Junyi Zhu1, Sishou Zhou2, Zhijun Qiao,3,41School of Mathematics and Statistics, Zhengzhou University, Zhengzhou, Henan 450001, China 2School of Mathematics and Statistics, Kashgar University, Kashgar, Xinjiang 844006, China 3School of Mathematical and Statistical Sciences, University of Texas Rio Grande Valley, Edinburg, TX 78539, United States of America
First author contact:4Author to whom any correspondence should be addressed. Received:2019-08-27Revised:2019-09-30Accepted:2019-11-6Online:2020-01-14
Abstract We generalize the $\bar{\partial }$-dressing method to investigate a (2+1)-dimensional lattice, which can be regarded as a forced (2+1)-dimensional discrete three-wave equation. The soliton solutions to the (2+1)-dimensional lattice are given through constructing different symmetry conditions. The asymptotic analysis of one-soliton solution is discussed. For the soliton solution, the forces are zero. Keywords：discrete (2 + 1)-dimensional three-wave equation;∂̄-dressing method;explicit solution
PDF (791KB)MetadataMetricsRelated articlesExportEndNote|Ris|BibtexFavorite Cite this article Junyi Zhu, Sishou Zhou, Zhijun Qiao. Forced (2+1)-dimensional discrete three-wave equation. Communications in Theoretical Physics, 2020, 72(1): 015004- doi:10.1088/1572-9494/ab5fb4
1. Introduction
The nonlinear interaction of several wave modes has attracted great interest, and particular attention has been focused on the (2+1)-dimensional three-wave equation [1–10] and the three-dimensional three-wave equation [11–14]. These equations can be applied to the problems of radiophysics and nonlinear optics [15]. Besides the continuous integrable system, the consideration of integrable discretization or discrete integrable equations is also important. To our knowledge, the references about (2+1)-dimensional discrete three-wave equation are few.
The feasible approach based on the $\bar{\partial }$ (Dbar)-problem [16–23] is a powerful tool to investigate the integrability of nonlinear PDEs, in particular for higher dimensional equations, and to find their explicit solutions, including soliton solutions. In addition, The Dbar-problem also provides a method to construct the symmetry conditions of integrable equations [24].
Extending the Zakharov–Shabat dressing method is one of the important methods to form and solve discrete integrable nonlinear equations [24–26]. Here, in this paper, we extend the Dbar-dressing method to investigate the discrete (2+1)-dimensional differential-difference equation$ \begin{eqnarray}\begin{array}{l}{\rm{i}}{\partial }_{t}[{a}_{q}{Q}_{{pq}}(n)-{a}_{p}{Q}_{{pq}}(n-1)]\\ \qquad -\,{\rm{i}}{\partial }_{y}[{b}_{q}{Q}_{{pq}}(n)-{b}_{p}{Q}_{{pq}}(n-1)]\\ \quad =\,{s}_{{pq}}[{R}_{p}(n){Q}_{{pq}}(n)-{Q}_{{pq}}(n-1){R}_{q}(n)]\\ \qquad +\,{s}_{{pq}}(1-{E}^{-}){Q}_{{pq}}(n)-{s}_{{pq}}{Q}_{{pr}}(n-1){Q}_{{rq}}(n)\\ \qquad +\,{s}_{{rq}}{Q}_{{pr}}(n){Q}_{{rq}}(n)+{s}_{{pr}}{Q}_{{pr}}(n-1){Q}_{{rq}}(n-1),\\ \quad r\ne p\ne q,\end{array}\end{eqnarray}$$ \begin{eqnarray}\begin{array}{l}(1-{E}^{-})[{{\rm{i}}{a}}_{p}{\partial }_{t}{Q}_{{pp}}(n)-{\rm{i}}{\partial }_{y}{Q}_{{pp}}(n)\\ \quad -\,\displaystyle \sum _{r\ne p}{s}_{{pr}}| {Q}_{{pr}}(n){| }^{2}]=0,\end{array}\end{eqnarray}$where ${E}^{-}$ is the backward shift, and aq, bq, (q=1, 2, 3) are real constants and$ \begin{eqnarray}{R}_{p}(n)={Q}_{{pp}}(n)-{Q}_{{pp}}(n-1),\quad {s}_{{pq}}={b}_{p}{a}_{q}-{a}_{p}{b}_{q}.\end{eqnarray}$Here and after, we have omitted the dependence of the variables y and t.
Let us impose the following constraint ${Q}^{\dagger }(n)=-Q(n)$ with $Q(n)={\left({Q}_{{pq}}(n\right)}_{3\times 3}$, which implies that Qpp(n) is purely imaginary. Here † denotes the Hermitian conjugate. It is noted that equations (1.1) and (1.2) were derived through the discrete version of Zakharov–Shabat dressing method [25]. We note that the linear equation (1.2) can be rewritten as a forced equation$ \begin{eqnarray}\begin{array}{c}\begin{array}{l}{\rm{i}}{ \mathcal L }\left(\begin{array}{c}{Q}_{11}(n)\\ {Q}_{22}(n)\\ {Q}_{33}(n)\end{array}\right)\,=\,S\left(\begin{array}{c}| {Q}_{12}(n){| }^{2}\\ | {Q}_{13}(n){| }^{2}\\ | {Q}_{23}(n){| }^{2}\end{array}\right)\,+\,F,\\ S=\left(\begin{array}{ccc}{s}_{12} & {s}_{13} & 0\\ -{s}_{12} & 0 & {s}_{23}\\ 0 & -{s}_{13} & -{s}_{23}\end{array}\right),\\ { \mathcal L }={\rm{diag}}(\begin{array}{c}{a}_{1}{{\rm{\partial }}}_{t}-{b}_{1}{{\rm{\partial }}}_{y},{a}_{2}{{\rm{\partial }}}_{t}-{b}_{2}{{\rm{\partial }}}_{y},{a}_{3}{{\rm{\partial }}}_{t}-{b}_{3}{{\rm{\partial }}}_{y},\end{array}),\\ F={\left({f}_{1},{f}_{2},{f}_{3}\right)}^{{\rm{T}}},\,detS=0.\end{array}\end{array}\,\end{eqnarray}$Here fp, (p=1, 2, 3) are real functions and are independent of the variable n. In addition, equation (1.4) implies the forced conservation law$ \begin{eqnarray}{{\rm{\partial }}}_{t}\left(\displaystyle \sum _{p=1}^{3}{{\rm{i}}a}_{p}{Q}_{{pp}}(n)\right)-{{\rm{\partial }}}_{y}\left(\displaystyle \sum _{p=1}^{3}{{\rm{i}}b}_{p}{Q}_{{pp}}(n)\right)=\displaystyle \sum _{p=1}^{3}{f}_{p},\end{eqnarray}$and Qpp(n) is representation of the energy functions $| {Q}_{{pq}}(n){| }^{2},$ (1≤p<q≤3) by virtue of $\det S=0$. More concisely, in the case of fp=0, Qpp(n) is determined only by the energy functions $| {Q}_{{pq}}{| }^{2}$ and $| {Q}_{{pr}}{| }^{2}$ in view of (1.4), where $p\ne q\ne r$. The total energy ${\sum }_{p=1}^{3}{{\rm{i}}a}_{p}{\int }_{-\infty }^{\infty }{Q}_{{pp}}(n){\rm{d}}y$ is conserved, if ${Q}_{{pp}}(n)\to 0$ as $| y| \to \infty $. Hence, equations (1.1) and (1.2) model a forced (2+1)-dimensional discrete three-wave (FD3W) equation.
In the paper, two covariant derivative operators and one backward translation operator are introduce to construct the Lax pair of the FD3W equation. A suitable symmetry condition is found to derive the soliton solutions. It is remarked that forces fp in (1.4) are zero for soliton solutions. We have to say that the picture of the forces to solution is not yet clear.
The whole paper is organized as follows. In section 2, the dressing method is presented to derive the Lax pair of the FD3W equation. In section 3, the nonlocal Dbar-problem and the symmetry condition are introduced. In section 4, the explicit solutions in several cases are given. In section 5, we give brief remarks about another form of FD3W equation and its soliton solution derived from another symmetry constraint.
2. Dressing approach
Given two diagonal constant matrices A and B, we consider the following covariant derivatives$ \begin{eqnarray}\begin{array}{c}\begin{array}{rcl}{{\bf{D}}}_{t}\chi & = & {\rm{i}}{\chi }_{t}+k\chi B,\,\,\,B={\rm{diag}}({b}_{1},{b}_{2},{b}_{3}),\\ {{\bf{D}}}_{y}\chi & = & {\rm{i}}{\chi }_{y}+k\chi A,\,\,\,A={\rm{diag}}({a}_{1},{a}_{2},{a}_{3}),\end{array}\end{array}\end{eqnarray}$and the backward translation operator$ \begin{eqnarray}{\bf{T}}\chi =({E}^{-}\chi )(1-k)I,\end{eqnarray}$where I is the unit matrix and ${E}^{\pm }$ is the shift operator ${E}^{\pm }\chi (n)=\chi (n\pm 1)$. Here χ=χ(n, y, t; k) and k is the complex spectral parameter.
Suppose χ has the following asymptotic behavior$ \begin{eqnarray}\chi =I+{k}^{-1}{\chi }^{(1)}+{k}^{-2}{\chi }^{(2)}\,+\,\cdots ,\quad k\to \infty ,\end{eqnarray}$where ${\chi }^{(j)}={\chi }^{(j)}(n,y,t)$ is independent of k. Introduce two new functions$ \begin{eqnarray}\begin{array}{rcl}v(n) & = & {BQ}(n-1)-Q(n)B,\\ u(n) & = & {AQ}(n-1)-Q(n)A,\end{array}\end{eqnarray}$where$ \begin{eqnarray}Q(n)={\chi }^{(1)}(n).\end{eqnarray}$Here we omit the variables y, t for convenience. After a regularization procedure, we find the following linear system$ \begin{eqnarray}\begin{array}{rcl}{{\bf{D}}}_{t}\chi +B({\bf{T}}\chi )-B\chi +v(n)\chi & = & 0,\\ {{\bf{D}}}_{y}\chi +A({\bf{T}}\chi )-A\chi +u(n)\chi & = & 0,\end{array}\end{eqnarray}$or$ \begin{eqnarray}\begin{array}{c}\begin{array}{l}{\rm{i}}{\chi }_{t}+k\chi B+B({E}^{-}\chi )(1-k)\\ \,-\,B\chi +(Q(n)B-{BQ}(n-1))\chi =0,\\ {\rm{i}}{\chi }_{y}+k\chi A+A({E}^{-}\chi )(1-k)\\ \,-\,A\chi +(Q(n)A-{AQ}(n-1))\chi =0.\end{array}\end{array}\end{eqnarray}$Substituting (2.3) into (2.7), the term of $O({k}^{-1})$ of system (2.7) implies$ \begin{eqnarray}\begin{array}{c}\begin{array}{l}{{\rm{i}}Q}_{y}(n)+{\chi }^{(2)}(n)A-A{\chi }^{(2)}(n-1)\\ \,+\,A[Q(n-1)-Q(n)]+u(n)Q(n)=0,\\ {{\rm{i}}Q}_{t}(n)+{\chi }^{(2)}(n)B-B{\chi }^{(2)}(n-1)\\ \,+\,B[Q(n-1)-Q(n)]+v(n)Q(n)=0,\end{array}\end{array}\end{eqnarray}$in terms of (2.5). Consider the derivative of u(n) in (2.4) with respect to t and v(n) with respect to y, and eliminate χ(2) by subtracting the results, we arrive at$ \begin{eqnarray}\begin{array}{c}\begin{array}{l}{{\rm{i}}u}_{t}(n)-{{\rm{i}}v}_{y}(n)\\ \,=\,A({E}^{+}-1)Q(n-1)B-B({E}^{+}-1)Q(n-1)A\\ \,+\,{BQ}(n-1)Q(n)A-{AQ}(n-1)Q(n)B\\ \,+\,Q(n)[{AQ}(n)B-{BQ}(n)A]\\ \,+\,[{AQ}(n-1)B-{BQ}(n-1)A]Q(n-1).\end{array}\end{array}\end{eqnarray}$The elements of equation (2.9) give rise to the FD3W equations (1.1) and (1.2).
3. Dbar-problem and symmetry condition
To obtain the solution of the FD3W equations (1.1) and (1.2), we introduce the following nonlocal Dbar-problem$ \begin{eqnarray}\bar{\partial }\chi (k,\bar{k})=\iint \chi (z,\bar{z})R(z,\bar{z};k,\bar{k}){\rm{d}}z\wedge {\rm{d}}\bar{z},\end{eqnarray}$with canonical normalization condition, where $R(z,\bar{z};k,\bar{k})$ is the spectral transformation matrix. This problem is equivalent to the following integral equation$ \begin{eqnarray}\begin{array}{l}\chi (k,\bar{k})=I+\displaystyle \frac{1}{2\pi {\rm{i}}}\iint \displaystyle \frac{{\rm{d}}\mu \wedge {\rm{d}}\bar{\mu }}{\mu -k}\\ \quad \times \iint \chi (\lambda ,\bar{\lambda })R(\lambda ,\bar{\lambda };\mu ,\bar{\mu }){\rm{d}}\lambda \wedge {\rm{d}}\bar{\lambda }.\end{array}\end{eqnarray}$Here and after, we take the domain to be the entire complex plane.
For the FD3W equations (1.1) and (1.2), if χ is a solution of the Dbar-problem (3.1), one needs to introduce the variables n, y, t into the spectral transformation matrix and take$ \begin{eqnarray}\begin{array}{l}R(z,\bar{z};k,\bar{k};n,y,t)\\ \quad =\,{\psi }_{0}(z;n,y,t){R}_{0}(z,\bar{z};k,\bar{k}){\psi }_{0}^{-1}(k;n,y,t),\end{array}\end{eqnarray}$where$ \begin{eqnarray}{\psi }_{0}(k;n,y,t)=\displaystyle \frac{1}{{\left(1-k\right)}^{n}}\exp \{-{\rm{i}}k\theta \},\,\theta ={Ay}+{Bt},\end{eqnarray}$satisfies the following condition$ \begin{eqnarray}{\psi }_{0}^{\dagger }(\bar{k};n,y,t)=\displaystyle \frac{1}{{\left(1-k\right)}^{2n}}{\psi }_{0}^{-1}(k;n,y,t).\end{eqnarray}$Here ${R}_{0}(z,\bar{z};k,\bar{k})$ is independent of the variables n, y, t and satisfies the following symmetry condition [24]$ \begin{eqnarray}{\left(\displaystyle \frac{1-k}{1-z}\right)}^{2n}{R}_{0}^{\dagger }(\bar{z},z;k,\bar{k})={R}_{0}(k,\bar{k};z,\bar{z}).\end{eqnarray}$
Furthermore, if set $ \begin{eqnarray*}\begin{array}{rcl}{f}_{j}(\lambda ,\bar{\lambda };n,y,t) & = & {\psi }_{0}(\lambda ;n,y,t){f}_{j0}(\lambda ,\bar{\lambda }),\\ {g}_{j}(\mu ,\bar{\mu };n,y,t) & = & {g}_{j0}(\mu ,\bar{\mu }){\psi }_{0}^{-1}(\mu ;n,y,t),\end{array}\end{eqnarray*}$ then we have $ \begin{eqnarray}{f}_{j0}(\lambda ,\bar{\lambda })={\left(1-\lambda \right)}^{2n}{g}_{j0}^{\dagger }(\bar{\lambda },\lambda ).\end{eqnarray}$in terms of the symmetry condition (3.6).
To obtain the soliton solutions of the FD3W equations (1.1) and (1.2), we first choose$ \begin{eqnarray}\begin{array}{c}\begin{array}{rcl}{g}_{j}(\mu ,\bar{\mu };n,y,t) & = & {C}_{j}\exp \{{\rm{i}}\mu \theta \}{\left(1-\mu \right)}^{n}\delta (\mu -{k}_{j}),\\ {f}_{j}(\lambda ,\bar{\lambda };n,y,t) & = & \exp \{-{\rm{i}}\lambda \theta \}{C}_{j}^{\dagger }{\left(1-\lambda \right)}^{n}\delta (\lambda -{\bar{k}}_{j}),\end{array}\end{array}\end{eqnarray}$where θ is defined in (3.4) and Cj is a constant matrix. Substituting (4.6) into (4.4), we obtain$ \begin{eqnarray}{\xi }_{j}=-{{\rm{i}}Z}_{j}^{\dagger },\,{\eta }_{j}=-{{\rm{i}}Z}_{j},\,{M}_{{jl}}=\displaystyle \frac{1}{2\pi {\rm{i}}}\displaystyle \frac{{Z}_{j}{Z}_{l}^{\dagger }}{{k}_{j}-{\bar{k}}_{l}},\end{eqnarray}$where$ \begin{eqnarray}{Z}_{j}=2{C}_{j}{\left(1-{k}_{j}\right)}^{n}\exp \{{{\rm{i}}k}_{j}\theta \}.\end{eqnarray}$Note that ${\xi }_{j}=-{\eta }_{j}^{\dagger }$ and ${M}_{{lj}}^{\dagger }={M}_{{jl}}$, which imply that ${{\chi }^{(1)}}^{\dagger }=-{\chi }^{(1)}$ in view of (4.3).
In general, if choose the element of matrix gj and fj as$ \begin{eqnarray}\begin{array}{c}\begin{array}{rcl}{g}_{j}^{({pq})}(\mu ,\bar{\mu };n,y,t) & = & {C}_{j}^{({pq})}{{\rm{e}}}^{{\rm{i}}\mu {\theta }_{q}}{\left(1-\mu \right)}^{n}\delta (\mu -{k}_{j}^{({pq})}),\\ {f}_{j}^{({pq})}(\lambda ,\bar{\lambda };n,y,t) & = & {{\rm{e}}}^{-{\rm{i}}\lambda {\theta }_{p}}{\bar{C}}_{j}^{({qp})}{\left(1-\lambda \right)}^{n}\delta (\lambda -{\bar{k}}_{j}^{({qp})}),\end{array}\end{array}\,\end{eqnarray}$then we have (p, q=1, 2, 3)$ \begin{eqnarray}\begin{array}{c}\begin{array}{c}\begin{array}{rcl}{\eta }_{j}^{\left({pq}\right)} & = & -{{\rm{i}}Z}_{j}^{\left({pq}\right)},\,{\xi }_{j}^{\left({qp}\right)}=-{\rm{i}}{\bar{Z}}_{j}^{\left({qp}\right)},\\ {M}_{{jl}}^{\left({pq}\right)} & = & \displaystyle \frac{1}{2\pi {\rm{i}}}\displaystyle \sum _{r=1}^{3}\displaystyle \frac{{Z}_{j}^{\left({pr}\right)}{\bar{Z}}_{l}^{\left({qr}\right)}}{{k}_{j}^{\left({pr}\right)}-{\bar{k}}_{l}^{\left({qr}\right)}},\end{array}\end{array}\end{array}\end{eqnarray}$where ${\theta }_{q}={a}_{q}y+{b}_{q}t$ and$ \begin{eqnarray}{Z}_{j}^{({pq})}=2{C}_{j}^{({pq})}{{\rm{e}}}^{{{\rm{i}}k}_{j}^{({pq})}{\theta }_{q}}{\left(1-{k}_{j}^{({pq})}\right)}^{n}.\end{eqnarray}$
Thus, in both case of (4.6) and (4.9), the solution of the FD3W equations (1.1) and (1.2) takes the form$ \begin{eqnarray}\begin{array}{c}Q=\displaystyle \frac{1}{2\pi {\rm{i}}}{\check{Z}}^{\dagger }{\left(I+M\right)}^{-1}\check{Z},\,\check{Z}={\left(\begin{array}{c}{Z}_{1}\\ \vdots \\ {Z}_{N}\end{array}\right)}_{(3N)\times 3}.\end{array}\end{eqnarray}$
Furthermore, the elements in qth column of the matrix Zj are ${Z}_{j}^{(* q)}$, and the collection of these elements composes the qth column of the matrix $\check{Z}$, denoted by ${\check{Z}}^{(q)}$. Note that ${\check{Z}}^{(q)}$ is a (3N)×1 matrix. Using these notations, the element of the solution Q can be rewritten as$ \begin{eqnarray}\begin{array}{l}{Q}_{{pq}}=-\displaystyle \frac{1}{2\pi {\rm{i}}}\displaystyle \frac{\det {\left(I+M\right)}_{{pq}}^{(a)}}{\det (I+M)},\\ {\left(I+M\right)}_{{pq}}^{(a)}=\left(\begin{array}{cc}0 & {\left({\check{Z}}^{(p)}\right)}^{\dagger }\\ {\check{Z}}^{(q)} & (I+M)\end{array}\right).\end{array}\end{eqnarray}$We note that representation for higher-order solution can be found in [27, 28].
For the general case (4.9), If take $1-{k}_{j}^{({pq})}={{\rm{e}}}^{{\sigma }_{j}^{({pq})}+{\rm{i}}{\phi }_{j}^{({pq})}}$, then ${k}_{j}^{({pq})}=1-{{\rm{e}}}^{{\sigma }_{j}^{({pq})}}\cos {\phi }_{j}^{({pq})}-{\rm{i}}{{\rm{e}}}^{{\sigma }_{j}^{({pq})}}\sin {\phi }_{j}^{({pq})}$. In this case, ${Z}_{j}^{({pq})}$ in (4.11) can be denoted as$ \begin{eqnarray}\begin{array}{rcl}{Z}_{j}^{({pq})} & = & 2{{\rm{e}}}^{{X}_{j}^{({pq})}+{{\rm{i}}{T}}_{j}^{({pq})}},\\ {X}_{j}^{({pq})} & = & n{\sigma }_{j}^{({pq})}+{{\rm{e}}}^{{\sigma }_{j}^{({pq})}}\sin {\phi }_{j}^{({pq})}\cdot {\theta }_{q}+{X}_{j0}^{({pq})},\\ {T}_{j}^{({pq})} & = & n{\phi }_{j}^{({pq})}+(1-{{\rm{e}}}^{{\sigma }_{j}^{({pq})}}\cos {\phi }_{j}^{({pq})}){\theta }_{q}+{T}_{j0}^{({pq})}.\end{array}\end{eqnarray}$Here we denote ${C}_{j}^{\left({pq}\right)}={{\rm{e}}}^{{X}_{j0}^{\left({pq}\right)}+{{\rm{i}}T}_{j0}^{\left({pq}\right)}}$.
In particular, if we choose the constants ${C}_{j},(j=1,\,\cdots ,\,N)$ as row vectors, or ${C}_{j}=({C}_{j}^{(1)},{C}_{j}^{(2)},{C}_{j}^{(3)})$. In this case, the elements of the row vector gj will be$ \begin{eqnarray}{g}_{j}^{(q)}={C}_{j}^{(q)}{\left(1-k\right)}^{n}{{\rm{e}}}^{{\rm{i}}k{\theta }_{q}}\delta (k-{k}_{j}^{(q)}),\,q\,=\,1,2,3,\end{eqnarray}$which imply that the element ${Z}_{j}^{(q)}$ of the row vector Zj takes the same form as these in (4.14), but with the superscript (q) instead of (pq). In addition, the matrix M in (4.13) reduces to a N×N matrix, with element Mjl taking the following form$ \begin{eqnarray}\begin{array}{c}\begin{array}{l}{M}_{{jl}}=\displaystyle \frac{2}{{\rm{i}}\pi }\displaystyle \sum _{q=1}^{3}\displaystyle \frac{1}{{k}_{j}^{(q)}-{\bar{k}}_{l}^{(q)}}{{\rm{e}}}^{{\rm{i}}({T}_{j}^{(q)}-{T}_{l}^{(q)})}{{\rm{e}}}^{{X}_{j}^{(q)}+{X}_{l}^{(q)})},\\ \,(j,l=1,\,\cdots N),\end{array}\end{array}\end{eqnarray}$Here ${X}_{j}^{(q)}$ and ${T}_{j}^{(q)}$ have the same definitions as those in (4.14), but with the superscript (q) instead of (pq). We note that the discrete spectrum ${k}_{j}^{(q)},(q=1,2,3)$ can not be real, because the diagonal part of the matrix M is undefined.
For N=1, equation (4.13) reduces to the one soliton solution of the FD3W equations (1.1) and (1.2)$ \begin{eqnarray}{Q}_{{pq}}=\displaystyle \frac{2}{{\rm{i}}\pi }{{\rm{e}}}^{{\rm{i}}({T}_{1}^{(q)}-{T}_{1}^{(p)})}\displaystyle \frac{{{\rm{e}}}^{{X}_{1}^{(p)}+{X}_{1}^{(q)}}}{1\,+\,M},\,M=\displaystyle \frac{1}{\pi }\displaystyle \sum _{r=1}^{3}\displaystyle \frac{{{\rm{e}}}^{2{X}_{1}^{(r)}}}{{{\rm{e}}}^{{\sigma }_{1}^{(r)}}\sin {\phi }_{1}^{(r)}}.\end{eqnarray}$Note that, for the soliton solution, the forces fp in (1.4) are zero.
If the imaginary part of the discrete spectrum admits ${\mathfrak{I}}({k}_{1}^{(q)})\gt 0,(q=1,2,3)$, then $\sin ({\phi }_{1}^{(q)})\gt 0$. Now we introduce the notations$ \begin{eqnarray}\begin{array}{rcl}{\tau }_{1}^{(q)} & = & \displaystyle \frac{1}{2}({\sigma }_{1}^{(q)}+\mathrm{ln}(\pi \sin ({\phi }_{1}^{(q)}))),\\ {\hat{X}}^{(q)} & = & {X}_{1}^{(q)}-{\tau }_{1}^{(q)},\quad q=1,2,3.\end{array}\end{eqnarray}$The one soliton solution can be rewritten as$ \begin{eqnarray}{Q}_{{pq}}=\displaystyle \frac{1}{{\rm{i}}\pi }\displaystyle \frac{{{\rm{e}}}^{{\rm{i}}({T}_{1}^{(q)}-{T}_{1}^{(p)})}{{\rm{e}}}^{{X}^{(p)}+{X}^{(q)}}}{{{\rm{e}}}^{{\hat{X}}^{(p)}+{\hat{X}}^{(q)}}\cosh ({\hat{X}}^{(p)}-{\hat{X}}^{(q)})+{{\rm{e}}}^{{\hat{X}}^{(r)}}\cosh ({\hat{X}}^{(r)})},\end{eqnarray}$and$ \begin{eqnarray}{Q}_{{qq}}=\displaystyle \frac{1}{{\rm{i}}\pi }\displaystyle \frac{{{\rm{e}}}^{2{X}^{(q)}}}{{{\rm{e}}}^{{\hat{X}}^{(r)}+{\hat{X}}^{(p)}}\cosh ({\hat{X}}^{(r)}-{\hat{X}}^{(p)})+{{\rm{e}}}^{{\hat{X}}^{(q)}}\cosh ({\hat{X}}^{(q)})},\end{eqnarray}$where $1\leqslant p,q,r\leqslant 3$ and $r\ne p\ne q$. Since the relationship between the two set solutions (1.4) are not simple, we still discuss them separately. In fact, the two set soliton solutions take different asymptotic analysis. In deed, along the ‘line’ ${\hat{X}}^{(p)}-{\hat{X}}^{(q)}=0$, we find $ \begin{eqnarray*}\begin{array}{c}\begin{array}{rcl}{Q}_{{pq}} & \to & \left\{\begin{array}{cl}0, & {\hat{X}}^{(r)}\to +\infty ,\\ \tfrac{{{\rm{\Omega }}}_{{pq}}}{{{\rm{e}}}^{-({\hat{X}}^{(p)}+{\hat{X}}^{(q)})}+2}, & {\hat{X}}^{(r)}\to -\infty ,\end{array}\right.\\ {Q}_{{rr}} & \to & \left\{\begin{array}{cl}{\rm{const}}, & {\hat{X}}^{(r)}\to +\infty ,\\ 0, & {\hat{X}}^{(r)}\to -\infty ,\end{array}\right.\end{array}\end{array}\end{eqnarray*}$ where $1\leqslant p\lt q\leqslant 3,1\leqslant r\leqslant 3$ and $r\ne p,q$. Here ${{\rm{\Omega }}}_{{pq}}$ is a bounded function. If we further impose that ${\hat{X}}^{(p)}={\hat{X}}^{(q)}=c$ is a constant, then $| {Q}_{{pq}}| $ tends to a certain constant in the direction of ${\hat{X}}^{(r)}\to -\infty $. Figure 1 shows the soliton solutions Qpq in (4.19) and figure 2 shows Qqq in (4.20).
Figure 1.
New window|Download| PPT slide Figure 1.One-soliton solution Qpq at y=0 in (4.19) with the parameters chosen as ${\sigma }_{1}^{(1)}=0.1$, ${\phi }_{1}^{(1)}=\pi /6;{\sigma }_{1}^{(2)}=0.2$, ${\phi }_{1}^{(1)}=\pi /2;{\sigma }_{1}^{(3)}=0.3$, ${\phi }_{1}^{(3)}=2\pi /3$ and ${a}_{1}=0.2,{a}_{2}=0.4$, ${a}_{2}=0.6;{b}_{1}=0.3$, ${b}_{2}=0.5,{b}_{3}=0.7$.
Figure 2.
New window|Download| PPT slide Figure 2.One-soliton solution Qqq at y=0 in (4.20) with the parameters chosen as ${\sigma }_{1}^{(1)}=0.1,{\phi }_{1}^{(1)}=\pi /6;{\sigma }_{1}^{(2)}=0.2$, ${\phi }_{1}^{(1)}=\pi /2;{\sigma }_{1}^{(3)}=0.3$, ${\phi }_{1}^{(3)}=2\pi /3$ and ${a}_{1}=0.2,{a}_{2}=0.4$, ${a}_{2}=0.6;{b}_{1}=0.3$, ${b}_{2}=0.5,{b}_{3}=0.7$.
Similarly, if the discrete spectrum satisfies ${\mathfrak{I}}({k}_{1}^{(q)})\lt 0,(q=1,2,3)$, or $\sin ({\phi }_{1}^{(q)})\lt 0$. We introduce the notations$ \begin{eqnarray}\begin{array}{rcl}{\tilde{\tau }}_{1}^{(q)} & = & \displaystyle \frac{1}{2}({\sigma }_{1}^{(q)}+\mathrm{ln}(\pi | \sin ({\phi }_{1}^{(q)})| ),\\ {\tilde{X}}^{(q)} & = & {X}_{1}^{(q)}-{\tilde{\tau }}_{1}^{(q)},\quad q=1,2,3.\end{array}\end{eqnarray}$We have another type of one soliton solution$ \begin{eqnarray}{Q}_{{pq}}=\displaystyle \frac{-1}{{\rm{i}}\pi }\displaystyle \frac{{{\rm{e}}}^{{\rm{i}}({T}_{1}^{(q)}-{T}_{1}^{(p)})}{{\rm{e}}}^{{X}^{(p)}+{X}^{(q)}}}{{{\rm{e}}}^{{\tilde{X}}^{(p)}+{\tilde{X}}^{(q)}}\cosh ({\tilde{X}}^{(p)}-{\tilde{X}}^{(q)})+{{\rm{e}}}^{{\tilde{X}}^{(r)}}\sinh ({\tilde{X}}^{(r)})},\end{eqnarray}$and$ \begin{eqnarray}{Q}_{{qq}}=\displaystyle \frac{-1}{{\rm{i}}\pi }\displaystyle \frac{{{\rm{e}}}^{2{X}^{(q)}}}{{{\rm{e}}}^{{\tilde{X}}^{(r)}+{\tilde{X}}^{(p)}}\cosh ({\tilde{X}}^{(r)}-{\tilde{X}}^{(p)})+{{\rm{e}}}^{{\tilde{X}}^{(q)}}\sinh ({\tilde{X}}^{(q)})},\end{eqnarray}$where $1\leqslant p,q,r\leqslant 3$ and $r\ne p\ne q$. The asymptotic behaviors can be discussed similarly.
The other cases of the discrete spectrum can be discussed in the same way. For example, for $1\leqslant p,q,r\leqslant 3,p\lt q,{\mathfrak{I}}({k}_{1}^{(p)})\gt 0,{\mathfrak{I}}({k}_{1}^{(r)})\gt 0$ and ${\mathfrak{I}}({k}_{1}^{(q)})\lt 0$, we have$ \begin{eqnarray}{Q}_{{pq}}=\displaystyle \frac{1}{{\rm{i}}\pi }\displaystyle \frac{{{\rm{e}}}^{{\rm{i}}({T}_{1}^{(q)}-{T}_{1}^{(p)})}{{\rm{e}}}^{{X}^{(p)}+{X}^{(q)}}}{{{\rm{e}}}^{{\hat{X}}^{(p)}+{\tilde{X}}^{(q)}}\sinh ({\hat{X}}^{(p)}-{\tilde{X}}^{(q)})+{{\rm{e}}}^{{\hat{X}}^{(r)}}\cosh ({\hat{X}}^{(r)})},\end{eqnarray}$and$ \begin{eqnarray}{Q}_{{qq}}=\displaystyle \frac{1}{{\rm{i}}\pi }\displaystyle \frac{{{\rm{e}}}^{2{X}^{(q)}}}{{{\rm{e}}}^{{\hat{X}}^{(r)}+{\hat{X}}^{(p)}}\cosh ({\hat{X}}^{(r)}-{\hat{X}}^{(p)})-{{\rm{e}}}^{{\tilde{X}}^{(q)}}\sinh ({\tilde{X}}^{(q)})},\end{eqnarray}$where $1\leqslant p,q,r\leqslant 3$ and $r\ne p\ne q$. Here ${\tilde{\tau }}_{1}^{(q)},{\tilde{X}}^{(q)}$ are defined in (4.21) and ${\tau }_{1}^{(p)},{\hat{X}}^{(p)},{\hat{X}}^{(r)}$ are defined in (4.18).
For N=2 and for simplicity, we consider the case of (4.6), and choose ${g}_{j}(\mu ,\bar{\mu };n,y,t)=({g}_{j}^{(1)},{g}_{j}^{(2)},{g}_{j}^{(3)}),(j=1,2)$with $ \begin{eqnarray*}{g}_{j}^{(q)}={C}_{j}^{(q)}(1-\mu ){{\rm{e}}}^{{\rm{i}}\mu {\theta }_{q}}\delta (\mu -{k}_{j}),\,q=1,2,3,\end{eqnarray*}$ then${Z}_{j}=({Z}_{j}^{(1)},{Z}_{j}^{(2)},{Z}_{j}^{(3)})$with $ \begin{eqnarray*}\begin{array}{c}\begin{array}{rcl}{Z}_{j}^{(q)} & = & 2{{\rm{e}}}^{{X}_{j}^{(q)}+{{\rm{i}}T}_{j}^{(q)}},\\ {X}_{j}^{(q)} & = & n{\sigma }_{j}+{{\rm{e}}}^{{\sigma }_{j}}\sin ({\phi }_{j})\cdot {\theta }_{q}+{X}_{j0}^{(q)},\\ {T}_{j}^{(q)} & = & n{\phi }_{j}+(1-{{\rm{e}}}^{{\sigma }_{j}}\cos ({\phi }_{j})){\theta }_{q}+{T}_{j0}^{(q)},\end{array}\end{array}\end{eqnarray*}$ and $M=({M}_{{jl}}),(j,l=1,2)$ with $ \begin{eqnarray*}{M}_{{jl}}=\displaystyle \frac{2}{{\rm{i}}\pi }\displaystyle \frac{1}{{k}_{j}-\bar{{k}_{l}}}\displaystyle \sum _{q=1}^{3}{{\rm{e}}}^{{X}_{j}^{(q)}+{X}_{l}^{(q)}}{{\rm{e}}}^{{\rm{i}}({T}_{j}^{(q)}+{T}_{l}^{(q)})}.\end{eqnarray*}$ Here we have taken ${k}_{j}=1-{{\rm{e}}}^{{\sigma }_{j}}\cos ({\phi }_{j})-{\rm{i}}{{\rm{e}}}^{{\sigma }_{j}}\sin ({\phi }_{j})$. In this case, we have$ \begin{eqnarray}\begin{array}{rcl}\det (I+M) & = & 1+\displaystyle \frac{1}{\pi }\displaystyle \sum _{j=1}^{2}\displaystyle \frac{-1}{{\mathfrak{I}}({k}_{j})}\displaystyle \sum _{q=1}^{3}{{\rm{e}}}^{2{X}_{j}^{(q)}}\\ & & +\displaystyle \sum _{p\ne q}{{\rm{e}}}^{{X}_{1}^{(p)}+{X}_{2}^{(p)}+{X}_{1}^{(q)}+{X}_{2}^{(q)}}\\ & & \times [{a}_{12}\cosh ({X}_{1}^{(p)}+{X}_{2}^{(p)}-{X}_{1}^{(q)}-{X}_{2}^{(q)})\\ & & +{b}_{12}\cosh ({X}_{1}^{(p)}-{X}_{2}^{(p)}-{X}_{1}^{(q)}+{X}_{2}^{(q)})\\ & & -{c}_{12}\cos ({T}_{1}^{(p)}-{T}_{2}^{(p)}-{T}_{1}^{(q)}+{T}_{2}^{(q)})],\end{array}\end{eqnarray}$and$ \begin{eqnarray}\begin{array}{c}\begin{array}{c}\begin{array}{c}\begin{array}{c}\begin{array}{rcl}det{\left(I+M\right)}_{{pq}}^{\left(a\right)} & = & -4\displaystyle \sum _{j=1}^{2}{{\rm{e}}}^{{X}_{1}^{\left(p\right)}+{X}_{j}^{\left(q\right)}}{{\rm{e}}}^{{\rm{i}}\left({T}_{j}^{\left(q\right)}-{T}_{j}^{\left(p\right)}\right)}\\ & & -\displaystyle \frac{8}{{\rm{i}}\pi }\displaystyle \sum _{j,l=1}^{2}\displaystyle \frac{{\left(-1\right)}^{j+l}}{{k}_{j}-{\bar{k}}_{l}}{{\rm{e}}}^{{X}_{j}^{\left(p\right)}+{X}_{l}^{\left(q\right)}}{{\rm{e}}}^{{\rm{i}}\left({T}_{l}^{\left(q\right)}-{T}_{j}^{\left(p\right)}\right)}\\ & & \times \displaystyle \sum _{s=1}^{3}{{\rm{e}}}^{{X}_{j}^{\left(s\right)}+{X}_{l}^{\left(s\right)}}{{\rm{e}}}^{{\rm{i}}\left({T}_{j}^{\left(s\right)}-{T}_{l}^{\left(s\right)}\right)},\end{array}\end{array}\end{array}\end{array}\,\end{array}\end{eqnarray}$where ${a}_{12}=\tfrac{1}{2}({b}_{12}-{c}_{12})$ and $ \begin{eqnarray*}{b}_{12}=\displaystyle \frac{2}{{\pi }^{2}{\mathfrak{I}}({k}_{1}){\mathfrak{I}}({k}_{2})},\quad {c}_{12}=\displaystyle \frac{8}{{\pi }^{2}| {k}_{2}-{\bar{k}}_{1}{| }^{2}}.\end{eqnarray*}$ Then (4.13) with (4.26) and (4.27) gives the two-soliton solution to FD3W equation. A typical solution is shown in figures 3 and 4.
Figure 3.
New window|Download| PPT slide Figure 3.Two-soliton solution Qpq at y=0 in the case of (4.6) with the parameters chosen as ${\sigma }_{1}=0.1,{\phi }_{1}=\pi /6;{\sigma }_{2}=0.2,{\phi }_{2}=2\pi /3;$ and ${a}_{1}=0.2,{a}_{2}=0.4,{a}_{3}=0.6;{b}_{1}=0.3,{b}_{2}=0.5,{b}_{3}=0.7$.
Figure 4.
New window|Download| PPT slide Figure 4.Two-soliton solution Qqq at y=0 in the case of (4.6) with the parameters chosen as ${\sigma }_{1}=0.1,{\phi }_{1}=\pi /6;{\sigma }_{2}=0.2,{\phi }_{2}=2\pi /3$ and ${a}_{1}=0.2,{a}_{2}=0.4,{a}_{3}=0.6;{b}_{1}=0.3,{b}_{2}=0.5,{b}_{3}=0.7$.
For N=2 and in the case of (4.15), one can construct the two-soliton solution similarly. The wave profiles will be various, because of more discrete spectrums, see figures 5 and 6.
Figure 5.
New window|Download| PPT slide Figure 5.Two-soliton solution Qpq at y=0 in the case of (4.6) with the parameters chosen as ${\sigma }_{1}^{(1)}=0.1,{\phi }_{1}^{(1)}=\pi /6;{\sigma }_{1}^{(2)}=0.2$, ${\phi }_{1}^{(2)}=\pi /2;{\sigma }_{1}^{(3)}=0.3$, ${\phi }_{1}^{(3)}=2\pi /3;{\sigma }_{2}^{(1)}=0.4$, ${\phi }_{2}^{(1)}=\pi /4;{\sigma }_{2}^{(2)}=0.5$, ${\phi }_{2}^{(2)}=\pi /3;{\sigma }_{2}^{(3)}=0.6$, ${\phi }_{2}^{(3)}=\pi /2$ and ${a}_{1}=0.2,{a}_{2}=0.4$, ${a}_{3}=0.6;{b}_{1}=0.3$, ${b}_{2}=0.5,{b}_{3}=0.7$.
Figure 6.
New window|Download| PPT slide Figure 6.Two-soliton solution Qqq at y=0 in the case of (4.6) with the parameters chosen as ${\sigma }_{1}^{(1)}=0.1,{\phi }_{1}^{(1)}=\pi /6;{\sigma }_{1}^{(2)}=0.2$, ${\phi }_{1}^{(2)}=\pi /2;{\sigma }_{1}^{(3)}=0.3$, ${\phi }_{1}^{(3)}=2\pi /3;{\sigma }_{2}^{(1)}=0.4$, ${\phi }_{2}^{(1)}=\pi /4;{\sigma }_{2}^{(2)}=0.5$, ${\phi }_{2}^{(2)}=\pi /3;{\sigma }_{2}^{(3)}=0.6$, ${\phi }_{2}^{(3)}=\pi /2$ and ${a}_{1}=0.2,{a}_{2}=0.4$, ${a}_{3}=0.6;{b}_{1}=0.3$, ${b}_{2}=0.5,{b}_{3}=0.7$.
5. Remarks
If introduce a new matrix function $\tilde{Q}(n)$ by ${\chi }^{(1)}(n)={\rm{i}}\tilde{Q}(n)$, instead of (2.5), then we find, from equation (2.6), another form of (2+1)-dimensional discrete three wave equation$ \begin{eqnarray}\begin{array}{c}\begin{array}{l}{{\rm{\partial }}}_{t}[{a}_{q}{\tilde{Q}}_{{pq}}(n)-{a}_{p}{\tilde{Q}}_{{pq}}(n-1)]\\ \,-\,{{\rm{\partial }}}_{y}[{b}_{q}{\tilde{Q}}_{{pq}}(n)-{b}_{p}{\tilde{Q}}_{{pq}}(n-1)]\\ \,=\,{s}_{{pq}}[{\tilde{R}}_{p}(n){\tilde{Q}}_{{pq}}(n)-{\tilde{Q}}_{{pq}}(n-1){\tilde{R}}_{q}(n)]\\ \,\,+\,{{\rm{i}}s}_{{pq}}({E}^{-}-1){\tilde{Q}}_{{pq}}(n)-{s}_{{pq}}{\tilde{Q}}_{{pr}}(n-1){\tilde{Q}}_{{rq}}(n)\\ \,\,+\,{s}_{{rq}}{\tilde{Q}}_{{pr}}(n){\tilde{Q}}_{{rq}}(n)+{s}_{{pr}}{\tilde{Q}}_{{pr}}(n-1){\tilde{Q}}_{{rq}}(n-1),\\ \,r\ne p\ne q,\end{array}\end{array}\,\end{eqnarray}$$ \begin{eqnarray}\begin{array}{l}(1-{E}^{-})\left[{a}_{p}{\partial }_{t}{\tilde{Q}}_{{pp}}(n)-{b}_{p}{\partial }_{y}{\tilde{Q}}_{{pp}}(n)\right.\\ \quad \left.+\,\displaystyle \sum _{r\ne p}{s}_{{pr}}| {\tilde{Q}}_{{pr}}(n){| }^{2}\right]=0,\end{array}\end{eqnarray}$where ${\tilde{R}}_{p}(n)$ have the same definition as in (2.3). Note that ${({\chi }^{(1)})}^{\dagger }(n)=-{\chi }^{(1)}(n)$ implies the constraint ${\tilde{Q}}^{\dagger }(n)=\tilde{Q}(n)$, which was imposed in [25]. The N-soliton solution takes the following form$ \begin{eqnarray}\tilde{Q}=\displaystyle \frac{-1}{2\pi }{\check{Z}}^{\dagger }{\left(I+M\right)}^{-1}\check{Z},\end{eqnarray}$where $\check{Z}$ is defined in (4.12).
Acknowledgments
Project 11471295 was supported by the National Natural Science Foundation of China. The third author’s work is partially supported by the President’s Endowed Professorship program of the University of Texas system.
,3,41School of Mathematics and Statistics, 
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