⇦Chinese Physics Letters, 2017, Vol. 34, No. 7, Article code 070202 Bright-Dark Mixed $N$-Soliton Solution of the Two-Dimensional Maccari System * Zhong Han(韩众)1, Yong Chen(陈勇)1,2** Affiliations 1Shanghai Key Laboratory of Trustworthy Computing, East China Normal University, Shanghai 200062 2Department of Physics, Zhejiang Normal University, Jinhua 321004 Received 29 March 2017 *Supported by the Global Change Research Program of China under Grant No 2015CB953904, the National Natural Science Foundation of China under Grant Nos 11675054 and 11435005, and the Shanghai Collaborative Innovation Center of Trustworthy Software for Internet of Things under Grant No ZF1213.
**Corresponding author. Email: ychen@sei.ecnu.edu.cn Citation Text: Han Z and Chen Y 2017 Chin. Phys. Lett. 34 070202 Abstract The general bright-dark mixed $N$-soliton solution of the two-dimensional Maccari system is obtained with the KP hierarchy reduction method. The dynamics of single and two solitons are discussed in detail. Asymptotic analysis shows that two solitons undergo elastic collision accompanied by a position shift. Furthermore, our analysis on mixed soliton bound states shows that arbitrary higher-order soliton bound states can take place. DOI:10.1088/0256-307X/34/7/070202 PACS:02.30.Ik, 05.45.Yv, 02.30.Jr © 2017 Chinese Physics Society Article Text The two-dimensional Maccari system is of the form[1] $$\begin{align} &iA_t+A_{xx}+LA=0,~~ \tag {1} \end{align} $$ $$\begin{align} &iB_t+B_{xx}+LB=0,~~ \tag {2} \end{align} $$ $$\begin{align} &L_y=(AA^*+\mu BB^*)_x ,~~ \tag {3} \end{align} $$ where $\mu \neq 0, \pm 1$ is a real constant, $A(x,y,t)$ and $B(x,y,t)$ are complex while $L(x,y,t)$ is real, and the asterisk means complex conjugate. This system is usually used to describe the motion of isolated waves, localized in a small part of space, in some fields such as plasma physics, nonlinear optics and hydrodynamics. It reduces to the nonlinear Schrödinger (NLS) equation[2] when $y=x$. The reduction $y=t$ leads to the system of the coupled long-wave resonance equation.[3] When $A=B^*$, it becomes the so-called simplest (2+1)-dimensional extension of the NLS equation introduced by Fokas.[4] Many studies have been carried out on this system. Uthayakumar et al.[5] have studied the integrability property of Eqs. (1)-(3) using singularity structure analysis. Based on the technique of coalescence of wavenumbers, its two-dromion solutions were obtained by Lai and Chow.[6] By means of the variable separation approach,[7-9] many coherent soliton structures such as dromions, breathers, foldon and solitoff were obtained by Zhang et al.[10,11] Most recently, Yuan et al.[12] constructed various rational solutions of Eqs. (1)-(3) through Hirota's bilinear method. The goal of this work is to construct the mixed $N$-soliton solution of Eqs. (1)-(3) via the KP hierarchy reduction method. The KP hierarchy reduction method was first developed by the Kyoto school,[13] and has been widely used to derive soliton[14-17] and rogue wave[18,19] solutions of many integrable systems. Assuming that the $A$ component is of bright type and the $B$ component is of dark type, the following dependent variable transformations are introduced, $$\begin{alignat}{1} A=\frac{g}{f},~B=\rho e^{i(\alpha x-\alpha^2 t)}\frac{h}{f},~L=2 (\log f)_{xx},~~ \tag {4} \end{alignat} $$ where $f(x,y,t)$ is real, $g(x,y,t)$ and $h(x,y,t)$ are complex, $\alpha$ and $\rho$ are two real constants. Then the two-dimensional (2D) Maccari system (1)-(3) can be converted into the bilinear form $$\begin{align} & (D^2_x+i D_t)g \cdot f=0,~~ \tag {5} \end{align} $$ $$\begin{align} & (D^2_x+2 i \alpha D_x+i D_t)h \cdot f=0,~~ \tag {6} \end{align} $$ $$\begin{align} & D_xD_yf\cdot f= gg^\ast-\mu \rho^2 (f^2-hh^\ast),~~ \tag {7} \end{align} $$ where $D$ is the usual Hirota bilinear operator.[20] Consider the Gram determinant tau functions of a two-component KP hierarchy $$\begin{align} \tau_{0}(k)=\,&\left| \begin{matrix} {\boldsymbol A} & {\boldsymbol I}\\ -{\boldsymbol I} & {\boldsymbol B} \end{matrix} \right|,~\tau_{1}(k)=\left|\begin{matrix} {\boldsymbol A} & {\boldsymbol I} & {\it \Omega}^{\rm T}\\ -{\boldsymbol I} & {\boldsymbol B} & {\boldsymbol 0}^{\rm T}\\ {\boldsymbol 0} & -\bar{\it \Psi} & 0 \end{matrix} \right|,\\ \tau_{-1}(k)=\,&\left| \begin{matrix} {\boldsymbol A} & {\boldsymbol I} & {\boldsymbol 0}^{\rm T}\\ -{\boldsymbol I} & {\boldsymbol B} & {\it \Psi}^{\rm T}\\ -\bar{\it \Omega} & {\boldsymbol 0} & 0 \end{matrix} \right|,~~ \tag {8} \end{align} $$ where $I$ is an $N \times N$ identity matrix, ${\boldsymbol 0}$ is an $N$-component zero-row vector, ${\boldsymbol A}$ and ${\boldsymbol B}$ are $N \times N$ matrices, and ${\it \Omega}$, ${\it \Psi}$, $\bar{\it \Omega}$ and $\bar{\it \Psi}$ are $N$-component row vectors whose elements are given respectively by $$\begin{align} a_{ij}(k)=\,&\frac{1}{p_i+\bar{p}_j} \Big(-\frac{p_i-c}{\bar{p}_j+c}\Big)^{k} e^{\xi_i+\bar{\xi}_j},\\ b_{ij}=\,&\frac{1}{q_i+\bar{q}_j}e^{\eta_i+\bar{\eta}_j},\\ {\it \Omega}=\,&(e^{\xi_1},e^{\xi_2},\ldots,e^{\xi_N}),~{\it \Psi}=(e^{\eta_1},e^{\eta_2},\ldots,e^{\eta_N}),\\ \bar{\it \Omega}=\,&(e^{\bar{\xi}_1},e^{\bar{\xi}_2}, \ldots,e^{\bar{\xi}_N}),~\bar{\it \Psi}=(e^{\bar{\eta}_1}, e^{\bar{\eta}_2},\ldots,e^{\bar{\eta}_N}), \end{align} $$ with $$\begin{align} \xi_i=\,&\frac{1}{p_i-c}x_{-1}+p_ix_1+p^2_ix_2+\xi_{i0},~\eta_i=q_iy_1+\eta_{i0},\\ \bar{\xi}_j=\,&\frac{1}{\bar{p}_j+c}x_{-1}+\bar{p}_jx_1-\bar{p}^2_jx_2 +\bar{\xi}_{j0},~\bar{\eta}_j=\bar{q}_jy_1+\bar{\eta}_{j0}, \end{align} $$ in which $p_i$, $\bar{p}_j$, $q_i$, $\bar{q}_j$, $\xi_{i0}$, $\bar{\xi}_{j0}$, $\eta_{i0}$, $\bar{\eta}_{j0}$ and $c$ are complex constants. Based on the Sato theory for KP hierarchy,[13] the tau functions (8) satisfy the bilinear equations $$\begin{align} &(D^2_{x_1}-D_{x_2})\tau_{1}(k) \cdot \tau_{0}(k)=0,~~ \tag {9} \end{align} $$ $$\begin{align} &(D^2_{x_1}-D_{x_2}+2cD_{x_1})\tau_{0}(k+1) \cdot \tau_{0}(k)=0,~~ \tag {10} \end{align} $$ $$\begin{align} &D_{x_1}D_{y_1}\tau_{0}(k) \cdot \tau_{0}(k)=-2\tau_{1}(k) \tau_{-1}(k),~~ \tag {11} \end{align} $$ $$\begin{align} &(D_{x_1}D_{x_{-1}}-2)\tau_{0}(k) \cdot \tau_{0}(k)=-2\tau_{0}(k+1) \tau_{0}(k-1),~~ \tag {12} \end{align} $$ which can be proved using the Grammian technique.[20] Assuming that $x_1$, $x_{-1}$ and $y_1$ are real; $x_2$ and $c$ are pure imaginary and taking $p^*_j=\bar{p}_j$, $q^*_j=\bar{q}_j$, $\xi^*_{j0}=\bar{\xi}_{j0}$, and $\eta^*_{j0}=\bar{\eta}_{j0}$, one can check that $$ a_{ij}(k)=a^*_{ji}(k),~~b_{ij}=b^*_{ji}. $$ Moreover, we let $$ f=\tau_{0}(0),~g=\tau_{1}(0),~h=\tau_{0}(1), $$ hence, $f$ is real and $$ g^{*}=-\tau_{-1}(0),~h^{*}=\tau_{0}(-1), $$ thus the bilinear Eqs. (9)-(12) become $$\begin{align} & (D^2_{x_1}-D_{x_2})g \cdot f=0,~~ \tag {13} \end{align} $$ $$\begin{align} & (D^2_{x_1}-D_{x_2}+2cD_{x_1})h \cdot f=0,~~ \tag {14} \end{align} $$ $$\begin{align} & D_{x_1}D_{y_1}f \cdot f=2gg^{*},~~ \tag {15} \end{align} $$ $$\begin{align} & (D_{x_1}D_{x_{-1}}-2)f \cdot f=-2h h^{*}.~~ \tag {16} \end{align} $$ By row and column operations, $f$ can be rewritten as $$\begin{align} f=\left| \begin{matrix} A' & I\\ -I & B' \end{matrix} \right|,~~ \tag {17} \end{align} $$ where the entries in $A'$ and $B'$ are $$ a'_{ij}=\frac{1}{p_i+p^*_j},~~ b'_{ij}=\frac{1}{q_i+q^*_j}e^{\eta_i+\eta^*_j+\xi^*_i+\xi_j}, $$ with $$\begin{align} \eta_i+\xi^*_i=\,&q_iy_1+\frac{1}{p^*_i+c}x_{-1}+p^*_ix_1\\ &-{p_i^*}^2x_2 +\xi^*_{i0}+\eta_{i0},\\ \eta^*_j+\xi_j=\,&q^*_jy_1+\frac{1}{p_j-c}x_{-1}+p_jx_1\\ &+p_j^2x_2 +\xi_{j0}+\eta^*_{j0}. \end{align} $$ Under the reduction conditions $$\begin{alignat}{1} -i{p_i^*}^2=q_i-\frac{\mu \rho^2}{p_i^*+c},~ip^2_j=q^*_j-\frac{\mu \rho^2}{p_j-c},~~ \tag {18} \end{alignat} $$ the following relation holds $$\begin{align} i\partial_{x_2}b'_{ij}=(\partial_{y_1}-\mu \rho^2\partial_{x_{-1}})b'_{ij},~~ \tag {19} \end{align} $$ hence we have $$\begin{align} if_{x_2}=f_{y_1}-\mu \rho^2f_{x_{-1}},~~ \tag {20} \end{align} $$ which also implies $$\begin{align} if_{x_1x_2}=f_{x_1y_1}-\mu \rho^2f_{x_1x_{-1}}.~~ \tag {21} \end{align} $$ In addition, Eqs. (15) and (16) can be expanded as $$\begin{align} f_{x_1y_1}f-f_{x_1}f_{y_{1}}=gg^{*},~~ \tag {22} \end{align} $$ and $$\begin{align} f_{x_1x_{-1}}f-f_{x_1}f_{x_{-1}}-f^2=-h h^{*},~~ \tag {23} \end{align} $$ respectively. By using relations (20) and (21), from Eqs. (22) and (23), we arrive at $$\begin{alignat}{1} i(f_{x_1x_2}f-f_{x_1}f_{x_2})=gg^{*}-\mu \rho^2(f^2-hh^{*}).~~ \tag {24} \end{alignat} $$ Through the variable transformations $$\begin{align} x_1=x,~x_2=i(t+\frac{1}{2}y),~~ \tag {25} \end{align} $$ Eqs. (13) and (14) become Eqs. (5) and (6) by taking $c=i\alpha$, and Eq. (24) is nothing but Eq. (7). With the variable transformations (25), the variables $x_{-1}$ and $y_1$ become dummy variables, hence they can be treated as constants. Therefore, we define $e^{\eta_{i}}=c^*_i$, $e^{\eta^*_{i}}=c_i$, $(i=1,2,\ldots,N)$ and let $C=-(c_1,c_2,\ldots,c_N)$, thus we have obtained the mixed $N$-soliton solution $$\begin{align} f=\,&\left| \begin{matrix} {\boldsymbol A} & {\boldsymbol I}\\ -{\boldsymbol I} & {\boldsymbol B} \end{matrix} \right|,~ g=\left| \begin{matrix} {\boldsymbol A} & {\boldsymbol I} & {\it \Omega}^{\rm T}\\ -{\boldsymbol I} & {\boldsymbol B} & {\boldsymbol 0}^{\rm T}\\ {\boldsymbol 0} & C & 0 \end{matrix} \right|,\\ h=\,&\left| \begin{matrix} {\boldsymbol A}^{(1)} & {\boldsymbol I}\\ -{\boldsymbol I} & {\boldsymbol B} \end{matrix} \right|,~~ \tag {26} \end{align} $$ where the entries in ${\boldsymbol A},{\boldsymbol A}^{(1)}$ and ${\boldsymbol B}$ are $$\begin{align} a_{ij}=\,&\frac{1}{p_i+p^*_j} e^{\xi_i+\xi^*_j},\\ a^{(1)}_{ij}=\,&\frac{1}{p_i+p^*_j} \Big(-\frac{p_i-i\alpha}{p^*_j+i\alpha}\Big) e^{\xi_i+\xi^*_j},\\ b_{ij}=\,&c_i^{*}c_j\Big[i(-{p_i^*}^2+p_j^2) +\frac{\mu\rho^2(p_i^*+p_j)}{(p_i^*+i\alpha)(p_j-i\alpha)}\Big]^{-1}, \end{align} $$ meanwhile, ${\it \Omega}$ and $C$ are given by $$\begin{align} {\it \Omega}=(e^{\xi_1},e^{\xi_2},\ldots,e^{\xi_N}),~C=-(c_1,c_2,\ldots,c_N), \end{align} $$ with $\xi_i=p_ix +ip^2_i(t+\frac{1}{2}y)+\xi_{i0}$, and $p_i$, $\xi_{i0}$ and $c_i$ $(i=1,2,\ldots,N)$ are complex constants. Take $N=1$ in the formula (26) and we can obtain one-soliton solution. In this case, the tau functions can be rewritten as $$\begin{align} f=\,&1+E_{11^*}e^{\xi_1+\xi^*_1},~g=c_1e^{\xi_1},\\ h=\,&1+F_{11^*}e^{\xi_1+\xi^*_1},~~ \tag {27} \end{align} $$ where $$\begin{align} E_{11^*}=\,&c_1c_1^{*}\Big[i(p_1+p_1^*)^2(p_1-p_1^*) \\ &+\frac{\mu\rho^2(p_1+p_1^*)^2}{(p_1^*+i\alpha)(p_1-i\alpha)}\Big]^{-1},\\ F_{11^*}=\,&e^{2i\phi}E_{11^*},~~e^{2i\phi_1}=-\frac{p_1-i\alpha}{p^*_1+i\alpha}. \end{align} $$ Note that this solution is nonsingular only if $E_{11^*}>0$. The tau functions (27) give the one-soliton solution $$\begin{align} A=\,&\frac{c_1}{2} e^{-\theta_1} e^{i\xi_{\rm 1I}} {\rm sech}(\xi_{\rm 1R}+\theta_1),\\ B=\,&\frac{\rho}{2} e^{i(\alpha x-\alpha^2 t)} [1+e^{2i\phi_1}\\ &+(e^{2i\phi_1}-1)\tanh(\xi_{\rm 1R}+\theta_1)],\\ L=\,&2p^2_{\rm 1R}{\rm sech}^2(\xi_{\rm 1R}+\theta_1), \end{align} $$ where $e^{2 \theta_1}=E_{11^*}$, $\xi_1=\xi_{\rm 1R}+i\xi_{\rm 1I}$, and the suffixes R and I denote the real and imaginary parts, respectively. Thus the amplitude of the bright soliton in $A$ component is $\frac{|c_1|}{2} e^{-\theta_1}$ while the amplitude of the bright soliton in $L$ component is $2p^2_{\rm 1R}$. For the dark soliton in $B$ component, $|B|$ approaches $|\rho|$ as $x,y \rightarrow \pm \infty$. Moreover, the intensity of the dark soliton is $|\rho|\cos\phi_1$. The mixed one-soliton at time $t=0$ is displayed in Fig. 1 with the parametric choice $p_1=1-\frac{1}{2}i$, $c_1=1+2i$, $\rho=\alpha=1$, $\xi_{10}=y=0$ and $\mu=2$.
cpl-34-7-070202-fig1.png
Fig. 1. Mixed one-soliton of the 2D Maccari system.
As a matter of fact, the interaction of nonlinear waves may present some novel phenomena.[17,21] To study the collision of two solitons, we take $N=2$ in the formula (26). The tau functions can be rewritten as $$\begin{align} f=\,&1+E_{11^*}e^{\xi_1+\xi^*_1}+E_{12^*}e^{\xi_1+\xi^*_2}+E_{21^*}e^{\xi_2+\xi^*_1}\\ &+E_{22^*}e^{\xi_2+\xi^*_2}+E_{121^*2^*}e^{\xi_1+\xi_2+\xi^*_1+\xi^*_2},\\ g=\,&c_1e^{\xi_1}+c_2e^{\xi_2}+G_{121^*}e^{\xi_1+\xi_2+\xi^*_1}\\ &+G_{122^*}e^{\xi_1+\xi_2+\xi^*_2},\\ h=\,&1+F_{11^*}e^{\xi_1+\xi^*_1}+F_{1,2^*}e^{\xi_1+\xi^*_2}+F_{21^*}e^{\xi_2 +\xi^*_1}\\ &+F_{22^*}e^{\xi_2+\xi^*_2}+F_{121^*2^*}e^{\xi_1+\xi_2+\xi^*_1+\xi^*_2},~~ \tag {28} \end{align} $$ where $$\begin{align} E_{ij^*}=\,&c_ic_j^{*}\Big[i(p_i+p_j^*)^2(p_i-p_j^*)\\ &+\frac{\mu\rho^2(p_i+p_j^*)^2} {(p_i-i\alpha)(p_j^*+i\alpha)}\Big]^{-1},\\ E_{121^*2^*}=\,&|p_1-p_2|^2\Big[\frac{E_{11^*}E_{22^*}}{(p_1+p_2^*)(p_2+p_1^*)}\\ &-\frac{E_{12^*}E_{21^*}}{(p_1+p_1^*)(p_2+p_2^*)}\Big],\\ F_{ij^*}=\,&-\frac{p_i-i\alpha_1}{p^*_j+i\alpha_1} E_{ij^*},\\ F_{121^*2^*}=\,&\frac{(p_1-i\alpha_1)(p_2-i\alpha_1)}{(p^*_1+i\alpha_1) (p^*_2+i\alpha_1)}E_{121^*2^*},\\ G_{12i^*}=\,&(p_1-p_2)\Big(\frac{c_1E_{2i^*}}{p_1+p_i^*} -\frac{c_2E_{1i^*}}{p_2+p_i^*}\Big). \end{align} $$ To obtain nonsingular solutions, the denominator $f$ in Eq. (28) must be nonzero. For this purpose, we rewrite $f$ as $$\begin{align} f=\,&2e^{\xi_{\rm 1R}+\xi_{\rm 2R}}[e^{\theta_1+\theta_2} \cosh(\xi_{\rm 1R}-\xi_{\rm 2R}+\theta_1-\theta_2)\\ &+e^{\theta_3} \cosh(\xi_{\rm 1R}-\xi_{\rm 2R}+\theta_3)\\ &+e^{\zeta_{R}} \cos(\xi_{\rm 1I}-\xi_{\rm 2I}+\zeta_{I})], \end{align} $$ where $$\begin{align} e^{2\theta_1}=\,&E_{11^*},~~e^{2\theta_2}=E_{22^*},\\ e^{2\theta_3}=\,&E_{121^*2^*},~~e^{\zeta_{R}+i\zeta_{I}}=E_{12^*}. \end{align} $$ Hence, $e^{\theta_1+\theta_2}+e^{\theta_3}>e^{\zeta_{R}}$ is a sufficient condition to guarantee nonsingular solutions. The asymptotic forms of the soliton $s_1$, ($s_2$) before and after collision are of the form: (1) Before collision ($x,y\rightarrow -\infty$) Soliton $s_1$ $$\begin{align} A_{1}^{-} \simeq\,&\frac{c_1}{2}e^{-\theta_1}e^{i\xi_{\rm 1I}}{\rm sech}(\xi_{\rm 1R}+\theta_1),\\ B_{1}^{-} \simeq\,&\frac{\rho}{2} e^{i(\alpha x-\alpha^2t)}\\ &\cdot[1+e^{2i\phi_1}+(e^{2i\phi_1}-1)\tanh(\xi_{\rm 1R}+\theta_1)],\\ L_{1}^{-} \simeq\,&2p^2_{\rm 1R}{\rm sech}^2(\xi_{\rm 1R}+\theta_1). \end{align} $$ Soliton $s_2$ $$\begin{align} A_{2}^{-} \simeq\,&\frac{1}{2}e^{-\theta_1-\theta_3}G_{121^*}e^{i\xi_{\rm 2I}}{\rm sech}(\xi_{\rm 2R}+\theta_3-\theta_1),\\ B_{2}^{-} \simeq\,&\frac{\rho}{2} e^{i(\alpha x-\alpha^2 t+2\phi_1)}\\ &\cdot[1+e^{2i\phi_2}+(e^{2i\phi_2}-1)\tanh(\xi_{\rm 2R}+\theta_3-\theta_1)],\\ L_{2}^{-} \simeq\,&2p^2_{\rm 2R}{\rm sech}^2(\xi_{\rm 2R}+\theta_3-\theta_1). \end{align} $$ (2) After collision ($x,y\rightarrow +\infty$) Soliton $s_1$ $$\begin{align} A_{1}^{+} \simeq\,&\frac{1}{2}e^{-\theta_2-\theta_3}G_{122^*}e^{i\xi_{\rm 1I}}{\rm sech}(\xi_{\rm 1R}+\theta_3-\theta_2),\\ B_{1}^{+} \simeq\,&\frac{\rho}{2} e^{i(\alpha x-\alpha^2t+2\phi_2)}\\ &\cdot[1+e^{2i\phi_1}+(e^{2i\phi_1}-1)\tanh(\xi_{\rm 1R}+\theta_3-\theta_2)],\\ L_{1}^{+} \simeq\,&2p^2_{\rm 1R}{\rm sech}^2(\xi_{\rm 1R}+\theta_3-\theta_2). \end{align} $$ Soliton $s_2$ $$\begin{align} A_{2}^{+} \simeq\,&\frac{c_2}{2}e^{-\theta_2}e^{i\xi_{\rm 2I}}{\rm sech}(\xi_{\rm 2R}+\theta_2),\\ B_{2}^{+} \simeq\,&\frac{\rho}{2} e^{i(\alpha x-\alpha^2t)}\\ &\cdot[1+e^{2i\phi_2}+(e^{2i\phi_2}-1)\tanh(\xi_{\rm 2R}+\theta_2)],\\ L_{2}^{+} \simeq\,&2p^2_{\rm 2R}{\rm sech}^2(\xi_{\rm 2R}+\theta_2). \end{align} $$ In the above expressions, $e^{2i\phi_j}=-(p_j-i\alpha_1)/(p^*_j+i\alpha_1)$. The amplitudes of the bright solitons in $A$ component before interaction are $(\frac{c_1}{2}e^{-\theta_1},\frac{1}{2}e^{-\theta_1-\theta_3}G_{121^*})$, and the amplitudes after interaction are $(\frac{1}{2}e^{-\theta_2-\theta_3}G_{122^*},\frac{c_2}{2}e^{-\theta_2})$. Substituting various quantities, we can find that $|\frac{c_1}{2}e^{-\theta_1}|=|\frac{1}{2}e^{-\theta_2-\theta_3}G_{122^*}|$ and $|\frac{1}{2}e^{-\theta_1-\theta_3}G_{121^*}|=|\frac{c_2}{2}e^{-\theta_2}|$, which indicate that the intensities of the bright solitons in $A$ component are the same before and after collision. Similarly, the dark solitons in $B$ component and the bright solitons in $L$ component also undergo elastic collision. In addition, both the bright and dark solitons admit the same magnitude position shift. The position shift of soliton $s_1$, ($s_2$) is ${\it \Lambda}_1=\theta_3-\theta_1-\theta_2$, (${\it \Lambda}_2=-{\it \Lambda}_1$). The phase shifts of the dark solitons $s_1$ and $s_2$ in $B$ component are $2\phi_2$ and $-2\phi_1$, respectively. The above analysis clearly reveals that the collision of two solitons is elastic and energies of solitons in different components completely transmit through. In Fig. 2, the collision of two solitons is displayed for the parameters chosen as $p_1=1-\frac{3}{4}i$, $p_2=2-\frac{1}{4}i$, $c_1=1+\frac{1}{2}i$, $c_2=\frac{1}{2}+i$, $\rho=\alpha=1$, $\xi_{10}=\xi_{20}=y=0$ and $\mu=2$. It is obvious that the solitons in all the components undergo elastic collision without shape change but just accompanied by a position shift.
cpl-34-7-070202-fig2.png
Fig. 2. Collision of mixed two solitons of the 2D Maccari system: (a) the $A$ component, (b) the $B$ component, and (c) the $L$ component.
Soliton bound states are another fascinating class of multi-soliton solutions, which can be viewed as composite solitons moving with the same velocity. Suppose that the wave number of the $i$th soliton is $p_i=p_{i{\rm R}}+ip_{i{\rm I}}$, then one can obtain the mixed two-soliton bound state from Eq. (28) with the restriction $p_{\rm 1I}=p_{\rm 2I}$. Such a bound state is displayed in Fig. 3 with the parametric choice $p_1=1-\frac{1}{4}i$, $p_2=2-\frac{1}{4}i$, $c_1=1+\frac{1}{2}i$, $c_2=\frac{1}{2}+i$, $\rho=\alpha=1$, $\xi_{10}=\xi_{20}=y=0$ and $\mu=2$. What is more, the bound state can exist up to an arbitrary order since the same $p_{i{\rm I}}$ value can give as many distinct $p_{i{\rm R}}$ values as we want.
cpl-34-7-070202-fig3.png
Fig. 3. Mixed two-soliton bound states of the 2D Maccari system: (a) the $A$ component, (b) the $B$ component, and (c) the $L$ component.
We would like to express our sincere thanks to Lou S Y and Chen J C for their valuable comments. References Universal and integrable nonlinear evolution systems of equations in 2+1 dimensionsOn the simplest integrable equation in 2+1Soliton Solution of New (2+1) Dimensional Nonlinear Partial Differential EquationsCoalescence of Ripplons, Breathers, Dromions and Dark SolitonsRevisitation of the localized excitations of the (2+1)-dimensional KdV equationLocalized excitations in (2+1)-dimensional systemsGeneralized Dromion Structures of New (2+1)-Dimensional Nonlinear Evolution EquationSolitons and infinite-dimensional Lie algebrasGeneral?N-Dark-Dark Solitons in the Coupled Nonlinear Schr?dinger EquationsGeneral N -soliton solution to a vector nonlinear Schr?dinger equationMulti-Dark Soliton Solutions of the Two-Dimensional Multi-Component Yajima–Oikawa SystemsGeneral Mixed Multi-Soliton Solutions to One-Dimensional Multicomponent Yajima–Oikawa SystemRogue waves in the Davey-Stewartson I equationDynamics of rogue waves in the Davey?Stewartson II equationRogue wave and a pair of resonance stripe solitons to a reduced (3+1)-dimensional Jimbo?Miwa equation
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