⇦Chinese Physics Letters, 2021, Vol. 38, No. 9, Article code 090201 Inelastic Interaction of Double-Valley Dark Solitons for the Hirota Equation Xiao-Man Zhang (张小曼)1, Yan-Hong Qin (秦艳红)1, Li-Ming Ling (凌黎明)1*, and Li-Chen Zhao (赵立臣)2,3,4* Affiliations 1School of Mathematics, South China University of Technology, Guangzhou 510640, China 2School of Physics, Northwest University, Xi'an 710127, China 3Shaanxi Key Laboratory for Theoretical Physics Frontiers, Xi'an 710127, China 4Peng Huanwu Center for Fundamental Theory, Xi'an 710127, China Received 8 July 2021; accepted 8 August 2021; published online 2 September 2021 Supported by the National Natural Science Foundation of China (Grant Nos. 11771151, 12022513, and 11775176), the Guangzhou Science and Technology Program (Grant No. 201904010362), the Fundamental Research Funds for Central Universities (Grant No. 2019MS110), and the Major Basic Research Program of Natural Science of Shaanxi Province (Grant No. 2018KJXX-094).
*Corresponding authors. Email: linglm@scut.edu.cn; zhaolichen3@nwu.edu.cn Citation Text: Zhang X M, Qin Y H, Ling L M, and Zhao L C 2021 Chin. Phys. Lett. 38 090201    Abstract For a scalar integrable model, it is generally believed that the solitons interact with each other elastically, for instance, multi-bright solitons from the nonlinear Schrödinger equation and the Korteweg-de Vries equation, etc. We obtain double-valley dark solitons from the defocusing Hirota equation by the Darboux transformation. Particularly, we report a remarkable phenomenon for the inelastic interaction of the double-valley dark solitons, in contrast to the solitons interacting with each other elastically for a scalar integrable model in previous works. Furthermore, we give the explicit conditions for the elastic collision based on the asymptotic analysis results. It is shown that the double-valley dark solitons could also admit elastic interaction under the special parameters settings. DOI:10.1088/0256-307X/38/9/090201 © 2021 Chinese Physics Society Article Text It is well known that the nonlinear Schrödinger equation (NLSE) plays an important role in many fields of nonlinear science such as water waves,[1] plasma,[2] Bose–Einstein condensates,[3] and nonlinear optics.[4,5] Particularly, in optical fibers the NLSE can describe a picosecond optical pulse propagation.[4,6] However, for propagation of a subpicosecond or femtosecond pulse, it is necessary to involve higher-order nonlinear and dispersive effects like third-order dispersion (TOD), self-steepening (SS), and intrapulse Raman scattering. When the TOD and SS are taken into account, the ultrashort light pulses in optical fibers can be governed by the integrable Hirota equation (HE):[7–10] $$ {i}q_t+\frac{1}{2}q_{xx}-\sigma |q|^2 q+{i}\alpha(-q_{xxx}+6\sigma |q |^2 q_{x})=0,~~ \tag {1} $$ where $\sigma=\pm 1$ means the focusing or defocusing type, and $\alpha $ denotes the coefficients of high order effects. When $\alpha=0 $, Eq. (1) can be reduced to the well-known NLSE. Due to the integrability for the HE, many kinds of exact localized waves solutions, such as the multisolitons, rogue waves and breathers, were obtained by distinct methods.[11–18] For the defocusing case, generally, it admits single-valley dark solitons (SVDSs).[17,19–21] Very recently, multivalley dark soliton solutions were obtained in multicomponent Manakov systems.[22] To the best of our knowledge, double-valley dark solitons (DVDSs) of the HE have never been reported before. Actually the DVDS here is closely related to the soliton molecules,[23–25] especially the dark soliton molecules derived by Lou et al.[26] The details of their relationship will be disclosed in the following main text. What is more, we find that the interaction between SVDS and DVDS or between DVDSs is inelastic generally. We know that there have already been works showing the inelastic interaction in the coupled nonlinear equation,[27–31] but few works reported the inelastic collision in the scalar case. Recently, the inelastic interaction phenomena of solitons in the scalar Sasa–Satsuma equation corresponding to a $3\times3 $ spectral problem has been reported in some works.[32,33] As far as we know, the inelastic interaction of double-valley dark solitons is revealed in the integrable systems with a $2\times 2$ spectral problem for the first time. In this Letter, we consider the defocusing HE (1) with $\sigma=1$. Compared with the soliton in the NLSE,[34,35] the soliton in the HE has the different velocity due to the high order dispersion and time-delay corrections, and its integrable properties and solutions can be studied by the same way as treating the NLSE. Darboux transformation (DT) is an effective and powerful technique to formulate multi-soliton solutions for the nonlinear integrable equation.[36–40] We would like to utilize the DT method[41] to derive the multi-dark soliton solution, which can be reduced to the DVDS solution to Eq. (1) under the special parameter setting. The DVDS solution can be classified into the asymmetric and symmetric ones. By the complicated asymptotic analysis, we show the inelastic collision between two DVDSs or between one DVDS and one SVDS. Meanwhile, we obtain the special conditions to keep the elastic interaction. Starting from the seed solution $q^{[0]}=c\,{e}^{{i}\theta}, \,\, \theta=ax-bt,$ where $b=\alpha(a^2+6c^2)a+\frac{1}{2}a^2+c^2 $, the $n$-dark soliton solutions on the plane wave background can be derived by applying the $n$-fold DT.[41] Letting the spectral parameter $\lambda_i=-\frac{a}{2}+c\cos{z_i},\,z_i\in(0,\pi),\,\,i=1,\ldots,n, $ the $n$-dark soliton solution can be expressed as $$\begin{alignat}{1} &q^{[n]}=c\frac{\det({\boldsymbol M}_1)}{\det({\boldsymbol M})}{e}^{{i}\theta},\\ &{\boldsymbol M}=\Big(\frac{{e}^{{i}(X_j-\bar{X}_m)}+d\delta_{[m,j]}}{{e}^{{i}z_j}-{e}^{-{i}z_m}}\Big)_{1\leq m,j\leq n}, \\ &{\boldsymbol M}_1=\Big(\frac{{e}^{{i}(X_j+z_j-\bar{X}_m+z_m)}+d\delta_{[m,j]}}{{e}^{{i}z_j}-{e}^{-{i}z_m}}\Big)_{1\leq m,j\leq n},~~ \tag {2} \end{alignat} $$ with $$\begin{aligned} &X_j ={i}c\sin(z_j)(x-v_j t), \\ &\delta_{[m,j]} =\begin{cases} 0 ,& m\neq j, \\ {e}^{2c\sin(z_m)\gamma_m} ,& m=j, \end{cases}\,\,\,\,j=1,2,\ldots,n, \end{aligned} $$ where $\gamma_m$ and $d $ are real parameters related to the position of solitons, the velocity of solitons $v_j$ is governed by $$\begin{align} v_j=\,&\alpha[4 c^2 {\cos}(z_j)^2-6ac{\cos}(z_j)+3a^2+2c^2]\\ &-c\cos(z_j)+a. \end{align} $$ Now we proceed to study the DVDS, which can be obtained by Eq. (2) under the setting $n=2 $ with the same velocity of two incoherent solitons. Due to $v_1=v_2\equiv v $, i.e., $$ \cos(z_2)=-\cos(z_1)+3a/(2c)+1/(4c\,\alpha), $$ the range of $z_1$ belongs to $(0,\pi)\cap(\arccos[-1+3a/(2c)+1/(4c\,\alpha)], \arccos[1+3a/(2c)+1/(4c\,\alpha)])$. Then the expression of the DVDS can be simplified to $$ q^{[2]}=c\frac{N}{D}{e}^{{i}\theta},~~ \tag {3} $$ where $$\begin{alignat}{1} N={}&\omega_{[1,2]}^2{e}^{-\eta_1-\eta_2}+d^2{e}^{\eta_1+\eta_2}+2d\cosh(\eta_1-\eta_2),\\ D={}&\omega_{[1,2]}^2{e}^{-\xi_1-\xi_2}+d^2{e}^{\xi_1+\xi_2}+2d\cosh(\xi_1-\xi_2),~~ \tag {4} \end{alignat} $$ and $$\begin{align} &\omega_{[1,2]}(z_1,z_2)=\frac{\sin[\frac{1}{2}(z_1-z_2)]}{\sin[\frac{1}{2}(z_1+z_2)]},\\ &\xi_{i}=c\sin(z_i)(x-vt+\gamma_i),~~\eta_i=\xi_i+{i} z_i. \end{align} $$ Here the parameters $\gamma_1,\gamma_2 $ should be close enough to make the two valleys of the DVDS not completely separate with each other. If $|\gamma_1-\gamma_2|$ is large, the two valleys of the DVDS will be far from each other, in which we regard them as two SDVSs or the dark soliton molecule[26] rather than a DVDS. Furthermore, if we choose $d=\omega_{[1,2]} $ in the solution (3), then the formula (3) will be consistent with the dark soliton molecules derived by the bilinear method in Ref. [26]. Our choice $d=1 $ can ensure the denominator to be positive definite and to avoid the singularity. The DVDS solution can admit the asymmetric one and symmetric one. By the tedious calculations, we obtain the condition $S(\gamma_1,\gamma_2)=0$, where $$\begin{alignat}{1} S(\gamma_1,\gamma_2)\equiv \gamma_1-\gamma_2+\frac{\sin(z_1)-\sin(z_2)}{2c\sin(z_1)\sin(z_2)}\ln|\omega_{[1,2]}|, ~~~~~ \tag {5} \end{alignat} $$ which can ensure the soliton to be a symmetric one and the corresponding symmetric axis to be $x_1=v t-\gamma_1+\frac{\ln|\omega_{[1,2]}|}{2c\sin(z_1)}$. We show an example of asymmetric DVDS in Fig. 1(a) (blue solid line), and by choosing proper phase parameters $\gamma_1 $ and $\gamma_2 $ which satisfy Eq. (5), we can obtain the symmetric DVDS in Fig. 1(b) (blue dashed line).
cpl-38-9-090201-fig1.png
Fig. 1. The density profiles of DVDSs. (a) An asymmetric DVDS when $t=0 $, with $\gamma_1=0.5$ and $\gamma_2=0.6124$. (b) A symmetric DVDS when $t=0 $, with $\gamma_1=0.625$ and $\gamma_2=0.6577$. The other parameters are $a=1$, $c=5$, $\alpha=0.1$, $z_1=\arccos(3/5)$, $z_2=\arccos(1/5)$.
cpl-38-9-090201-fig2.png
Fig. 2. (a) The collision dynamics of an SDVS and a DVDS. (b) The density profile of $q^{[3]} $ when $t=8$. (c) The density profile of $q^{[3]} $ when $t=-16$. The parameters are chosen as $a=1$, $c=2$, $\alpha=-1/2 $, $z_1=\pi/2$, $z_2=\pi/3$, $z_3=1.5$, $\gamma_1=1$, $\gamma_2=0.949$, $\gamma_3=-0.5$.
Next we would like to investigate the interaction between the DVDSs. We consider the following two cases: (1) a collision between a DVDS and an SVDS, (2) a collision between two DVDSs. For the first case, we implement a three-fold DT with the spectral parameters $\lambda_1=-\frac{a}{2}+c\cos{z_1}$, $\lambda_2=-\frac{a}{2}+c\cos{z_2} $ (generating a DVDS), and $\lambda_3=-\frac{a}{2}+c\cos{z_3} $ (generating an SVDS). We choose a symmetric soliton as the DVDS $(q_{_{\scriptstyle S_1}}) $. Figure 2(a) shows the collision between a DVDS $(q_{_{\scriptstyle S_1}}) $ and an SDVS $(q_{_{\scriptstyle S_2}}) $. To show the collision clearly, we use the coordinate transformation $X=x-\frac{v_1+v_3}{2}t$, where $v_1$ and $v_3$ correspond to the velocities of DVDS $(q_{_{\scriptstyle S_1}}) $ and SDVS $(q_{_{\scriptstyle S_2}}) $, respectively. By plotting the density profile of the $3$-dark soliton $q^{[3]}$ before ($t=-16 $) and after the collision ($t=8 $) [Figs. 2(b) and 2(c)], we can find that the symmetric DVDS is not symmetric after the collision, but the SDVS does not change its shape. Thus the collision is inelastic for the DVDS, which has not been observed before in the scalar integrable system.
cpl-38-9-090201-fig3.png
Fig. 3. (a) The collision dynamics of two DVDSs. (b) The density profile of $q^{[4]}$ when $t=3$. (c) The density profile of $q^{[4]}$ when $t=-3$. Here $z_3=\arccos(7/10)$, $z_4=\arccos(1/10)$, $\gamma_3=0.8402$, $\gamma_4=0.3015$, and the other parameters are the same as those in Fig. 1(b).
Similarly, the collision between two DVDSs can be obtained by implementing a four-fold DT with the spectral parameters $\lambda_1=-\frac{a}{2}+c\cos{z_1}$, $\lambda_2=-\frac{a}{2}+c\cos{z_2} $ (generating a DVDS), and $\lambda_3=-\frac{a}{2}+c\cos{z_3}$, $\lambda_4=-\frac{a}{2}+c\cos{z_4} $ (generating another DVDS). We give an example in Fig. 3, it is shown that the symmetric DVDS $(q_{_{\scriptstyle S_1}}) $ is not symmetric after the collision, and the DVDS $(q_{_{\scriptstyle S_2}}) $ also changes its shape. To analyze the mechanism of inelastic interaction deeply, we will give the asymptotic analysis for the exact soliton solutions $q^{[3]}$ and $q^{[4]}$. For the $q^{[3]}$, which is involved of a DVDS $(q_{_{\scriptstyle S_1}}) $ and an SDVS $(q_{_{\scriptstyle S_2}}) $, we take $0 < v_1=v_2 < v_3$. Then, when $t\rightarrow \pm\infty $, the DVDS solution has the following asymptotic form: $$ q_{_{\scriptstyle S_1}}^\pm=c\frac{N_1^\pm}{D_1^\pm}{e}^{{i}\theta},~~ \tag {6} $$ where $$\begin{align} N_1^\pm=\,&F^\pm {e}^{-\eta_1-\eta_{2}}+G^\pm{e}^{\eta_{1}-\eta_2}+H^\pm{e}^{\eta_{2}-\eta_{1}} \\ &+{e}^{\eta_1+\eta_{2}},\\ D_1^\pm=\,&F^\pm{e}^{-\xi_1-\xi_{2}}+G^\pm{e}^{\xi_{1}-\xi_2}+H^\pm{e}^{\xi_{2}-\xi_{1}} \\ &+{e}^{\xi_1+\xi_{2}}, \end{align} $$ and $$\begin{align} &\omega_{[i,j]}=\frac{\sin[\frac{1}{2}(z_i-z_j)]}{\sin[\frac{1}{2}(z_i+z_j)]},\\ &F^-=\omega^2_{[1,2]},\,\,G^-=1,\,\,\,\ H^-=1,\\ &F^+=F^-G^+H^+,\ \,\,G^+=\omega^2_{[2,3]},\ \,\,H^+=\omega^2_{[1,3]}. \end{align} $$ Similarly, the asymptotic expression of the SVDS solution reads $$ q_{_{\scriptstyle S_2}}^\pm=c[1-B_1+B_1\tanh(Y_1)]{e}^{{i}\theta},\,\, t\to\pm\infty,~~ \tag {7} $$ where $$\begin{align} &B_1 ={i}{e}^{-{i} z_3}\sin{z_3},\\ &Y_1 =c\sin{z_3}(x-v_3t+\gamma_3)-\ln(K^\pm),\\ &K^-=\prod_{i=1}^{2}|\omega_{[i,3]}|,\,\,K^+=1. \end{align} $$ It can be easily checked that $$ \Big|q_{_{\scriptstyle S_2}}^-\Big[x+\frac{\ln(K^-)}{c\sin(z_3)}\Big]\Big|= \Big|q_{_{\scriptstyle S_2}}^+\Big[x+\frac{\ln(K^+)}{c\sin(z_3)}\Big]\Big|, $$ so the SDVS $(q_{_{\scriptstyle S_2}}) $ recovers its shape after the collision with a phase shift. However, by analyzing the expression (6) we can see that the collision may not be elastic for the DVDS $(q_{_{\scriptstyle S_1}}) $ in most cases, and one of the conditions which make the collision be elastic is given by $$ \frac{\ln(G^+)}{\ln(H^+)}=\frac{\sin(z_{2})}{\sin(z_{1})}.~~ \tag {8} $$ Therefore, when the parameters $z_i\,(i=1,2,3)$ satisfy Eq. (8), the DVDS and the SDVS would admit elastic collision. Fixing the DVDS with parameters $z_1,z_2 $, we can adjust $z_3$ to satisfy the condition (8). Thus by setting the parameter $z_3=1.339$ and the other parameters as the same as the inelastic collision in Fig. 2, the new solution $q^{[3]}$ admits elastic collision which has been shown in Figs. 4(a)–4(c).
cpl-38-9-090201-fig4.png
Fig. 4. (a) The elastic collision dynamics of an SDVS and a DVDS. (b) The density profile of $q^{[3]} $ when $t=6$. (c) The density profile of $q^{[3]} $ when $t=-10$. Here $z_3=1.339$, and the other parameters are the same as those in Fig. 2.
Next, we study the $4$-dark soliton solution $q^{[4]}$ which involves two DVDSs $(q_{_{\scriptstyle S_1}},q_{_{\scriptstyle S_2}})$. Suppose that the velocity of soliton is $0 < v_1=v_2 < v_3=v_4$. Then, when $t\rightarrow \pm\infty$, for the $q_{_{\scriptstyle S_k}} $ we have $$ q_{_{\scriptstyle S_k}}^\pm=c\frac{N_k^\pm}{D_k^\pm}{e}^{{i}\theta},\,\,k=1,2,~~ \tag {9} $$ with $$\begin{align} N_k^\pm=\,&F_k^\pm {e}^{-\eta_{2k-1}-\eta_{2k}}+G^\pm_k{e}^{\eta_{2k-1}-\eta_{2k}}\\ &+H^\pm_k{e}^{\eta_{2k}-\eta_{2k-1}}+{e}^{\eta_{2k-1}+\eta_{2k}},\\ D_k^\pm=\,&F_k^\pm{e}^{-\xi_{2k-1}-\xi_{2k}}+G^\pm_k{e}^{\xi_{2k-1}-\xi_{2k}} \\ &+H^\pm_k{e}^{\xi_{2k}-\xi_{2k-1}}+{e}^{\xi_{2k-1}+\xi_{2k}}, \end{align} $$ and $$\begin{align} &F_k^-=\prod_{j=2k-1}^{2k}\prod_{i=1}^{j-1}\omega^2_{[i,j]},\ \,\, F_k^+=\prod_{j=2k-1}^{2k}\prod_{i=j+1}^{4}\omega^2_{[i,j]},\\ &G^-_k=\prod_{i=1}^{2k-2}\omega^2_{[i,2k]},\ \ \,\,G^+_k=\prod_{i=2k+1}^{4}\omega^2_{[i,2k]},\\ &H^-_k=\prod_{i=1}^{2k-2}\omega^2_{[i,2k-1]},\ \,\,\,H^+_k=\prod_{i=2k+1}^{4}\omega^2_{[i,2k-1]}. \end{align} $$ From the asymptotic expression (9) we can also derive the elastic condition for the DVDS $q_{_{\scriptstyle S_k}} $: $$ \frac{\ln(G_k^+)-\ln(G_k^-)}{\ln(H_k^+)-\ln(H_k^-)}=\frac{\sin(z_{2k})}{\sin(z_{2k-1})}.~~ \tag {10} $$ Therefore, when the parameters $z_i\,(i=1,2,3,4) $ satisfy Eq. (10) with $k=1,2$, both the DVDSs would admit elastic collision. However, both the conditions are not easy to satisfy at the same time, and we would like to explore them in the future. To give an example, we fix the DVDS $(q_{_{\scriptstyle S_1}}) $ with parameters $z_1,z_2 $, and adjust the rest parameters $z_3,z_4 $ to satisfy the conditions (8) if only $k=1 $, thus the collision will be elastic for $q_{_{\scriptstyle S_1}} $. As shown in Fig. 5, we adjust the parameters $z_3,z_4 $ in Fig. 3 to satisfy the condition (8) approximately, then the symmetric DVDS $(q_{_{\scriptstyle S_1}}) $ still obtains its symmetry after the collision ($t=2$), thus the collision becomes elastic for the $q_{_{\scriptstyle S_1}} $ now.
cpl-38-9-090201-fig5.png
Fig. 5. (a) The elastic collision dynamics of two DVDSs. (b) The density profile of $q^{[4]}$ when $t=2$. (c) The density profile of $q^{[4]}$ when $t=-2$. Here $z_3=\arccos(109/125)$, $z_4=\arccos(-9/125)$, and the other parameters are the same as those in Fig. 3.
Actually due to the specificity of symmetric DVDS, we can calculate the distance it moves as the collision is elastic for it. Taking the collision in Fig. 5 as an example, from Eq. (9) we can obtain $$ |q_{_{\scriptstyle S_1}}^+(x;\gamma_1,\gamma_{2})|^2=|q_{_{\scriptstyle S_1}}^-(x;\hat{\gamma}_1,\hat{\gamma}_{2})|^2, $$ where $$\begin{aligned} &\hat{\gamma}_1=\gamma_1+\frac{\ln(H_1^+)}{2c\sin(z_1)},~\,\,\hat{\gamma}_{2}=\gamma_{2}+\frac{\ln(G_1^+)}{2c\sin(z_{2})}. \end{aligned}~~ \tag {11} $$ Thus $q_{_{\scriptstyle S_1}} $ would keep the symmetry after interaction when the new phase parameters $\hat{\gamma}_{2} $ and $\hat{\gamma}_{1} $ also satisfy the symmetric condition $S(\hat{\gamma}_1,\hat{\gamma}_2)=0$. Substituting Eq. (11) into $S(\hat{\gamma}_1,\hat{\gamma}_2)=0$ and comparing it with Eq. (5), it exactly turns out to be the elastic condition (10), which means that the collision is elastic for the symmetric DVDS when it keeps symmetric after the interaction. The symmetric axis has also changed after the collision, which turns to $x_2=vt-\hat{\gamma}_1+\frac{\ln|\omega_{[1,2]}|}{2c\sin(z_1)}$. Thus, the symmetric soliton does not change its shape but only moves along $x$-axis by $c=x_2-x_1=v(t^+-t^-)+\frac{\ln(H_1^+)}{2c\sin(z_1)}$. Thus the symmetric DVDS $(q_{_{\scriptstyle S_1}}) $ in Fig. 5 moves along the $X$-axis by $c=X_2-X_1=\frac{v_1-v_3}{2}(t^+-t^-)+\frac{\ln(H_1^+)}{2c\sin(z_1)} \approx-4.1979$. By moving the $X$-axis of the density profile after the collision by $c$, we can find the figure of the density profile of $q_{_{\scriptstyle S_1}} $ before the collision [green solid line in Fig. 6(b)] overlap the figure after the collision [red dashed line in Fig. 6(b)] very well. We also move the $X$-axis in Fig. 3 and show the inelastic collision for $q_{_{\scriptstyle S_1}}$ [Fig. 6(a)] to pose a contrast with the elastic condition in Fig. 6(b).
cpl-38-9-090201-fig6.png
Fig. 6. (a) Density profile of $q_{_{\scriptstyle S_1}} $ in Fig. 3 when $t=-3 $ (green solid line) and $t=3 $ (red dashed line), for the inelastic collision. (b) Density profile of $q_{_{\scriptstyle S_1}} $ in Fig. 5 when $t=-2 $ (green solid line) and $t=2 $ (red dashed line), for the elastic collision.
In summary, we have given the DVDS solution for the scalar defocusing HE (1) using the DT and reported the inelastic collision between the DVDSs. The spectral parameter conditions for elastic collision have been discussed in detail by performing the asymptotic analysis. These results will deepen and enrich our understanding of the dark solitons of the HE. Furthermore, the dark soliton and soliton molecules have been demonstrated experimentally in optical fibers[42–44] and especially in femtosecond pulse.[45–50] We expect that the inelastic phenomenon obtained in this study will be verified and observed in the physical experiments in future. References The Propagation of Nonlinear Wave EnvelopesNonlinear aspects of quantum plasma physicsTheory of Bose-Einstein condensation in trapped gasesExact envelope‐soliton solutions of a nonlinear wave equationSoliton solutions to coupled higher‐order nonlinear Schrödinger equationsNonlinear pulse propagation in a monomode dielectric guideCompositionally modulated phase in crystalline n -alkane mixturesConstruction of Multi-soliton Solutions of the N -Coupled Hirota Equations in an Optical Fiber *Rogue waves and rational solutions of the Hirota equationMultisolitons, breathers, and rogue waves for the Hirota equation generated by the Darboux transformationDarboux transformation of the general Hirota equation: multisoliton solutions, breather solutions, and rogue wave solutionsThe Hirota equation: Darboux transform of the Riemann–Hilbert problem and higher-order rogue wavesMoving breathers and breather-to-soliton conversions for the Hirota equationDark soliton solutions of the defocusing Hirota equation by the binary Darboux transformationAsymptotic analysis of high-order solitons for the Hirota equationDark soliton solutions of the coupled Hirota equation in nonlinear fiberPropagation of dark solitons with higher-order effects in optical fibersA Direct Derivation of the Dark Soliton Excitation EnergyMultivalley dark solitons in multicomponent Bose-Einstein condensates with repulsive interactionsPolarized dark solitons in isotropic Kerr mediaSoliton molecules in dipolar Bose-Einstein condensatesSoliton molecules and asymmetric solitons in fluid systems via velocity resonanceDark soliton molecules in nonlinear opticsGeneral N -soliton solution to a vector nonlinear Schrödinger equationElastic and inelastic collisions of the semirational solutions for the coupled Hirota equations in a birefringent fiberNondegenerate Solitons in Manakov SystemNondegenerate solitons and their collisions in Manakov systemsNondegenerate soliton solutions in certain coupled nonlinear Schrödinger systemsCertain bright soliton interactions of the Sasa-Satsuma equation in a monomode optical fiberSoliton and breather solutions of the Sasa–Satsuma equation via the Darboux transformationBound-State Soliton Solutions of the Nonlinear Schrödinger Equation and Their Asymmetric DecompositionsSoliton Molecules and Some Hybrid Solutions for the Nonlinear Schrödinger Equation *Breather Interaction Properties Induced by Self-Steepening and Space-Time CorrectionSoliton, Breather and Rogue Wave Solutions for the Nonlinear Schrödinger Equation Coupled to a Multiple Self-Induced Transparency SystemDarboux transformation and multi-dark soliton for N -component nonlinear Schrödinger equationsDark-Pulse Propagation in Optical FibersGeneration of dark solitons in erbium-doped fiber lasers based Sb_2Te_3 saturable absorbersExperimental Observation of Temporal Soliton MoleculesGeneration of 33-fsec pulses at 132 μm through a high-order soliton effect in a single-mode optical fiberExperimental Observation of the Fundamental Dark Soliton in Optical FibersCoherent quantum control of two-photon transitions by a femtosecond laser pulseUltrafast photonics of two dimensional AuTe2Se4/3 in fiber lasersTungsten disulfide saturable absorbers for 67 fs mode-locked erbium-doped fiber lasersReal-time spectral interferometry probes the internal dynamics of femtosecond soliton molecules
[1] Benney D J and Newell A C 1967 J. Math. Phys. 46 133
[2] Shukla P K and Eliasson B 2010 Phys. -Usp. 53 51
[3] Dalfovo F, Giorgini S, Pitaevskii L P, and Stringari S 1999 Rev. Mod. Phys. 71 463
[4]Agrawal G P 2001 Nonlinear Fiber Optics 3rd edn (San Diego: Academic Press)
[5]Hasegawa A and Kodama Y 1995 Solitons in Optical Communications (Oxford: Clarendon Press)
[6]Kivshar Y S and Agrawal G P 2003 Optical Solitons: From Fibers to Photonic Crystals (San Diego: Academic Press)
[7] Hirota R 1973 J. Math. Phys. 14 805
[8] Tasgal R S and Potasek M J 1992 J. Math. Phys. 33 1208
[9] Kodama Y and Hasegawa A 1987 IEEE J. Quantum Electron. 23 510
[10] Sirota E B 2001 Phys. Rev. E 64 050701
[11] Kang Z Z and Xia T C 2019 Chin. Phys. Lett. 36 110201
[12] Ankiewicz A, Soto-Crespo J M, and Akhmediev N 2010 Phys. Rev. E 81 046602
[13] Tao Y S and He J S 2012 Phys. Rev. E 85 026601
[14] Wang D S, Chen F, and Wen X Y 2016 Adv. Differ. Eq. 2016 67
[15] Chen S Y and Yan Z Y 2019 Appl. Math. Lett. 95 65
[16] Chowdury A, Ankiewicz A, and Akhmediev N 2015 Proc. R. Soc. A 471 20150130
[17] Zhang H Q and Yuan S S 2017 Nonlinear Dyn. 89 531
[18] Zhang X E and Ling L M 2021 Physica D 426 132982
[19] Bindu S G, Mahalingam A, and Porsezian K 2001 Phys. Lett. A 286 321
[20] Mahalingam A and Porsezian K 2001 Phys. Rev. E 64 046608
[21] Zhao L C, Qin Y H, Wang W L, Yang Z Y 2020 Chin. Phys. Lett. 37 050502
[22] Qin Y H, Zhao L C, Yang Z Q, and Ling L M 2021 Phys. Rev. E 104 014201
[23] Sheppard A P and Kivshar Y S 1997 Phys. Rev. E 55 4773
[24] Lakomy K, Nath R, and Santos L 2012 Phys. Rev. A 86 013610
[25] Lou S Y 2019 arXiv:1909.03399v1 [nlin.SI]
[26] Xu D H and Lou S Y 2020 Acta Phys. Sin. 69 014208 (in Chinese)
[27] Feng B F 2014 J. Phys. A 47 355203
[28] Xie X Y and Liu X B 2020 Appl. Math. Lett. 105 106291
[29] Stalin S, Ramakrishnan R, Senthilvelan M, and Lakshmanan M 2019 Phys. Rev. Lett. 122 043901
[30] Ramakrishnan R, Stalin S, and Lakshmanan M 2020 Phys. Rev. E 102 042212
[31] Stalin S, Ramakrishnan R, and Lakshmanan M 2020 Phys. Lett. A 384 126201
[32] Liu L, Tian B, Chai H P, and Yuan Y Q 2017 Phys. Rev. E 95 032202
[33] Xu T, Wang D, Li M, and Liang H 2014 Phys. Scr. 89 075207
[34] Zhang Y S and He J S 2019 Chin. Phys. Lett. 36 030201
[35] Wang B, Zhang Z, and Li B 2020 Chin. Phys. Lett. 37 030501
[36]Rogers C and Schief W K 2002 Bäcklund and Darboux Transformations: Geometry and Modern Applications in Soliton Theory (Cambridge: Cambridge University Press)
[37]Matveev V and Salle M A 1991 Darboux Transformations and Solitons (Berlin: Springer)
[38]Gu C H, Hu H S, and Zhou Z X 2005 Darboux Transformations in Integrable Systems: Theory and Their Applications to Geometry (Dordrecht: Springer)
[39] Wu Y H, Liu C, Yang Z Y, and Yang W L 2020 Chin. Phys. Lett. 37 040501
[40] Wang X and Wang L 2018 Chin. Phys. Lett. 35 030201
[41] Ling L M, Zhao L C, and Guo B L 2015 Nonlinearity 28 3243
[42] Krökel D, Halas N, Giuliani G, and Grischkowsky D 1988 Phys. Rev. Lett. 60 29
[43] Liu W J et al. 2015 Opt. Express 23 26023
[44] Stratmann M, Pagel T, and Mitschke F 2005 Phys. Rev. Lett. 95 143902
[45] Gouveia-Neto A S, Gomes A S L, and Taylor J R 1987 Opt. Lett. 12 395
[46] Weiner A M et al. 1988 Phys. Rev. Lett. 61 2445
[47] Meshulach D and Silberberg Y 1998 Nature 396 239
[48] Liu W J et al. 2020 Commun. Phys. 3 15
[49] Liu W J et al. 2017 Opt. Express 25 2950
[50] Herink G et al. 2017 Science 356 50