrable NLEEs originated in the works of Manakov, Shabat and Zakharov carried out in 1975–1979, and afterwards it was applied extensively to verify that many NLEEs are in possession of abundant multi-soliton solutions, such as the general coupled nonlinear Schrödinger equations,[14] the coupled derivative Schrödinger equation,[15] the two-component Gerdjikov–Ivanov equation,[16] the coupled Gerdjikov–Ivanov derivative nonlinear Schrödinger equation,[17] the coupled modified Korteweg–de Vries equation,[18] the Kundu–Eckhaus equation,[19] the six-component fourth-order AKNS system,[20] and the Chen–Lee–Liu equation.[21] In this work, we focus on the $N$-coupled Hirota equations[22] $$\begin{align} {{q}_{jt}}=\,&i\Big[\frac{1}{2}{{q}_{jxx}} +\Big(\sum\limits_{r=1}^{N}{{{|{{q}_{r}}|}^{2}}}\Big){{q}_{j}}\Big]+\epsilon \Big[{{q}_{jxxx}}\\ &+3\Big(\sum\limits_{r=1}^{N}{{{|{{q}_{r}}|}^{2}}}\Big){{q}_{jx}} +3\Big(\sum\limits_{r=1}^{N}{q_{r}^{*}{{q}_{rx}}}\Big){{q}_{j}}\Big],~~ \tag {1} \end{align} $$ with $j=1,2,\ldots,N$, which govern the nonlinear wave propagation of simultaneous $N$ fields in an optical fiber with the effects of group velocity dispersion, self-phase modulation, higher-order dispersion, and self-steepening. Here, $q_{j}$ represents the complex amplitude of the pulse envelope, the subscripts of $q_{j}$ denote the partial derivatives with respect to the scaled spatial coordinate $x$ and time coordinate $t$ correspondingly, while $\epsilon$ is a real constant, and the asterisk represents the complex conjugate. When $\epsilon=0$, Eq. (1) reduces to the $N$-coupled nonlinear Schrödinger equations. There has been much work on investigating Eq. (1) so far. In Ref. [22], some one-soliton solutions to Eq. (1) were worked out through drawing on the Bäcklund transformation method. The Lax pair and integrability were revealed as well. By employing Hirota's method and some auxiliary functions, the improved bilinear forms and some novel types of soliton solutions[23] were explored for Eq. (1). Inspired by the work,[24] we would like to study Eq. (1) under the zero boundary condition at infinity by the Riemann–Hilbert method. The first focus for us is on determination of a matrix Riemann–Hilbert problem associated with Eq. (1), which will lay the foundation for subsequent computation. We start with the Lax pair[22] $$\begin{align} {{\phi}_{x}}=\,&{{U}}\phi=(i\lambda\sigma+Q)\phi,~~ \tag {2a}\\ {{\phi}_{t}}=\,&{{V}}\phi=[(i{{\lambda }^{2}}-4i\epsilon {{\lambda}^{3}})\sigma+\tilde{V}]\phi,~~ \tag {2b} \end{align} $$ where $\phi=({{\phi}_{1}},{{\phi}_{2}},\ldots,{{\phi}_{N+1}})^{\rm T}$ is the spectral function, $\lambda\in \mathbb{C}$ is a spectral parameter, the superscript T is the transpose of the vector, and $$\begin{align} \tilde{V}=\,&(\lambda-4\epsilon {{\lambda }^{2}})Q+\Big(\frac{i}{2}-2i\epsilon \lambda\Big){{Q}_{1}}+\epsilon {{Q}_{2}},\\ \sigma=\,&{\rm diag}(-1,1,\ldots,1), \end{align} $$ with $$ Q=\left(\begin{matrix} 0 & {{q}_{1}} & {{q}_{2}} & \cdots & {{q}_{N}} \\ -q_{1}^{*} & 0 & 0 & \cdots & 0 \\ -q_{2}^{*} & 0 & 0 & \cdots & 0 \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ -q_{N}^{*} & 0 & 0 & \cdots & 0 \\ \end{matrix}\right),~~~~{{Q}_{1}}=\left(\begin{matrix} \sum\limits_{r=1}^{N}{{{|{{q}_{r}}|}^{2}}} & {{q}_{1x}} & {{q}_{2x}} & \cdots & {{q}_{Nx}} \\ q_{1x}^{*} & -{{|{{q}_{1}}|}^{2}} & -{{q}_{2}}q_{1}^{*} & \cdots & -{{q}_{N}}q_{1}^{*} \\ q_{2x}^{*} & -{{q}_{1}}q_{2}^{*} & -{{|{{q}_{2}}|}^{2}} & \cdots & -{{q}_{N}}q_{2}^{*} \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ q_{Nx}^{*} & -{{q}_{1}}q_{N}^{*} & -{{q}_{2}}q_{N}^{*} & \cdots & -{{|{{q}_{N}}|}^{2}} \\ \end{matrix}\right), $$ $$ {{Q}_{2}}=\left(\begin{matrix} \sum\limits_{r=1}^{N}{({{q}_{rx}}q_{r}^{*}-{{q}_{r}}q_{rx}^{*})} & {{q}_{1xx}}+2{{q}_{1}}\sum\limits_{r=1}^{N}{{{|{{q}_{r}}|}^{2}}} & {{q}_{2xx}}+2{{q}_{2}}\sum\limits_{r=1}^{N}{{{|{{q}_{r}}|}^{2}}} & \cdots & {{q}_{Nxx}}+2{{q}_{N}}\sum\limits_{r=1}^{N}{{{|{{q}_{r}}|}^{2}}} \\ -q_{1xx}^{*}-2q_{1}^{*}\sum\limits_{r=1}^{N}{{{|{{q}_{r}}|}^{2}}} & -({{q}_{1x}}q_{1}^{*}-{{q}_{1}}q_{1x}^{*}) & -({{q}_{2x}}q_{1}^{*}-{{q}_{2}}q_{1x}^{*}) & \cdots & -({{q}_{Nx}}q_{1}^{*}-{{q}_{N}}q_{1x}^{*}) \\ -q_{2xx}^{*}-2q_{2}^{*}\sum\limits_{r=1}^{N}{{{|{{q}_{r}}|}^{2}}} & -({{q}_{1x}}q_{2}^{*}-{{q}_{1}}q_{2x}^{*}) & -({{q}_{2x}}q_{2}^{*}-{{q}_{2}}q_{2x}^{*}) & \cdots & -({{q}_{Nx}}q_{2}^{*}-{{q}_{N}}q_{2x}^{*}) \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ -q_{Nxx}^{*}-2q_{N}^{*}\sum\limits_{r=1}^{N}{{{|{{q}_{r}}|}^{2}}} & -({{q}_{1x}}q_{N}^{*}-{{q}_{1}}q_{Nx}^{*}) & -({{q}_{2x}}q_{N}^{*}-{{q}_{2}}q_{Nx}^{*}) & \cdots & -({{q}_{Nx}}q_{N}^{*}-{{q}_{N}}q_{Nx}^{*}) \\ \end{matrix}\right). $$ In our analysis, we suppose that all the potential functions $q_{j}(1\leq j\leq N)$ decay to zero sufficiently fast at infinity. Therefore, a variable transformation $\phi=\eta {{e}^{i\lambda \sigma x+(i{{\lambda }^{2}}-4i\epsilon {{\lambda }^{3}})\sigma t}} $ is introduced to map Eqs. (2a) and (2b) into the form $$\begin{align} {{\eta }_{x}}=\,&i\lambda [\sigma,\eta]+Q\eta,~~ \tag {3a}\\ {{\eta }_{t}}=\,&(i{{\lambda }^{2}}-4i\epsilon {{\lambda }^{3}})[\sigma,\eta]+\tilde{V}\eta,~~ \tag {3b} \end{align} $$ where $[\sigma,\eta]\equiv\sigma \eta-\eta\sigma$ denotes the commutator. In what follows, we concentrate on Eq. (3a) to carry out the spectral analysis, where the variable $t$ enters as a dummy variable and is omitted. Concerning Eq. (3a), we write its two matrix Jost solutions $\eta_{\pm}$ in terms of a collection of columns, $$\begin{align} {{\eta }_{-}}=\,&({{[{{\eta }_{-}}]_{1}}},{{[{{\eta }_{-}}]_{2}}},\ldots,{{[{{\eta }_{-}}]_{N+1}}}),\\ {{\eta }_{+}}=\,&({{[{{\eta }_{+}}]_{1}}},{{[{{\eta }_{+}}]_{2}}},\ldots,{{[{{\eta }_{+}}]_{N+1}}}),~~ \tag {4} \end{align} $$ obeying the asymptotic conditions $$ {\eta_{\pm}}\to {I}_{N+1},~x\to \pm\infty,~~ \tag {5} $$ where the subscripts of $\eta$ indicated refer to which end of the $x$-axis the boundary conditions are set, and ${I}_{N+1}$ is the identity matrix of rank $N+1$. Indeed, the Jost solutions ${\eta_{\pm }}$ are uniquely determined by the Volterra integral equations $$\begin{align} {{\eta }_{-}}(x,\lambda)= {I}_{N+1}+\int_{-\infty }^{x}&{{{e}}^{i\lambda \sigma (x-y)}}Q(y){{\eta }_{-}}(y,\lambda)\\ &\cdot{{{e}}^{-i\lambda \sigma (x-y)}}dy,~~ \tag {6a}\\ {{\eta }_{+}}(x,\lambda)= {I}_{N+1}-\int_{x}^{+\infty }&{{{e}}^{i\lambda \sigma (x-y)}}Q(y){{\eta }_{+}}(y,\lambda)\\ &\cdot{{{e}}^{-i\lambda \sigma (x-y)}}dy.~~ \tag {6b} \end{align} $$ The analysis on Eqs. (6a) and (6b) reveals that ${{[{{\eta }_{+}}]_{1}}},{{[{{\eta }_{-}}]_{2}}},\ldots,{{[{{\eta }_{-}}]_{N+1}}}$ are analytic for $\lambda \in {\mathbb{C}^{-}}$ and continuous for $\lambda \in {\mathbb{C}^{-}}\cup \mathbb{R}$, while ${{[{{\eta }_{-}}]_{1}}},{{[{{\eta }_{+}}]_{2}}},\ldots,{{[{{\eta }_{+}}]_{N+1}}}$ are analytic for $\lambda\in {\mathbb{C}^{+}}$ and continuous for $\lambda \in {\mathbb{C}^{+}}\cup \mathbb{R}$, where the domains ${\mathbb{C}^{-}}$ and ${\mathbb{C}^{+}}$ are the lower and upper half $\lambda$-plane, respectively. Owing to ${\rm tr}Q=0$, applying Abel's identity, as well as recalling the asymptotics (5), it can be shown that $\det {\eta_{\pm }}=1$ for all $x$ and $\lambda \in \mathbb{R}$. In addition, both ${\eta_{-}}E$ and ${\eta_{+}}E$ are the matrix solutions to Eq. (2a), they must be linearly associated, namely, $$ {{\eta }_{-}}E={{\eta }_{+}}ES(\lambda),\quad E={{{e}}^{i\lambda \sigma x}},~~ \tag {7} $$ where $S(\lambda)={{({{s}_{kj}})_{(N+1)\times (N+1)}}}$ and $\det S(\lambda)=1$. Then, we shall determine two matrix functions, which are analytically continued to the upper and lower half-planes, respectively. In the light of the analytic properties of ${\eta_{\pm}}$, we define the first analytic function of $\lambda$ in ${\mathbb{C}^{+}}$ as $$ {{{\it \Psi}}_{1}}=({{[{{\eta }_{-}}]_{1}}},{{[{{\eta }_{+}}]_{2}}},\ldots,{{[{{\eta }_{+}}]_{N+1}}}).~~ \tag {8} $$ We can obtain the asymptotic behavior ${{\it \Psi}_{1}}\to {I}_{N+1}$ as $\lambda \in {\mathbb{C}^{+}}\to \infty$. To present a matrix Riemann–Hilbert problem, we also need to formulate the analytic counterpart of ${\it \Psi}_{1}$ in ${\mathbb{C}^{-}}$. We consider the adjoint scattering equation related to Eq. (3a), $$ {{\chi}_{x}}=i\lambda [\sigma,\chi]-\chi Q.~~ \tag {9} $$ It can be known that $\eta_{\pm}^{-1}$ meet the adjoint Eq. (9) and follow the boundary conditions $\eta_{\pm}^{-1}\rightarrow{I}$ as $x\rightarrow\pm\infty$. By setting ${{[\eta _{\pm }^{-1}]^{l}}}$ being the $l$th row of $\eta _{\pm }^{-1}$, we obtain $$ \eta _{\pm }^{-1}=\left(\begin{matrix} {{[\eta _{\pm }^{-1}]^{1}}} \\ {{[\eta _{\pm }^{-1}]^{2}}} \\ \vdots \\ {{[\eta _{\pm }^{-1}]^{N+1}}} \\ \end{matrix}\right).~~ \tag {10} $$ Utilizing the same techniques as before, we have $$ {{\it \Psi}_{2}}=\left(\begin{matrix} {{[\eta _{-}^{-1}]^{1}}} \\ {{[\eta _{+}^{-1}]^{2}}} \\ \vdots \\ {{[\eta _{+}^{-1}]^{N+1}}} \\ \end{matrix}\right),~~ \tag {11} $$ which is analytic in ${\mathbb{C}^{-}}$. Similar to ${{\it \Psi}_{1}}$, the asymptotic behavior of ${{\it \Psi}_{2}}$ turns out to be ${{\it \Psi}_{2}}\to {I}$ as $\lambda \in {\mathbb{C}^{-}}\to \infty$. From Eq. (7), we have $$ \eta _{-}^{-1}={E}{R}(\lambda){{E}^{-1}}\eta _{+}^{-1},~~ \tag {12} $$ with $R(\lambda)=S^{-1}(\lambda)={{({{r}_{kj}})_{(N+1)\times (N+1)}}}$. We insert Eq. (4) into Eq. (7) and obtain $$\begin{align} {{[{{\eta }_{-}}]_{1}}}=\,&{{s}_{11}}{{[{{\eta }_{+}}]_{1}}}+{{s}_{21}}{{{e}}^{2i\lambda x}}{{[{{\eta }_{+}}]_{2}}}+{{s}_{31}}{{{e}}^{2i\lambda x}}{{[{{\eta }_{+}}]_{3}}}\\ &+\cdots +{{s}_{N+1,1}}{{{e}}^{2i\lambda x}}{{[{{\eta }_{+}}]_{N+1}}}. \end{align} $$ Hence, ${{\it \Psi}_{1}}$ is expressed as $$\begin{align} {{\it \Psi}_{1}}=\,&({{[{{\eta }_{+}}]_{1}}},{{[{{\eta }_{+}}]_{2}}},{{[{{\eta }_{+}}]_{3}}},\ldots,{{[{{\eta }_{+}}]_{N+1}}})\\ &\cdot\left(\begin{matrix} {{s}_{11}} & 0 & 0 & \cdots & 0 \\ {{s}_{21}}{{{e}}^{2i\lambda x}} & 1 & 0 & \cdots & 0 \\ {{s}_{31}}{{{e}}^{2i\lambda x}} & 0 & 1 & \ddots & \vdots \\ \vdots & \vdots & \ddots & \ddots & 0 \\ {{s}_{N+1,1}}{{{e}}^{2i\lambda x}} & 0 & \cdots & 0 & 1 \\ \end{matrix}\right). \end{align} $$ After carrying Eq. (10) into Eq. (12), we find $$\begin{align} {{[\eta _{-}^{-1}]^{1}}}=\,&{{r}_{11}}{{[\eta _{+}^{-1}]^{1}}}+{{r}_{12}}{{{e}}^{-2i\lambda x}}{{[\eta _{+}^{-1}]^{2}}}\\ &+{{r}_{13}}{{{e}}^{-2i\lambda x}}{{[\eta _{+}^{-1}]^{3}}}+\cdots \\ &+{{r}_{1,N+1}}{{{e}}^{-2i\lambda x}}{{[\eta _{+}^{-1}]^{N+1}}}. \end{align} $$ Consequently, ${{\it \Psi}_{2}}$ is represented as $$\begin{align} {{\it \Psi}_{2}}\!=\!&\left( \begin{matrix} {{r}_{11}}\! \!&\!\! {{r}_{12}}{{{e}}^{-2i\lambda x}} \!&\! {{r}_{13}}{{{e}}^{-2i\lambda x}} \!&\! \cdots \!&\! {{r}_{1,N+1}}{{{e}}^{-2i\lambda x}} \\ 0 \!&\! 1 \!&\! 0 \!&\! \cdots \!&\! 0 \\ 0 \!&\! 0 \!&\! 1 \!&\! \ddots \!&\! \vdots \\ \vdots \!&\! \vdots \!&\! \ddots \!&\! \ddots \!&\! 0 \\ 0 \!&\! 0 \!&\! \cdots \!&\! 0 \!&\! 1 \\ \end{matrix}\right)\\ &\cdot\left(\begin{matrix} {{[\eta _{+}^{-1}]^{1}}} \\ {{[\eta _{+}^{-1}]^{2}}} \\ {{[\eta _{+}^{-1}]^{3}}} \\ \vdots \\ {{[\eta _{+}^{-1}]^{N+1}}} \\ \end{matrix}\right). \end{align} $$ Having constructed two matrix functions ${{\it \Psi}_{1}}$ and ${{\it \Psi}_{2}}$, which are analytic in ${\mathbb{C}^{+}}$ and ${\mathbb{C}^{-}}$, respectively, we now describe a matrix Riemann–Hilbert problem for Eq. (1). With the notations of ${{\it \Psi}_{1}}\rightarrow{{\it \Psi}^{+}}$ as $\lambda\in {\mathbb{C}^{+}}\rightarrow\mathbb{R}$ and ${{\it \Psi}_{2}}\rightarrow{{\it \Psi}^{-}}$ as $\lambda \in {\mathbb{C}^{-}}\rightarrow\mathbb{R}$, a matrix Riemann–Hilbert problem can be expressed as $$ {{\it \Psi}^{-}}(x,\lambda){{\it \Psi}^{+}}(x,\lambda)=\mathcal G(x,\lambda),~\lambda \in \mathbb{R},~~ \tag {13} $$ where $$\begin{align} &\mathcal G(x,\lambda)\!\!=\!\!\!\left(\begin{matrix} 1 \!\!&\!\! {{r}_{12}}{{{e}}^{-2i\lambda x}} \!\!&\!\! {{r}_{13}}{{{e}}^{-2i\lambda x}} \!\!&\!\! \cdots \!\!&\!\! \kappa_{1,N+1}\\ {{s}_{21}}{{{e}}^{2i\lambda x}} \!\!&\!\! 1 \!\!&\!\! 0 \!\!&\!\! \cdots \!\!&\!\! 0 \\ {{s}_{31}}{{{e}}^{2i\lambda x}} \!\!&\!\! 0 \!\!&\!\! 1 \!\!&\!\! \ddots \!\!&\!\! \vdots \\ \vdots \!\!&\!\! \vdots \!\!&\!\! \ddots \!\!&\!\! \ddots \!\!&\!\! 0 \\ \kappa_{N+1,1} \!\!&\!\! 0 \!\!&\!\! \cdots \!\!&\!\! 0 \!\!&\!\! 1 \\ \end{matrix}\right),\\ &~~~\kappa_{1,N+1}={{r}_{1,N+1}}{{{e}}^{-2i\lambda x}},~~\kappa_{N+1,1}={{s}_{N+1,1}}{{{e}}^{2i\lambda x}}, \end{align} $$ with its canonical normalization conditions given by $$\begin{align} &{{\it \Psi}_{1}}(x,\lambda)\to {I}_{N+1},~ \lambda \in {\mathbb{C}^{+}}\to \infty, \\ &{{\it \Psi}_{2}}(x,\lambda)\to {I}_{N+1},~ \lambda \in {\mathbb{C}^{-}}\to \infty, \end{align} $$ and $\sum\nolimits_{\iota=1}^{N+1}{{r}_{1\iota}}{{s}_{\iota1}}=1$. The second focus is on explicit construction of soliton solutions to Eq. (1) based on the Riemann–Hilbert problem presented above. We now suppose the Riemann–Hilbert problem (13) to be irregular, which reveals that both $\det {{\it \Psi}_{1}}$ and $\det {{\it \Psi}_{2}}$ are in possession of some zeros in their own analytic domains. According to the definitions of ${{\it \Psi}_{1}}$ and ${{\it \Psi}_{2}}$, as well as the scattering relation (7), we have $$ \det {{\it \Psi}_{1}}(\lambda)={{s}_{11}}(\lambda),~\det {{\it \Psi}_{2}}(\lambda)={{r}_{11}}(\lambda). $$ For further analysis, we now specify the zeros. The potential matrix $Q$ is found to be anti-Hermitian, that is, $Q^{† }=-Q$. Using this property, we arrive at $$ \eta_{\pm }^{† }({{\lambda}^{*}})=\eta_{\pm }^{-1}(\lambda).~~ \tag {14} $$ It is rather convenient to introduce the following matrices $$ {A_{1}}={\rm diag}(1,\underbrace{0,0,\cdots,0}_{N}),~ {A_{2}}={\rm diag}(0,\underbrace{1,1,\cdots,1}_{N}), $$ to rewrite Eqs. (8) and (11) as $$ {{\it \Psi}_{1}}={{\eta }_{-}}{A_{1}}+{{\eta }_{+}}{A_{2}},~ {{\it \Psi}_{2}}={A_{1}}\eta _{-}^{-1}+{A_{2}}\eta _{+}^{-1}.~~ \tag {15} $$ Taking the Hermitian of the first equation of (15) and using Eq. (14), we can find $$ {\it \Psi}_{1}^{† }({{\lambda}^{*}})={{\it \Psi}_{2}}(\lambda),\quad {{S}^{† }}({{\lambda}^{*}})={{S}^{-1}}(\lambda),~~ \tag {16} $$ for $\lambda \in {\mathbb{C}^{-}}$. From the second equation of (16), we further have $s_{11}^{*}({{\lambda}^{*}})={{r}_{11}}(\lambda)$. Therefore, we assume that in the generic case, $\det {{\it \Psi}_{1}}$ has $n$ simple zeros $\{{{\lambda}_{l}}\}_{l=1}^{n}$ in ${\mathbb{C}^{+}}$ and $\det {{\it \Psi}_{2}}$ has $n$ simple zeros $\{{{\hat{\lambda}}_{l}}\}_{l=1}^{n}$ in ${\mathbb{C}^{-}}$, where ${{\hat{\lambda}}_{l}}=\lambda_{l}^{*}$. Each of $\ker {{\it \Psi}_{1}}({{\lambda }_{l}})$ includes only a single basis column vector ${{\omega}_{l}}$, and each of $\ker {{\it \Psi}_{2}}({{\hat{\lambda }}_{l}})$ includes only a single basis row vector ${{\hat{\omega}}_{l}}$, $$ {{\it \Psi}_{1}}({{\lambda}_{l}}){{\omega}_{l}}=0,~{{\hat{\omega}}_{l}}{{\it \Psi}_{2}}({{\hat{\lambda}}_{l}})=0.~~ \tag {17} $$ Taking the Hermitian of the first equation of (17) and using (16), we find $$ {{\hat{\omega}}_{l}}=\omega_{l}^{† },~1\le l\le n.~~ \tag {18} $$ By differentiation of the first equation of (17) in $x$ and $t$ separately and using Eqs. (3a) and (3b), we derive $$\begin{align} {{\it \Psi}_{1}}({{\lambda }_{l}})\Big(\frac{\partial {{\omega}_{l}}}{\partial x}-i{{\lambda }_{l}}\sigma {{\omega}_{l}}\Big)=\,&0,\\ {{\it \Psi}_{1}}({{\lambda }_{l}})\Big(\frac{\partial {{\omega}_{l}}}{\partial t}-(i\lambda _{l}^{2}-4i\epsilon \lambda _{l}^{3} )\sigma {{\omega }_{l}}\Big)=\,&0. \end{align} $$ By computing, we obtain $$ {{\omega}_{l}}={{{e}}^{i{{\lambda }_{l}}\sigma x+(i\lambda _{l}^{2}-4i\epsilon \lambda _{l}^{3})\sigma t}}{{\upsilon}_{l}},\quad 1\le l\le n,~~ \tag {19} $$ with ${{\upsilon}_{l}}$ being arbitrary constant vectors. In consideration of the relation (18), we have $$ {{\hat{\omega}}_{l}}=\upsilon_{l}^{† }{{{e}}^{-i\lambda _{l}^{*}\sigma x+(4i\epsilon \lambda {{_{l}^{*}}^{3}}-i\lambda {{_{l}^{*}}^{2}})\sigma t}},~ 1\le l\le n.~~ \tag {20} $$ The vectors (19) and (20) together with the zeros constitute the full set of the generic discrete data. To present soliton solutions explicitly, here we take $\mathcal G={I}_{N+1}$, which indicates that no reflection exists in the scattering problem. Thus the solutions[25] to the Riemann–Hilbert problem (13) can be given by $$\begin{align} {{\it \Psi}_{1}}(\lambda)=\,&{I}_{N+1}-\sum\limits_{k=1}^{n} {\sum\limits_{l=1}^{n}{\frac{{{\omega}_{k}}{{{\hat{\omega}}}_{l}}{{({{M}^{-1}})}_{kl}}}{\lambda -{{{\hat{\lambda}}}_{l}}}}},\\ {{\it \Psi}_{2}}(\lambda)=\,&{I}_{N+1}+\sum\limits_{k=1}^{n} {\sum\limits_{l=1}^{n}{\frac{{{\omega}_{k}}{{{\hat{\omega}}}_{l}}{{({{M}^{-1}})}_{kl}}}{\lambda-{{\lambda }_{k}}}}},~~ \tag {21} \end{align} $$ where $M$ is defined by $$ M=({{M}_{kl}})_{n\times n}=\Big(\frac{{\hat{\omega}_{k}}{{{{\omega}}}_{l}}} {{{\lambda}_{l}}-{{{\hat{\lambda}}}_{k}}}\Big)_{n\times n},~ 1\le k,l\le n,~~ \tag {22} $$ and ${{({{M}^{-1}})}_{kl}}$ represents the $(k,l)$-element of ${M}^{-1}$. Since the matrix function ${\it \Psi}_{1}$ meets Eq. (3a), we substitute $$ {{\it \Psi}_{1}}(\lambda)={I}_{N+1}+{{\lambda }^{-1}}{\it \Psi}_{1}^{(1)}+{{\lambda }^{-2}}{\it \Psi}_{1}^{(2)}+O({{\lambda }^{-3}}),\quad \lambda \to \infty, $$ into Eq. (3a) and generate $$\begin{align} &Q=-i[\sigma,{\it \Psi}_{1}^{(1)}]\\ =\,&\left(\begin{matrix} 0 \!\!&\!\! 2i{{({\it \Psi}_{1}^{(1)})_{12}}} \!\!&\!\! \cdots \!\!&\!\! 2i{{({\it \Psi}_{1}^{(1)})_{1,N+1}}} \\ -2i{{({\it \Psi}_{1}^{(1)})_{21}}} \!\!&\!\! 0 \!\!&\!\! \cdots \!\!&\!\! 0 \\ \vdots \!\!&\!\! \vdots \!\!&\!\! \ddots \!\!&\!\! \vdots \\ -2i{{({\it \Psi}_{1}^{(1)})_{N+1,1}}} \!\!&\!\! 0 \!\!&\!\! \cdots \!\!&\!\! 0 \\ \end{matrix}\right). \end{align} $$ Hence, the potential functions are reconstructed as $$ {{q}_{j}}=2i{{({\it \Psi}_{1}^{(1)})_{1,j+1}}},~ 1\le j\le N.~~ \tag {23} $$ Now it evidently follows from Eq. (21) that $$ {\it \Psi}_{1}^{(1)}=-\sum\limits_{k=1}^{n}{\sum\limits_{l=1}^{n} {{{\omega}_{k}}{{{\hat{\omega}}}_{l}}{{({{M}^{-1}})_{kl}}}}}. $$ Consequently, the general bright multi-soliton solutions to Eq. (1) can be gained as $$ {{q}_{j}}=-2i\sum\limits_{k=1}^{n}{\sum\limits_{l=1}^{n}{{{\omega}_{k,1}} {{{\hat{\omega}}}_{l,j+1}}{{({{M}^{-1}})}_{kl}}}},~ 1\le j\le N, $$ where $M$ is given by Eq. (22), and ${{\omega}_{k}}={{( {{\omega}_{k,1}},{{\omega}_{k,2}},\ldots,{{\omega}_{k,N+1}})^{\rm T}}}$, ${{\hat{\omega}}_{k}}\!\!=\!\!{{({{{\hat{\omega}}}_{k,1}},{{{\hat{\omega}}}_{k,2}},\ldots,{{{\hat{\omega}}}_{k,N+1}})}}$, $1\le k\le n$, are determined by Eqs. (19) and (20).
cpl-36-11-110201-fig1.png
Fig. 1. Profiles of $|{{q}_{1}}|$ with $\epsilon={{\alpha }_{1}}=1$, ${{\beta }_{1}}=1/2$, ${{\gamma }_{1}}=1/3$, $\zeta=0$, ${{\lambda }_{1}}=3i/10$, and ${{\lambda }_{2}}=i/5$: (a) three-dimensional plot, and (b) $x$-curves.
As a particular reduction, we now consider the case of $N=3$, which corresponds to the three-coupled Hirota equations $$\begin{align} {{q}_{jt}}=\,&i\Big[\frac{1}{2}{{q}_{jxx}} +\Big(\sum\limits_{r=1}^{3}{{{|{{q}_{r}}|}^{2}}}\Big) {{q}_{j}}\Big]+\epsilon \Big[{{q}_{jxxx}}\\ &+3\Big(\sum\limits_{r=1}^{3}{{{|{{q}_{r}}|}^{2}}}\Big){{q}_{jx}} +3\Big(\sum\limits_{r=1}^{3}{q_{r}^{*}{{q}_{rx}}}\Big){{q}_{j}}\Big],~ j=1,2,3.~~ \tag {24} \end{align} $$
cpl-36-11-110201-fig2.png
Fig. 2. Profiles of $|{{q}_{2}}|$ with $\epsilon={{\alpha }_{1}}=1$, ${{\beta }_{1}}=1/2$, ${{\gamma }_{1}}=1/3$, $\zeta=0$, ${{\lambda }_{1}}=3i/10$, and ${{\lambda }_{2}}=i/5$: (a) three-dimensional plot, and (b) $x$-curves.
The formulas established above can be employed to write out soliton solutions to Eq. (24) explicitly. Particularly, if letting ${\upsilon}_{1}={{( {{\alpha}_{1}},{{\beta}_{1}},{{\gamma}_{1}},{{\delta}_{1}})^{\rm T}}}$ and ${\upsilon}_{2}={{( {{\alpha}_{2}},{{\beta}_{2}},{{\gamma}_{2}},{{\delta}_{2}})^{\rm T}}}$, then a set of two-soliton solutions is obtained as $$\begin{align} {{q}_{1}}=\,&\frac{2i}{{{M}_{12}}{{M}_{21}}-{{M}_{11}}{{M}_{22}}}({{\alpha }_{1}}\beta _{1}^{*}{{e}^{\vartheta_{1}^{*}-{{\vartheta}_{1}}}}{{M}_{22}}\\ &-{{\alpha }_{1}}\beta _{2}^{*}{{e}^{\vartheta_{2}^{*}-{{\vartheta}_{1}}}}{{M}_{12}}-{{\alpha }_{2}}\beta _{1}^{*}{{e}^{\vartheta_{1}^{*}-{{\vartheta}_{2}}}}{{M}_{21}}\\ &+{{\alpha }_{2}}\beta _{2}^{*}{{e}^{\vartheta_{2}^{*}-{{\vartheta}_{2}}}}{{M}_{11}} ), \\ {{q}_{2}}=\,&\frac{2i}{{{M}_{12}}{{M}_{21}}-{{M}_{11}}{{M}_{22}}}({{\alpha }_{1}}\gamma _{1}^{*}{{e}^{\vartheta_{1}^{*}-{{\vartheta}_{1}}}}{{M}_{22}}\\ &-{{\alpha }_{1}}\gamma _{2}^{*}{{e}^{\vartheta_{2}^{*}-{{\vartheta}_{1}}}}{{M}_{12}}-{{\alpha }_{2}}\gamma _{1}^{*}{{e}^{\vartheta_{1}^{*}-{{\vartheta}_{2}}}}{{M}_{21}}\\ &+{{\alpha }_{2}}\gamma _{2}^{*}{{e}^{\vartheta_{2}^{*}-{{\vartheta}_{2}}}}{{M}_{11}} ), \\ {{q}_{3}}=\,&\frac{2i}{{{M}_{12}}{{M}_{21}}-{{M}_{11}}{{M}_{22}}}({{\alpha }_{1}}\delta _{1}^{*}{{e}^{\vartheta_{1}^{*}-{{\vartheta}_{1}}}}{{M}_{22}}\\ &-{{\alpha }_{1}}\delta _{2}^{*}{{e}^{\vartheta_{2}^{*}-{{\vartheta}_{1}}}}{{M}_{12}}-{{\alpha }_{2}}\delta _{1}^{*}{{e}^{\vartheta_{1}^{*}-{{\vartheta}_{2}}}}{{M}_{21}}\\ &+{{\alpha }_{2}}\delta _{2}^{*}{{e}^{\vartheta_{2}^{*}-{{\vartheta}_{2}}}}{{M}_{11}} ),~~ \tag {25} \end{align} $$ in which $$\begin{align} {{M}_{11}}=\,&\frac{1}{{{\lambda }_{1}}-\lambda _{1}^{*}}({{|{{\alpha }_{1}}|}^{2}}{{e}^{-\vartheta_{1}^{*}-{{\vartheta}_{1}}}}+{{|{{\beta }_{1}}|}^{2}}{{e}^{\vartheta_{1}^{*}+{{\vartheta}_{1}}}}\\ &+{{|{{\gamma }_{1}}|}^{2}}{{e}^{\vartheta_{1}^{*}+{{\vartheta }_{1}}}}+{{|{{\delta }_{1}}|}^{2}}{{e}^{\vartheta_{1}^{*}+{{\vartheta}_{1}}}} ), \\ {{M}_{12}}=\,&\frac{1}{{{\lambda }_{2}}-\lambda _{1}^{*}}(\alpha _{1}^{*}{{\alpha }_{2}}{{e}^{-\vartheta_{1}^{*}-{{\vartheta}_{2}}}}+\beta _{1}^{*}{{\beta }_{2}}{{e}^{\vartheta_{1}^{*}+{{\vartheta}_{2}}}}\\ &+\gamma _{1}^{*}{{\gamma }_{2}}{{e}^{\vartheta_{1}^{*}+{{\vartheta}_{2}}}}+\delta _{1}^{*}{{\delta }_{2}}{{e}^{\vartheta_{1}^{*}+{{\vartheta}_{2}}}} ), \\ {{M}_{21}}=\,&\frac{1}{{{\lambda }_{1}}-\lambda _{2}^{*}}({{\alpha }_{1}}\alpha _{2}^{*}{{e}^{-{{\vartheta}_{1}}-\vartheta_{2}^{*}}}+{{\beta }_{1}}\beta _{2}^{*}{{e}^{{{\vartheta}_{1}}+\vartheta_{2}^{*}}}\\ &+{{\gamma }_{1}}\gamma _{2}^{*}{{e}^{{{\vartheta}_{1}}+\vartheta_{2}^{*}}}+{{\delta }_{1}}\delta _{2}^{*}{{e}^{{{\vartheta}_{1}}+\vartheta_{2}^{*}}} ), \\ {{M}_{22}}=\,&\frac{1}{{{\lambda }_{2}}-\lambda _{2}^{*}}({{|{{\alpha }_{2}}|}^{2}}{{e}^{-\vartheta_{2}^{*}-{{\vartheta}_{2}}}}+{{|{{\beta }_{2}}|}^{2}}{{e}^{\vartheta_{2}^{*}+{{\vartheta}_{2}}}}\\ &+{{|{{\gamma }_{2}}|}^{2}}{{e}^{\vartheta_{2}^{*}+{{\vartheta }_{2}}}}+{{|{{\delta }_{2}}|}^{2}}{{e}^{\vartheta_{2}^{*}+{{\vartheta}_{2}}}} ), \end{align} $$ and ${{\vartheta}_{1}}=i{{\lambda }_{1}}x+(i\lambda _{1}^{2}-4i\epsilon \lambda _{1}^{3})t,{{\vartheta}_{2}}=i{{\lambda }_{2}}x+(i\lambda _{2}^{2}-4i\epsilon \lambda _{2}^{3})t$.
cpl-36-11-110201-fig3.png
Fig. 3. Profiles of $|{{q}_{3}}|$ with $\epsilon={{\alpha }_{1}}=1$, ${{\beta }_{1}}=1/2$, ${{\gamma }_{1}}=1/3$, $\zeta=0$, ${{\lambda }_{1}}=3i/10$, and ${{\lambda }_{2}}=i/5$: (a) three-dimensional plot, and (b) $x$-curves.
Furthermore, by taking ${{\alpha }_{1}}={{\alpha }_{2}}=1$ and ${{\beta}_{1}}={{\beta}_{2}},{{\gamma}_{1}}={{\gamma}_{2}},{{\delta}_{1}}={{\delta}_{2}}$ in Eq. (25), we obtain $$\begin{align} {{q}_{1}}=\,&\frac{2i}{{{M}_{12}}{{M}_{21}}-{{M}_{11}}{{M}_{22}}}(\beta _{1}^{*}{{e}^{\vartheta_{1}^{*}-{{\vartheta}_{1}}}}{{M}_{22}}\\ &-\beta _{2}^{*}{{e}^{\vartheta_{2}^{*}-{{\vartheta}_{1}}}}{{M}_{12}}-\beta _{1}^{*}{{e}^{\vartheta_{1}^{*}-{{\vartheta}_{2}}}}{{M}_{21}}\\ &+\beta _{2}^{*}{{e}^{\vartheta _{2}^{*}-{{\vartheta}_{2}}}}{{M}_{11}} ), \\ {{q}_{2}}=\,&\frac{2i}{{{M}_{12}}{{M}_{21}}-{{M}_{11}}{{M}_{22}}}(\gamma _{1}^{*}{{e}^{\vartheta_{1}^{*}-{{\vartheta}_{1}}}}{{M}_{22}}\\ &-\gamma _{2}^{*}{{e}^{\vartheta_{2}^{*}-{{\vartheta}_{1}}}}{{M}_{12}}-\gamma _{1}^{*}{{e}^{\vartheta_{1}^{*}-{{\vartheta}_{2}}}}{{M}_{21}}\\ &+\gamma _{2}^{*}{{e}^{\vartheta _{2}^{*}-{{\vartheta}_{2}}}}{{M}_{11}}), \\ {{q}_{3}}=\,&\frac{2i}{{{M}_{12}}{{M}_{21}}-{{M}_{11}}{{M}_{22}}}(\delta _{1}^{*}{{e}^{\vartheta_{1}^{*}-{{\vartheta}_{1}}}}{{M}_{22}}\\ &-\delta _{2}^{*}{{e}^{\vartheta_{2}^{*}-{{\vartheta}_{1}}}}{{M}_{12}}-\delta _{1}^{*}{{e}^{\vartheta_{1}^{*}-{{\vartheta}_{2}}}}{{M}_{21}}\\ &+\delta _{2}^{*}{{e}^{\vartheta _{2}^{*}-{{\vartheta}_{2}}}}{{M}_{11}}),~~ \tag {26} \end{align} $$ with $$\begin{align} {{M}_{11}}=\,&\frac{2{{e}^{\zeta }}}{{{\lambda }_{1}}-\lambda _{1}^{*}}\cosh (\vartheta _{1}^{*}+{{\vartheta}_{1}}+\zeta), \end{align} $$ $$\begin{align} {{M}_{12}}=\,&\frac{2{{e}^{\zeta }}}{{{\lambda }_{2}}-\lambda _{1}^{*}}\cosh (\vartheta _{1}^{*}+{{\vartheta}_{2}}+\zeta), \\ {{M}_{21}}=\,&\frac{2{{e}^{\zeta }}}{{{\lambda }_{1}}-\lambda _{2}^{*}}\cosh ( {{\vartheta}_{1}}+\vartheta_{2}^{*}+\zeta),\\ {{M}_{22}}=\,&\frac{2{{e}^{\zeta }}}{{{\lambda }_{2}}-\lambda _{2}^{*}}\cosh (\vartheta_{2}^{*}+{{\vartheta}_{2}}+\zeta), \end{align} $$ where we have set ${{|{{\beta }_{1}}|}^{2}}+{{|{{\gamma }_{1}}|}^{2}}+{{|{{\delta }_{1}}|}^{2}}={{e}^{2\zeta }}$. By selecting appropriate values for the parameters, some three-dimensional plots and $x$-curves of solutions (26) are presented in Figs. 1–3. In summary, the aim of the present research is to seek bright multi-soliton solutions for the $N$-coupled Hirota equations in an optical fiber. To this end, we first perform the analysis on the given spectral problem and establish a matrix Riemann–Hilbert problem on the real axis. Second, based on the obtained Riemann–Hilbert problem treated by considering that no reflection exists in the scattering problem, the general bright multi-soliton solutions to the $N$-coupled Hirota equations are constructed explicitly. In addition, as a by-product, the three-coupled Hirota equations together with the bright two-soliton solutions are written out. Future research should be undertaken to investigate other multiple coupled equations using the Riemann–Hilbert method. References New families of exact soliton-like solutions and dromion solutions to the (2+1)-dimensional higher-order Broer–Kaup equationsDarboux transformation and soliton-like solutions of nonlinear Schrödinger equationsTwo new integrable fourth-order nonlinear equations: multiple soliton solutions and multiple complex soliton solutionsDarboux Transformations and N -soliton Solutions of Two (2+1)-Dimensional Nonlinear EquationsNonlocal symmetry and exact solutions of the (2+1)-dimensional modified Bogoyavlenskii–Schiff equationSoliton–cnoidal interactional wave solutions for the reduced Maxwell–Bloch equationsOn soliton solutions of the relativistic Lotka–Volterra hierarchyLong time behavior and soliton solution for the Harry Dym equationDark Sharma–Tasso–Olver Equations and Their Recursion OperatorsDarboux Transformations, Higher-Order Rational Solitons and Rogue Wave Solutions for a (2 + 1)-Dimensional Nonlinear Schrödinger EquationPeakon Solutions of Alice-Bob b -Family Equation and Novikov EquationNonlinear-Evolution Equations of Physical SignificanceEvidence for an ionic model of 3 d impurities in metalsIntegrable properties of the general coupled nonlinear Schrödinger equationsRiemann-Hilbert approach and N-soliton formula for coupled derivative Schrödinger equationRiemann-Hilbert method and N -soliton for two-component Gerdjikov-Ivanov equationIntegrability aspects and multi-soliton solutions of a new coupled Gerdjikov–Ivanov derivative nonlinear Schrödinger equationInverse scattering transform and soliton classification of the coupled modified Korteweg-de Vries equationLong-time asymptotics and the bright N -soliton solutions of the Kundu–Eckhaus equation via the Riemann–Hilbert approachRiemann–Hilbert problems of a six-component fourth-order AKNS system and its soliton solutionsRiemann–Hilbert approach and N -soliton solutions for the Chen–Lee–Liu equationExact soliton solutions for a family of N coupled nonlinear Schrödinger equations in optical fiber mediaMixed-type soliton solutions for the N -coupled Hirota system in an optical fiber
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