⇦Chinese Physics Letters, 2017, Vol. 34, No. 9, Article code 090201 Soliton Solutions to the Coupled Gerdjikov–Ivanov Equation with Rogue-Wave-Like Phenomena * Jian-Bing Zhang(张建兵)1**, Ying-Yin Gongye(公冶映茵)1, Shou-Ting Chen(陈守婷)2 Affiliations 1School of Mathematics and Statistics, Jiangsu Normal University, Xuzhou 221116 2School of Mathematics and Physical Science, Xuzhou Institute of Technology, Xuzhou 221008 Received 18 April 2017 *Supported by the National Natural Science Foundation of China under Grant Nos 11671177 and 11271168, the Jiangsu Qing Lan Project (2014), and the Six Talent Peaks Project of Jiangsu Province under Grant No 2016-JY-08.
**Corresponding author. Email: jbzmath@jsnu.edu.cn Citation Text: Zhang J B, Gongye Y Y and Chen S T 2017 Chin. Phys. Lett. 34 090201 Abstract Bilinear forms of the coupled Gerdjikov–Ivanov equation are derived. The $N$-soliton solutions to the equation are obtained by Hirota's method. It is interesting that the two-soliton solutions can generate the rogue-wave-like phenomena by selecting special parameters. The equation can be reduced to the Gerdjikov–Ivanov equation as well as its bilinear forms and its solutions. DOI:10.1088/0256-307X/34/9/090201 PACS:02.30.Ik, 05.45.Yv © 2017 Chinese Physics Society Article Text It is well known that the nonlinear Schrödinger equation (NLSE) is one of the most generic soliton equations, and arises from a wide variety of fields, such as quantum field theory, weakly nonlinear dispersive water waves and nonlinear optics.[1-3] To study the effect of higher-order perturbations, various modifications and generations of the NLSE have been proposed and studied.[3-9] Among them, there are three celebrated equations with derivative-type nonlinearities which are called the DNLSE. One is the Kaup–Newell equation[4] $$ iq_t+q_{xx}+i(|q|^2q)_x=0,~~ \tag {1} $$ which called DNLSE I. The second type is the Chen–Lee–Liu equation[5,6] $$ iq_t+q_{xx}+i|q|^2q_x=0,~~ \tag {2} $$ which is called DNLSE II. The last one takes the form $$ iq_t+q_{xx}-iq^2q^*_x+ \frac{1}{2}q^3{q^*}^2=0,~~ \tag {3} $$ which is called the Gerjikov–Ivanov (GI) equation or DNLSE III.[7,8] Here $i$ is the imaginary unit, and the symbol $*$ denotes complex conjugate. It is found that these three equations may be transformed to each other by a gauge transformation. In this work, we study the coupled DNLSE III, i.e., $$ \begin{cases} \!\! q_t-q_{xx}+2q^2r_x+2q^3r^2=0,\\\!\! r_t+r_{xx}+2 r^2q_x-2q^2r^3=0, \end{cases}~~ \tag {4} $$ with the help of Hirota's method. The Hirota direct method has taken an important role in the study of integrable systems. It can be used not only to find the solutions to soliton equations, but also to finite dimensional Hamiltonian systems.[10-12] Equation (4) has already been proved to be integrable in the Liouville sense by means of trace identity.[13,14] In Refs. [15,16], Fan et al. constructed an $N$-fold Darboux transformation (DT) of Eq. (3) and derived its soliton equations. It is generally known that some soliton equations can produce rogue wave phenomena.[17,18] He et al. investigated the rogue waves and breather solutions to the GI equation using DT.[19,20] The algebro-geometric solutions to the GI equation were given in Refs. [21,22]. In this work, we show that through a variable transformation the bilinear equations for Eq. (4) can be derived for constructing its $N$-soliton solutions. We also describe that the bilinear forms and multi-soliton solutions to Eq. (3) can be derived by reduction. Firstly, we deduce the Lax pairs of the GI Eq. (4), which usually assures the complete integrability of a nonlinear equation. From the spectral problem $$\begin{alignat}{1} \left(\begin{matrix}\phi_1\\\phi_2\end{matrix}\right)_x=\,&M \left(\begin{matrix}\phi_1\\\phi_2\end{matrix}\right),\\ M=\,&\left(\begin{matrix} -\frac{1}{2}(\eta^2-2qr)&\eta q\\ \eta r&\frac{1}{2}(\eta^2-2qr)\end{matrix}\right),~~ \tag {5a} \end{alignat} $$ the time evolution $$\begin{alignat}{1} \left(\begin{matrix}\phi_1\\\phi_2\end{matrix}\right)_t=N \left(\begin{matrix}\phi_1\\ \phi_2\end{matrix}\right),~ N=\left(\begin{matrix}A&B\\C&-A\end{matrix}\right),~~ \tag {5b} \end{alignat} $$ and the related zero curvature equation $$ M_t-N_x+[M,N]=0,~~ \tag {6} $$ one can derive the coupled GI Eq. (4). Its corresponding Lax pairs (5a) and (5b) are governed by $$\begin{align} A=\,&\frac{1}{2}\eta^4-qr\eta^2-q^2r^2+rq_x-qr_x,~~ \tag {7a}\\ B=\,&-q\eta^3+q_x\eta,~~ \tag {7b}\\ C=\,&-r\eta^3-r_x\eta.~~ \tag {7c} \end{align} $$ Secondly, we give the bilinear forms of the coupled GI equation and further its $N$-soliton solutions. By the variable transformation $$ q=\frac{gs}{f^2},~~ r=\frac{hf}{s^2},~~ \tag {8} $$ and the following identities $$\begin{align} &aa(D^2_x b\cdot b)-bb(D^2_x a\cdot a)\\ =\,&2D_x (D_x b\cdot a)\cdot ba,~~ \tag {9a}\\ &(D_x a\cdot b) cd+ ab (D_x c\cdot d)\\ =\,&(D_x a\cdot d)cb+ad(D_x c\cdot b),~~ \tag {9b}\\ &(D_x a\cdot b) cd-ab (D_x c\cdot d)\\ =\,&(D_x a\cdot c) bd-ac\cdot (D_x b\cdot d),~~ \tag {9c} \end{align} $$ Eq. (4) can be transformed into the following bilinear equations $$\begin{align} &(D_t-D^2_x) g\cdot f=0,~~ \tag {10a}\\ &(D_t-D^2_x) f\cdot s=0,~~ \tag {10b}\\ &(D_t+D^2_x) h\cdot s=0,~~ \tag {10c}\\ &D_x f\cdot s=-\frac{1}{3} gh,~~ \tag {10d} \end{align} $$ where $D$ is the well-known Hirota's bilinear operator defined by $$\begin{align} D^m_tD^n_x a\cdot b=\,&(\partial_t-\partial_{t'})^m(\partial_x-\partial_{x'})^n a(t,x)\\ &\cdot b(t',x')|_{t'=t,x'=x}.~~ \tag {11} \end{align} $$ We expand $f$, $g$, $h$ and $s$ as $$\begin{alignat}{1} f=\,&1+\sum^{\infty}_{j=1}f^{(2j)}\varepsilon^{2j},~ g=\sum^{\infty}_{j=1}g^{(2j-1)}\varepsilon^{2j-1},~~ \tag {12a}\\ s=\,&1+\sum^{\infty}_{j=1}s^{(2j)}\varepsilon^{2j},~~ h=\sum^{\infty}_{j=1}h^{(2j-1)}\varepsilon^{2j-1}.~~ \tag {12b} \end{alignat} $$ Substituting the expansions into Eq. (10) and comparing the coefficients of the same power of $\varepsilon$, we have $$\begin{align} g^{(1)}_t-g^{(1)}_{xx}=\,&0,~~ \tag {13a}\\ g^{(3)}_t-g^{(3)}_{xx}=\,&-(D_t-D^2_x)g^{(1)}\cdot f^{(2)},~~ \tag {13b}\\ g^{(5)}_t-g^{(5)}_{xx}=\,&-(D_t-D^2_x)(g^{(1)}\cdot f^{(4)}+g^{(3)}\cdot f^{(2)}),~~ \tag {13c} \end{align} $$ $$\begin{align} \ldots~~\,&\ldots\\ h^{(1)}_t+h^{(1)}_{xx}=\,&0,~~ \tag {14a}\\ h^{(3)}_t+h^{(3)}_{xx}=\,&-(D_t+D^2_x)h^{(1)}\cdot s^{(2)},~~ \tag {14b}\\ h^{(5)}_t+h^{(3)}_{xx}=\,&-(D_t+D^2_x)(h^{(1)}\cdot s^{(4)}+h^{(3)}\cdot s^{(2)}),~~ \tag {14c} \end{align} $$ $$\begin{align} \ldots~~\,&\ldots\\ f^{(2)}_t-s^{(2)}_t=\,&f^{(2)}_{xx}+s^{(2)}_{xx},~~ \tag {15a}\\ f^{(4)}_t-f^{(4)}_{xx}\,&-s^{(4)}_t-s^{(4)}_{xx}=-(D_t-D^2_x)f^{(2)}\cdot s^{(2)},~~ \tag {15b}\\ f^{(6)}_t-f^{(6)}_{xx}\,&-s^{(6)}_t-s^{(6)}_{xx}\\ =\,&-(D_t-D^2_x)(f^{(2)}\cdot s^{(4)}+f^{(4)}\cdot s^{(2)}),~~ \tag {15c} \end{align} $$ $$\begin{align} \ldots~~\,&\ldots\\ f^{(2)}_x-s^{(2)}_x=\,&-\frac{1}{3}g^{(1)}h^{(1)},~~ \tag {16a}\\ \end{align} $$ $$\begin{align} f^{(4)}_x-s^{(4)}_x=\,&-D_x f^{(2)}\cdot s^{(2)}-\frac{1}{3}(g^{(1)}h^{(3)}+g^{(3)}h^{(1)}),~~ \tag {16b}\\ f^{(6)}_x-s^{(6)}_x=\,&-D_x (f^{(4)}\cdot s^{(2)}+f^{(2)}\cdot s^{(4)})\\ &-\frac{1}{3}(g^{(1)}h^{(5)}+g^{(3)}h^{(3)}+g^{(5)}h^{(1)}),~~ \tag {16c}\\ \ldots~~\,&\ldots\\ \end{align} $$ To obtain the $N$-soliton solutions for the coupled GI equation, we take $$\begin{align} \!\!\!\!\!\!\!\!\!\!g^{(1)}=\,&\sum^{N}_{j=1}e^{\xi_j},~ \xi_j=k_jx+\omega_jt+\xi^{(0)}_j,~ \omega_j=k^2_j,~~ \tag {17a}\\ h^{(1)}=\,&\sum^{N}_{j=1}e^{\eta_j},~ \eta_j=-l_jx+m_jt\!+\!\eta^{(0)}_j,~ m_j=-l^2_j,~~ \tag {17b} \end{align} $$ where $\xi^{(0)}_j$, $\eta^{(0)}_j$ ($j=1, 2, \ldots, N)$ are all arbitrary constants. When $N=1$, Eq. (17) can be written as $$ g^{(1)}=e^{\xi_1},~ h^{(1)}=e^{\eta_1}.~~ \tag {18} $$ From Eqs. (15) and (16), we successively solve out $$ f^{(2)}=-\frac{1}{3} k_1 e^{\xi_1+\eta_1+\theta_{13}},~ s^{(2)}=-\frac{1}{3}l_1 e^{\xi_1+\eta_1+\theta_{13}},~~ \tag {19} $$ where $e^{\theta_{13}}=\frac{1}{(k_1-l_1)^2}$. Equations (13)-(16) admit $f^{(2j)}=g^{(2j-1)}=s^{(2j)}=h^{(2j-1)}=0$ ($j=2,3,\ldots$). Thus we obtain the one-soliton solution for Eq. (4), $$\begin{align} q=\,&\frac{(1-\frac{1}{3}l_1e^{\xi_1+\eta_1+\theta_{13}})e^{\xi_1}} {(1-\frac{1}{3}k_1e^{\xi_1+\eta_1+\theta_{13}})^2},~~ \tag {20a}\\ r=\,&\frac{(1-\frac{1}{3}k_1e^{\xi_1+\eta_1+\theta_{13}})e^{\eta_1}} {(1-\frac{1}{3}l_1e^{\xi_1+\eta_1+\theta_{13}})^2},~~ \tag {20b} \end{align} $$ where we have taken $\varepsilon=1$ in Eq. (12). We depict the shape of $|q(x,t)|$ in Fig. 1 and $|r(x,t)|$ in Fig. 2, respectively, where $|\cdot|$ denotes the module of a complex function. When $N=2$, Eqs. (13)-(16) give $$\begin{alignat}{1} g=\,&\sum\limits^2_{j=1}e^{\xi_j} -\sum\limits^4_{j=3}\frac{l_{j-2}}{3}\\ &\cdot\exp{[\sum\limits^2_{p=1}(\xi_p+\theta_{\rm pj})+\eta_j+\theta_{12}]},~~ \tag {21a}\\ h=\,&\sum\limits^4_{j=3}e^{\eta_{j-2}}-\sum\limits^2_{j=1}\frac{k_j}{3}\\ &\cdot\exp{[\sum\limits^4_{p=3}(\eta_{p-2}+\theta_{\rm jp})+\xi_j+\theta_{34}]},~~ \tag {21b}\\ f=\,&1-\sum\limits_{\stackrel {1\leq j\leq 2} {3\leq p\leq 4}}\frac{k_j}{3}\exp{(\xi_j+ \eta_{p-2}+\theta_{\rm jp})}\\ &+\frac{k_1k_2}{9}\exp[\sum\limits^2_{j=1}(\xi_j+\eta_j)+\sum\limits^{4}_{1\leq j < p}\theta_{j,p}],~~ \tag {21c}\\ \end{alignat} $$ $$\begin{alignat}{1} s=\,&1-\sum\limits_{\stackrel {1\leq j\leq 2} {3\leq p\leq 4}}\frac{ l_{p-2}}{3}\exp{(\xi_j+ \eta_{p-2}+\theta_{\rm jp})}\\ &+\frac{l_1l_2}{9}\exp{[\sum\limits^2_{j=1}(\xi_j+\eta_j)+\sum\limits^{4}_{1\leq j < p}\theta_{j,p}]},~~ \tag {21d} \end{alignat} $$ where $\xi_j$, $\eta_j$, $k_j$, $l_j$, $\omega_j$ and $m_j$ are defined by Eq. (17) and $e^{ \theta_{12}}=(k_1-k_2)^2$, $e^{ \theta_{34}}=(l_1-l_2)^2$, and $e^{\theta_{j,2+p}}=\frac{1}{(k_k-l_p)^2}$ ($j,p=1,2$).
cpl-34-9-090201-fig1.png
Fig. 1. (a) The 3D-shape of the one-soliton solution of $|q(x,t)|$ with $k_1=1+0.5i$, $l_1=-0.75-0.5i$, $\xi^{(0)}_1=\eta^{(0)}_1=\frac{\pi}{2}i$, and (b) the 2D-shape of the one-soliton solution of $|q(x,t)|$ with $k_1=1+0.5i$, $l_1=-0.75-0.5i$, $\xi^{(0)}_1=\eta^{(0)}_1=\frac{\pi}{2}i$, and $t=2$.
cpl-34-9-090201-fig2.png
Fig. 2. (a) The 3D-shape of the one-soliton solution of $|r(x,t)|$ with $k_1=1+0.5i$, $l_1=-0.75-0.5i$, $\xi^{(0)}_1=\eta^{(0)}_1=\frac{\pi}{2}i$, and (b) the 2D-shape of the one-soliton solution of $|r(x,t)|$ with $k_1=1+0.5i$, $l_1=-0.75-0.5i$, $\xi^{(0)}_1=\eta^{(0)}_1=\frac{\pi}{2}i$, and $t=1.5$.
In this case, $g$, $h$, $f$ and $s$ provide the two-soliton solutions to Eq. (4) through Eq. (8). Obviously, two-soliton solutions are singular when $f=0$ or $s=0$. However, we can eliminate the singularities by choosing suitable parameters and constants. For example, it is easy to found that $f>0$ and $s>0$, when $\xi_j=k_jx+k^2_jt+\frac{i}{2}\pi, k_j>0$ and $\eta_j=-l_jx-l^2_jt+\frac{i}{2}\pi$, $l_j>0$ ($j=1,2$), since $e^{i\pi}=-1$. It is interesting that the two-soliton solutions can generate the rogue-wave-like phenomenon by selecting special parameters $k_j$ and $l_j (j=1,2)$. We depict the phenomena in Figs. 3 and 4.
cpl-34-9-090201-fig3.png
Fig. 3. Shape of the two-soliton solution of $|q(x,t)|$ with $k_1=0.1+0.3i$, $k_2=0.1+0.9i$, $l_1=-0.2+0.2i$, $l_2=-0.1+0.8i$, and $\xi^{(0)}_1=\eta^{(0)}_1=\frac{\pi}{2}i$.
cpl-34-9-090201-fig4.png
Fig. 4. Shape of the two-soliton solution of $|r(x,t)|$ with $k_1=0.1+0.3i$, $k_2=0.1+0.9i$, $l_1=-0.2+0.2i,l_2=-0.1+0.8i$, and $\xi^{(0)}_1=\eta^{(0)}_1=\frac{\pi}{2}i$.
To express 3-soliton solutions succinctly, we define the following functions $$\begin{alignat}{1} &f_1(j,p)=-k_j\exp{(\xi_j+\eta_{p-3}+\theta_{\rm jp})}/3,\\ &s_1(j,p)=-l_{p-3}\exp{(\xi_j+\eta_{p-3}+\theta_{\rm jp})}/3,~~ \tag {22a}\\ &f_2(j_1,j_2,p_1,p_2)=f_1(j_1,p_1)f_1(j_2,p_2)\\ &\cdot\exp{(\theta_{j_1 j_2}+\theta_{j_1 p_2}+\theta_{j_2 p_1}+\theta_{p_1 p_2})},~~ \tag {22b}\\ &s_2(j_1,j_2,p_1,p_2)=s_1(j_1,p_1)s_1(j_2,p_2)\\ &\cdot\exp{(\theta_{j_1 j_2}+\theta_{j_1 p_2}+\theta_{j_2 p_1}+\theta_{p_1 p_2})},~~ \tag {22c}\\ &h_1(j,p)=-\exp{(\eta_{j-3}+\eta_{p-3}+\theta_{\rm jp})}\\ &\cdot\sum\limits^3_{y=1} k_y \exp{(\xi_y+\theta_{\rm yj}+\theta_{\rm yp})}/3,~~ \tag {22d}\\ &g_1(j,p)=-\exp{(\xi_j+\xi_p+\theta_{\rm jp})}\\ &\cdot\sum\limits^6_{y=4} l_{y-3}\exp{(\eta_{y-3}+\theta_{\rm jy}+\theta_{\rm py})}/3,~~ \tag {22e}\\ \end{alignat} $$ $$\begin{alignat}{1} &h_2(j,p)=\frac{k_{j}k_{p}}{9}\exp[\sum\limits^3_{y=1}(\eta_y+\theta_{j(y+3)}+\theta_{p(y+3)}) \\ &+\sum\limits_{4\leq y_1 < y_2\leq6}\theta_{y_1y_2}+\xi_j+\xi_p+\theta_{\rm jp}],~~ \tag {22f}\\ &g_2(j,p)=\frac{l_{j-3}l_{p-3}}{9}\exp[\sum\limits^3_{y=1}(\xi_y+\theta_{\rm yj}+\theta_{\rm yp})\\ &+\sum\limits_{1\leq y_1 < y_2\leq3}\theta_{y_1y_2}+\eta_{j-3}+\eta_{p-3} +\theta_{\rm jp}].~~ \tag {22g} \end{alignat} $$ When $N=3$, Eqs. (13)-(16) give $$\begin{alignat}{1} \!\!\!\!\!\!\!\!\!\!g=\,&\sum\limits^3_{j=1}e^{\xi_j}+\sum\limits_{1\leq j < p\leq3}g_1(j,p)+\sum\limits_{4\leq j < p\leq6}g_2(j,p),~~ \tag {23a}\\ \!\!\!\!\!\!\!\!\!\!h=\,&\sum\limits^3_{j=1}e^{\eta_j}\!+\!\sum\limits_{4\leq j < p\leq 6}h_1(j,p)+\sum\limits_{1\leq j < p\leq 3}h_2(j,p),~~ \tag {23b}\\ \!\!\!\!\!\!\!\!\!\!f=\,&1+\sum\limits^{3}_{j=1}\sum\limits^{6}_{p=4}f_1(jp)+\sum\limits_{\stackrel{1\leq j_1,j_2\leq 3}{4\leq p_1,p_2\leq 6}} f_2(j_1,j_2,p_1,p_2)\\ \!\!\!\!\!\!\!\!\!\!&-\frac{k_1k_2k_3}{27}\exp{[\sum\limits^3_{j=1}(\xi_j+\eta_j) +\sum\limits^6_{1\leq j < p\leq6}\theta_{\rm jp}]},~~ \tag {23c}\\ \!\!\!\!\!\!\!\!\!\!s=\,&1+\sum\limits^{3}_{j=1}\sum\limits^{6}_{p=4}s_1(jp)+\sum\limits_{\stackrel {1\leq j_1,j_2\leq 3}{4\leq p_1,p_2\leq 6}}s_2(j_1,j_2,p_1,p_2)\\ \!\!\!\!\!\!\!\!\!\!&-\frac{l_1l_2l_3}{27}\exp{[\sum\limits^3_{j=1} (\xi_j+\eta_j)+\sum\limits^6_{1\leq j < p\leq6}\theta_{\rm jp}]}.~~ \tag {23d} \end{alignat} $$ In general, the $N$-soliton solutions can be described as $$\begin{align} g_N(t,x)=\,&\sum_{\mu=0,1}A_2(\mu){\rm \exp} [\sum^{2N}_{j=1}\mu_j\xi^{'}_j+\sum^{2N}_{1\leq j < p}\mu_j\mu_p \theta_{\rm jp}],~~ \tag {24a}\\ f_N(t,x)=\,&\sum_{\mu=0,1}A_1(\mu){\rm \exp} [\sum^{2N}_{j=1}\mu_j\xi^{''}_j+\sum^{2N}_{1\leq j < p}\mu_j\mu_p \theta_{\rm jp} ],~~ \tag {24b}\\ h_N(t,x)=\,&\sum_{\mu=0,1}A_3(\mu){\rm \exp} [\sum^{2N}_{j=1}\mu_j\eta^{'}_j+\sum^{2N}_{1\leq j < p}\mu_j\mu_p \theta_{\rm jp} ],~~ \tag {24c}\\ s_N(t,x)=\,&\sum_{\mu=0,1}A_1(\mu){\rm \exp} [\sum^{2N}_{j=1}\mu_j\eta^{''}_j+\sum^{2N}_{1\leq j < p}\mu_j\mu_p \theta_{\rm jp} ],~~ \tag {24d} \end{align} $$ where $$\begin{alignat}{1} \xi^{'}_j=\,&\xi_j,~ \xi^{'}_{N+j}=\eta_j+\ln{(-\frac{1}{3}l_j)},\\ \xi^{''}_{j}=\,&\xi_j+\ln{(-\frac{1}{3}k_j)},~ \xi^{''}_{N+j}=\eta_j,~~ \tag {25a}\\ \eta^{'}_j=\,&\xi_j+\ln{(-\frac{1}{3}k_j)},~\eta^{'}_{N+j}=\eta_j,\\ \eta^{''}_{j}=\,&\xi_j,~\eta^{''}_{N+j}=\eta_j+\ln{(-\frac{1}{3}l_j)},~~ \tag {25b}\\ e^{\theta_{j,N+p}}=\,&\frac{1}{(k_j-l_p)^2},~ (j,p=1,2,\ldots,N),~~ \tag {25c}\\ \end{alignat} $$ $$\begin{align} e^{\theta_{j,p}}=\,&(k_j-k_p)^2,~ e^{\theta_{N+j,N+p}}=(l_j-l_p)^2,\\ &(j < p=2,3,\ldots,N).~~ \tag {25d} \end{align} $$ Here $A_1(\mu)$, $A_2(\mu)$ and $A_3(\mu)$ take over all possible combinations of $\mu_j=0,1 (j=1,2,\ldots, 2N)$ and satisfy the following conditions, respectively, $$\begin{align} \sum^N_{j=1}\mu_j=\,&\sum^N_{j=1}\mu_{N+j},\\ \sum^{N}_{j=1}\mu_j=\,&\sum^{N}_{j=1}\mu_{N+j}+1,\\ 1+\sum^N_{j=1}\mu_j=\,&\sum^N_{j=1}\mu_{N+j}.~~ \tag {26} \end{align} $$ We replace $e^{\xi^{(0)}_1}$, $e^{\xi^{(0)}_2}$, $e^{\eta^{(0)}_1}$ and $e^{\eta^{(0)}_2}$ by $\alpha e^{\xi^{(0)}_1}/(k_1-k_2)$, $\alpha e^{\xi^{(0)}_2}/(k_2-k_1)$, $\beta e^{\eta^{(0)}_1}/(l_1-l_2)$ and $\beta e^{\eta^{(0)}_2}/(l_2-l_1)$ ($\alpha$ and $\beta$ are arbitrary real constants), respectively. Then the two-soliton solutions (21) under the limits of $k_2\rightarrow k_1$ and $l_2\rightarrow l_1$ lead the limit solutions $$ q=\frac{\bar{g}\bar{s}}{\bar{f}^2},~ r=\frac{\bar{h}\bar{f}}{\bar{s}^2},~~ \tag {27} $$ where $$\begin{align} \bar{g}=\,&\alpha(2k_1t+x)e^{\xi_1}+\frac{ \alpha^2 \beta}{3}\frac{k_1+3l_1}{(l_1-k_1)^5}e^{2\xi_1+\eta_1}\\ &-\frac{\alpha^2\beta}{3} \frac{l_1(x+2l_1t)}{(k_1-l_1)^4}e^{2\xi_1+\eta_1},~~ \tag {28a}\\ \bar{h}=\,&-\beta(2l_1t+x)e^{\eta_1}-\frac{\alpha^2 \beta}{3}\frac{3k_1+l_1}{(k_1-l_1)^5}e^{\xi_1+2\eta_1} \\ &+\frac{\alpha\beta^2}{3}\frac{k_1(x+2k_1t)}{(l_1-k_1)^4} e^{\xi_1+2\eta_1},~~ \tag {28b}\\ \end{align} $$ $$\begin{align} \bar{f}=\,&1+\alpha\beta\Big[-\frac{ (x+2l_1t)+k_1(x+2k_1t)(x+2l_1t)}{(l_1-k_1)^2} \\ &+2\frac{1+k_1(x+2l_1t)+k_1(x+2k_1t)}{(k_1-l_1)^3}\\ &-\frac{6k_1}{(k_1-l_1)^4}\Big] e^{\xi_1+\eta_1}+\frac{\alpha^2 \beta^2 k^2_1}{9(l_1-k_1)^8} e^{2\xi_1+2\eta_1},~~ \tag {28c}\\ \bar{s}=\,&1+\alpha\beta\Big[\frac{(x+2k_1t)-l_1(x+2k_1t)(x+2l_1t)}{(l_1-k_1)^2} \\ &-2\frac{1-l_1(x+2k_1t)-l_1(x+2l_1t)}{(k_1-l_1)^3}\\ &-\frac{6l_1}{(k_1-l_1)^2}\Big]e^{\xi_1+\eta_1}+\frac{\alpha^2 \beta^2 l^2_1}{9(l_1-k_1)^8} e^{2\xi_1+2\eta_1}.~~ \tag {28d} \end{align} $$ These are the so-called one-double pole solutions. This kind of limit procedure can be found in Ref. [23], which builds a bridge between Hirota's approach and the inverse scattering transform on the level of double-pole solutions. Zhou et al. find that the limit solutions for classical $2N$-solitons are nothing but the $N$-double-pole solutions.[24]
cpl-34-9-090201-fig5.png
Fig. 5. (a) The 3D-shape of the one-soliton solution of $|q(x,t)|$ with $k_1=1+0.5i$, $\xi^{(0)}_1=\frac{\pi}{2}i$, and (b) the 2D-shape of the one-soliton solution of $|q(x,t)|$ with $k_1=1+0.5i$, $\xi^{(0)}_1=\frac{\pi}{2}i$, and $t=0$.
cpl-34-9-090201-fig6.png
Fig. 6. Shape of the two-soliton solution of $|q(x,t)|$ with $k_1=0.05+0.1i$, $k_2=0.05+0.2i$, $\xi^{(0)}_1=\eta^{(0)}_2=\frac{\pi}{2}i$, and $t=1.5$.
Now we consider the GI Eq. (3). We shall give its bilinear equations and $N$-soliton solutions by reduction. Setting $r=iq^{*}$, $q=\frac{1}{\sqrt{2}}q$ and replacing $t$ by $-it$ in Eq. (4), one can find that Eq. (4) reduces to DNLSE III. Taking $s=f^*$ and $h=ig^*$, replacing $t$ by $-it$, $g=\frac{1}{\sqrt{2}}g$ and $h=\frac{1}{\sqrt{2}}h$, Eqs. (13)-(16) reduce to the bilinear forms of the GI Eq. (3) as follows: $$\begin{align} &(iD_t+D^2_x) g\cdot f=0,~~ \tag {29a}\\ &(iD_t+D_x) f\cdot f^*=0,~~ \tag {29b}\\ &D_x f\cdot f^*=-\frac{i}{6} gg^*,~~ \tag {29c} \end{align} $$ which can also be directly obtained from Eq. (3) through the transformation $q=(gf^*/f^2)$. If we take $l_j=-k^*_j$, $\eta^{(0)}_j=\xi^{*(0)}$ in Eqs. (24) and (25), then $\eta_j=\xi^*_j$, $e^{\theta^*_{j,N+p}}=e^{\theta_{p,N+j}}$ and $e^{\theta^*_{j,p}=e^{\theta_{N+j,N+p}}}$. Thus we can also have $s=f^*$, $h=ig^*$, and obtain the $N$-soliton solutions to the GI Eq. (3) by reduction $$\begin{align} g_N(t,x)=\,&\sum_{\mu=0,1}A_2(\mu){\rm \exp} [\sum^{2N}_{j=1}\mu_j\xi^{'}_j+\sum^{2N}_{1\leq j < p}\mu_j\mu_p \theta_{\rm jp}],~~ \tag {30a}\\ f_N(t,x)=\,&\sum_{\mu=0,1}A_1(\mu){\rm \exp} [\sum^{2N}_{j=1}\mu_j\xi^{''}_j+\sum^{2N}_{1\leq j < p}\mu_j\mu_p \theta_{\rm jp}],~~ \tag {30b} \end{align} $$ where $$\begin{alignat}{1} \!\!\!\!\!\!\!\xi_j=\,&k_jx-ik^2_jt+\xi^{(0)}_j,\\ \!\!\!\!\!\!\!\xi^{'}_j=\,&\xi_j,~\xi^{'}_{N+j}=\xi^*_j+\ln{\Big(\frac{1}{3}k^*_j\Big)},~~ \tag {31a}\\ \!\!\!\!\!\!\!\xi^{''}_j=\,&\xi_j+\ln{\Big(-\frac{1}{3}k_j\Big)},\\ \!\!\!\!\!\!\!\xi^{''}_{N+j}=\,&\xi^*_j,~ (j=1,2,\ldots,N),~~ \tag {31b}\\ \!\!\!\!\!\!\!e^{\theta_{j,N+p}}=\,&\frac{1}{2(k_j+k^*_p)^2},~ (j,p=1,2,\ldots,N),~~ \tag {31c}\\ \!\!\!\!\!\!\!e^{\theta_{j,p}}=\,& 2(k_j-k_p)^2,~ (j < p=1,2,\ldots,N),~~ \tag {31d} \end{alignat} $$ $k_j$ and $\xi^{(0)}_j$ are arbitrary constants, $A_1(\mu)$ and $A_2(\mu)$ take over all possible combinations of $\mu_j=0,1 (j=1,2,\ldots,2N)$ and satisfy the condition (26). We depict the one-soliton and two-soliton shape of $|q(x,t)|$ in Figs. 5 and 6, respectively. In summary, we have presented multi-soliton solutions for the coupled GI equation by Hirota's approach. By reductions, we also directly obtain the multi-soliton solutions for the GI equation. We demonstrate the solitons of the coupled GI equation and the GI equation in elastic scattering. References On the Modulation of Water Waves in the Neighbourhood of kh ≈ 1.363Optical solitons in a monomode fiberExact solutions of the multidimensional derivative nonlinear Schrodinger equation for many-body systems of criticalityAn exact solution for a derivative nonlinear Schrödinger equationIntegrability of Nonlinear Hamiltonian Systems by Inverse Scattering MethodBilinearization of a Generalized Derivative Nonlinear Schrödinger EquationExact solutions to higher-order nonlinear equations through gauge transformationPainleve analysis of the non-linear Schrodinger family of equationsBilinear approaches for a finite-dimensional Hamiltonian systemBilinear approach for a symmetry constraint of the modified KdV equationSolving the KdV Equation Under Bargmann Constraint via Bilinear ApproachIntegrable evolution systems based on Gerdjikov–Ivanov equations, bi-Hamiltonian structure, finite-dimensional integrable systems and N-fold Darboux transformationNew Lax pairs of the Toda lattice and the nonlinearization under a higher-order Bargmann constraintDarboux transformation and soliton-like solutions for the Gerdjikov-Ivanov equationExplicit N -Fold Darboux Transformations and Soliton Solutions for Nonlinear Derivative Schrödinger EquationsRogue Waves in the (2+1)-Dimensional Nonlinear Schrödinger Equation with a Parity-Time-Symmetric PotentialRogue Waves in the Three-Dimensional Kadomtsev—Petviashvili EquationThe rogue wave and breather solution of the Gerdjikov-Ivanov equationThe higher order rogue wave solutions of the Gerdjikov–Ivanov equationVariable separation and algebro-geometric solutions of the Gerdjikov–Ivanov equationAlgebro-geometric solutions for the Gerdjikov-Ivanov hierarchyN Double Pole Solution for the Modified Korteweg-de Vries Equation by the Hirota's MethodBreathers and limit solutions of the nonlinear lumped self-dual network equation
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