Abstract We propose a systematic method to construct the Mel'nikov model of long-short wave interactions, which is a special case of the Kadomtsev-Petviashvili (KP) equation with self-consistent sources (KPSCS). We show details how the Cauchy matrix approach applies to Mel'nikov's model which is derived as a complex reduction of the KPSCS. As a new result we find that in the dispersion relation of a 1-soliton there is an arbitrary time-dependent function that has previously not reported in the literature about the Mel'nikov model. This function brings time variant velocity for the long wave and also governs the short-wave packet. The variety of interactions of waves resulting from the time-freedom in the dispersion relation is illustrated. Keywords：Mel'nikov model of long-short wave interaction;Cauchy matrix approach;self-consistent sources;KP equation
PDF (1064KB)MetadataMetricsRelated articlesExportEndNote|Ris|BibtexFavorite Cite this article Hong-Juan Tian, Da-Jun Zhang. Cauchy matrix structure of the Mel'nikov model of long-short wave interaction. Communications in Theoretical Physics, 2020, 72(12): 125006- doi:10.1088/1572-9494/abb7d4
1. Introduction
The following wave interaction model was first proposed and systematically studied by Mel'nikov [1-4]:$\begin{eqnarray}3{u}_{{yy}}-{u}_{{tx}}-{\left(3{u}^{2}+{u}_{{xx}}+8\delta | \varphi {| }^{2}\right)}_{{xx}}=0,\end{eqnarray}$$\begin{eqnarray}{\rm{i}}{\varphi }_{y}=u\varphi +{\varphi }_{{xx}},\end{eqnarray}$which describes the interaction of a long wave with a short-wave packet propagating on the $x,y$ plane at an angle to each other. In the above model u is the long wave amplitude, φ is the complex short-wave envelope, i is the imaginary unit and the parameter δ satisfies δ2=1. Now it is clear that (1.1) is a special case of the Kadomtsev-Petviashvili (KP) equation with self-consistent sources (KPSCS),$\begin{eqnarray}{u}_{{tx}}-3{u}_{{yy}}-{\left(3{u}^{2}+{u}_{{xx}}+8\delta \displaystyle \sum _{j=1}^{N}{\varphi }_{j}{\psi }_{j}\right)}_{{xx}}=0,\end{eqnarray}$$\begin{eqnarray}\begin{array}{rcl}{\varphi }_{j,y} & = & u{\varphi }_{j}+{\varphi }_{j,{xx}},\\ {\psi }_{j,y} & = & -u{\psi }_{j}-{\psi }_{j,{xx}},\,\,(j=1,2,\,\cdots \,,\,N^{\prime} ).\end{array}\end{eqnarray}$The link between equations (1.1) and (1.2) will be explained in section 2.3. It is also interesting that (1.1) can also be derived via a multi-scale expansion from the KP equation [5].
As a remarkable and typical structure of the solutions of Mel'nikov-type integrable equations with self-consistent sources, there is time-freedom in the dispersion relation. This can be understood from the following fact: the source term (e.g. ${\displaystyle \sum }_{j=1}^{N}{({\varphi }_{j}{\psi }_{j})}_{x}$) is a squared eigenfunction symmetry but restriction-free on time evolution. Such a time-freedom was found in 1986 [5] but not drawn attention to in early research [1-4, 6-8]. Later it was realized from the inverse scattering transform [9, 10] and bilinear approach [11-13]. Recently, by means of this typical feature, we developed the Cauchy matrix approach to construct the Yajima-Oikawa system [14]. Note that the system (1.1) was first solved by Mel'nikov in [2] using a direct method. The Cauchy matrix approach is also a direct method. It is proposed in [15] for constructing solutions of some quadrilateral lattice equations, and later in [16, 17] this approach is extended to a more general method. It has more advantages than Mel'nikov's direct method used in [2], in particular, it allows explicit expression for solutions corresponding to multiple discrete eigenvalues (cf. [14]).
The purpose of this paper is not only to show details of how the Cauchy matrix approach applies to equations (1.1) and (1.2), but also to investigate the variety of interactions of waves resulting from the time-freedom in dispersion relation and the resonance of short-wave packets.
The paper is organized as follows: in section 2 we derive the KPSCS (1.2) and the Mel'nikov model (1.1) from the Cauchy matrix approach. Then in section 3 the dynamics of obtained solutions are investigated. Finally, section 4 outlines concluding remarks.
2. Cauchy matrix approach
2.1. The KPSCS
In this subsection, as a demonstration of the Cauchy matrix approach to an integrable equation with self-consistent sources, we start from the Sylvester equation and dispersion relation to construct the KPSCS (1.2).
Consider the Sylvester equation$\begin{eqnarray}{\boldsymbol{LM}}+{\boldsymbol{MK}}={\boldsymbol{r}}{{\boldsymbol{s}}}^{{\rm{T}}},\end{eqnarray}$where ${\boldsymbol{M}}={({M}_{{ij}})}_{N\times N}\in {{\mathbb{C}}}_{N\times N}[x,y,t]$, i.e. each Mij is a function of (x, y, t), ${\boldsymbol{L}}$ and ${\boldsymbol{K}}$ are two constant matrices $\in {{\mathbb{C}}}_{N\times N}$,$\begin{eqnarray}{\boldsymbol{r}}={\left({r}_{1},{r}_{2},\cdots ,{r}_{N}\right)}^{{\rm{T}}},\,\,{\boldsymbol{s}}={\left({s}_{1},{s}_{2},\cdots ,{s}_{N}\right)}^{{\rm{T}}},\end{eqnarray}$are two column vectors in ${{\mathbb{C}}}_{N\times 1}[x,y,t]$. We suppose ${ \mathcal E }({\boldsymbol{L}})\bigcap { \mathcal E }(-{\boldsymbol{K}})=\varnothing $, where ${ \mathcal E }({\boldsymbol{A}})$ denotes the eigenvalue set ${\boldsymbol{A}}$. This can guarantee the Sylvester equation (2.1) has a unique solution ${\boldsymbol{M}}$ for given $({\boldsymbol{L}},{\boldsymbol{K}},{\boldsymbol{r}},{\boldsymbol{s}})$ (cf. [18]). Let us introduce dispersion relations,$\begin{eqnarray}{{\boldsymbol{r}}}_{x}={\boldsymbol{Lr}},\,{{\boldsymbol{s}}}_{x}={{\boldsymbol{K}}}^{{\rm{T}}}{\boldsymbol{s}},\end{eqnarray}$$\begin{eqnarray}{{\boldsymbol{r}}}_{y}=-{{\boldsymbol{L}}}^{2}{\boldsymbol{r}},\,{{\boldsymbol{s}}}_{y}={\left({{\boldsymbol{K}}}^{{\rm{T}}}\right)}^{2}{\boldsymbol{s}},\end{eqnarray}$$\begin{eqnarray}{{\boldsymbol{r}}}_{t}=4{{\boldsymbol{L}}}^{3}{\boldsymbol{r}}-\displaystyle \frac{1}{2}\delta \beta (t){\boldsymbol{r}},\,{{\boldsymbol{s}}}_{t}=4{\left({{\boldsymbol{K}}}^{{\rm{T}}}\right)}^{3}{\boldsymbol{s}}-\displaystyle \frac{1}{2}\delta {\beta }^{{\rm{T}}}(t){\boldsymbol{s}},\end{eqnarray}$where δ is a constant, $\beta (t)\in {{\mathbb{R}}}_{N\times N}[t]$ and commutates with ${\boldsymbol{L}}$ and ${\boldsymbol{K}}$, i.e.$\begin{eqnarray}\beta (t){\boldsymbol{L}}={\boldsymbol{L}}\beta (t),\,\beta (t){\boldsymbol{K}}={\boldsymbol{K}}\beta (t).\end{eqnarray}$
By virtue of the uniqueness of the solution of the Slyvester equation (2.1), we can find the following evolutions for ${\boldsymbol{M}}$.
When ${ \mathcal E }({\boldsymbol{L}})\bigcap { \mathcal E }(-{\boldsymbol{K}})=\varnothing $, the Slyvester equation (2.1) and dispersion relations (2.3) give rise to$\begin{eqnarray}{{\boldsymbol{M}}}_{x}={{\boldsymbol{rs}}}^{{\rm{T}}},\,\,{{\boldsymbol{M}}}_{y}=-{\boldsymbol{L}}{\boldsymbol{r}}{{\boldsymbol{s}}}^{{\rm{T}}}+{\boldsymbol{r}}{{\boldsymbol{s}}}^{{\rm{T}}}{\boldsymbol{K}},\end{eqnarray}$$\begin{eqnarray}\begin{array}{rcl}{{\boldsymbol{M}}}_{t} & = & 4({{\boldsymbol{L}}}^{2}{\boldsymbol{r}}{{\boldsymbol{s}}}^{{\rm{T}}}-{{\boldsymbol{Lrs}}}^{{\rm{T}}}{\boldsymbol{K}}+{\boldsymbol{r}}{{\boldsymbol{s}}}^{{\rm{T}}}{{\boldsymbol{K}}}^{2})\\ & & -\displaystyle \frac{\delta }{2}(\beta (t){\boldsymbol{M}}+{\boldsymbol{M}}\beta (t)).\end{array}\end{eqnarray}$
Differentiating the Sylvester equation (2.1) w.r.t. x and making use of the dispersion relations (2.3a), we have$\begin{eqnarray*}{\boldsymbol{L}}{{\boldsymbol{M}}}_{x}+{{\boldsymbol{M}}}_{x}{\boldsymbol{K}}={{\boldsymbol{r}}}_{x}{{\boldsymbol{s}}}^{{\rm{T}}}+{\boldsymbol{r}}{{\boldsymbol{s}}}_{x}^{{\rm{T}}}={\boldsymbol{Lr}}{{\boldsymbol{s}}}^{{\rm{T}}}+{\boldsymbol{r}}{{\boldsymbol{s}}}^{{\rm{T}}}{\boldsymbol{K}},\end{eqnarray*}$i.e.$\begin{eqnarray}{\boldsymbol{L}}({{\boldsymbol{M}}}_{x}-{\boldsymbol{r}}{{\boldsymbol{s}}}^{{\rm{T}}})+({{\boldsymbol{M}}}_{x}-{\boldsymbol{r}}{{\boldsymbol{s}}}^{{\rm{T}}}){\boldsymbol{K}}=0.\end{eqnarray}$When ${ \mathcal E }({\boldsymbol{L}})\bigcap { \mathcal E }(-{\boldsymbol{K}})=\varnothing $, the above equation has a unique zero solution ${{\boldsymbol{M}}}_{x}-{\boldsymbol{r}}{{\boldsymbol{s}}}^{{\rm{T}}}=0$, i.e. the first relation in (2.5a). The remaining relations can be similarly proved.
Next, let us introduce an infinite matrix ${\boldsymbol{S}}={({S}^{(i,j)})}_{\infty \times \infty }$, where$\begin{eqnarray}{S}^{(i,j)}={{\boldsymbol{s}}}^{{\rm{T}}}{{\boldsymbol{K}}}^{j}{\left({\boldsymbol{I}}+{\boldsymbol{M}}\right)}^{-1}{{\boldsymbol{L}}}^{i}{\boldsymbol{r}}={{\boldsymbol{s}}}^{{\rm{T}}}{{\boldsymbol{K}}}^{j}{{\boldsymbol{u}}}^{(i)}={{\boldsymbol{w}}}^{(j)}{{\boldsymbol{L}}}^{i}{\boldsymbol{r}}.\end{eqnarray}$Here ${{\boldsymbol{u}}}^{(i)}$ and ${{\boldsymbol{w}}}^{(j)}$ are auxiliary N-th order vectors defined as$\begin{eqnarray}{{\boldsymbol{u}}}^{(i)}={\left({\boldsymbol{I}}+{\boldsymbol{M}}\right)}^{-1}{{\boldsymbol{L}}}^{i}{\boldsymbol{r}},\,\,{{\boldsymbol{w}}}^{(j)}={{\boldsymbol{s}}}^{{\rm{T}}}{{\boldsymbol{K}}}^{j}{\left({\boldsymbol{I}}+{\boldsymbol{M}}\right)}^{-1}.\end{eqnarray}$We need to work out various evolutions of ${S}^{(i,j)}$. To achieve that, we make use of the above auxiliary vectors. First, rewriting ${{\boldsymbol{u}}}^{(i)}$ into$\begin{eqnarray*}({\boldsymbol{I}}+{\boldsymbol{M}}){{\boldsymbol{u}}}^{(i)}={{\boldsymbol{L}}}^{i}{\boldsymbol{r}},\end{eqnarray*}$and then taking derivative w.r.t. x, we get$\begin{eqnarray*}{{\boldsymbol{M}}}_{x}{{\boldsymbol{u}}}^{(i)}+({\boldsymbol{I}}+{\boldsymbol{M}}){{\boldsymbol{u}}}_{x}^{\left(i\right)}={{\boldsymbol{L}}}^{i}{{\boldsymbol{r}}}_{x}.\end{eqnarray*}$Making use of (2.5a) and (2.3a) yields$\begin{eqnarray*}\left({\boldsymbol{I}}+{\boldsymbol{M}}\right){{\boldsymbol{u}}}_{x}^{\left(i\right)}={{\boldsymbol{L}}}^{i+1}{\boldsymbol{r}}-{\boldsymbol{r}}{{\boldsymbol{s}}}^{{\rm{T}}}{{\boldsymbol{u}}}^{(i)}.\end{eqnarray*}$Then, multiplying ${\left({\boldsymbol{I}}+{\boldsymbol{M}}\right)}^{-1}$ from the left, we get$\begin{eqnarray}\begin{array}{rcl}{{\boldsymbol{u}}}_{x}^{(i)} & = & {\left({\boldsymbol{I}}+{\boldsymbol{M}}\right)}^{-1}{{\boldsymbol{L}}}^{i+1}{\boldsymbol{r}}-{\left({\boldsymbol{I}}+{\boldsymbol{M}}\right)}^{-1}{\boldsymbol{r}}{{\boldsymbol{s}}}^{{\rm{T}}}{{\boldsymbol{u}}}^{(i)}\\ & = & {{\boldsymbol{u}}}^{\left(i+1\right)}-{{\boldsymbol{u}}}^{\left(0\right)}{S}^{\left(i,0\right)}.\end{array}\end{eqnarray}$In the same manner, we have$\begin{eqnarray}{{\boldsymbol{u}}}_{y}^{(i)}=-{{\boldsymbol{u}}}^{(i+2)}+{S}^{(i,0)}{{\boldsymbol{u}}}^{(1)}-{S}^{(i,1)}{{\boldsymbol{u}}}^{(0)},\end{eqnarray}$$\begin{eqnarray}\begin{array}{rcl}{{\boldsymbol{u}}}_{t}^{(i)} & = & 4({{\boldsymbol{u}}}^{(i+3)}-{S}^{(i,0)}{{\boldsymbol{u}}}^{(2)}+{S}^{(i,1)}{{\boldsymbol{u}}}^{(1)}-{S}^{(i,2)}{{\boldsymbol{u}}}^{(0)})\\ & & +\displaystyle \frac{\delta }{2}\beta (t){{\boldsymbol{u}}}^{(i)}-\delta {\left({\boldsymbol{I}}+{\boldsymbol{M}}\right)}^{-1}\beta (t){{\boldsymbol{u}}}^{(i)},\end{array}\end{eqnarray}$$\begin{eqnarray}{{\boldsymbol{w}}}_{x}^{\left(j\right)}={{\boldsymbol{w}}}^{(j+1)}-{{\boldsymbol{w}}}^{(0)}{S}^{(0,j)},\end{eqnarray}$$\begin{eqnarray}{{\boldsymbol{w}}}_{y}^{\left(j\right)}={{\boldsymbol{w}}}^{(j+2)}-{{\boldsymbol{w}}}^{(1)}{S}^{(0,j)}+{{\boldsymbol{w}}}^{(0)}{S}^{(1,j)}.\end{eqnarray}$Now, for ${S}^{(i,j)}$ defined in (2.7), we have$\begin{eqnarray*}\begin{array}{rcl}{S}_{x}^{\left(i,j\right)} & = & {{\boldsymbol{s}}}_{x}^{{\rm{T}}}{{\boldsymbol{K}}}^{j}{{\boldsymbol{u}}}^{(i)}+{{\boldsymbol{s}}}^{{\rm{T}}}{{\boldsymbol{K}}}^{j}{{\boldsymbol{u}}}_{x}^{\left(i\right)}\\ & = & {{\boldsymbol{s}}}^{{\rm{T}}}{{\boldsymbol{K}}}^{j+1}{{\boldsymbol{u}}}^{(i)}+{{\boldsymbol{s}}}^{{\rm{T}}}{{\boldsymbol{K}}}^{j}({{\boldsymbol{u}}}^{(i+1)}-{{\boldsymbol{u}}}^{(0)}{S}^{(i,0)}),\end{array}\end{eqnarray*}$which yields$\begin{eqnarray}{S}_{x}^{\left(i,j\right)}={S}^{(i,j+1)}+{S}^{(i+1,j)}-{S}^{(0,j)}{S}^{(i,0)}.\end{eqnarray}$In a similar way we can calculate derivatives of ${S}^{(i,j)}$ w.r.t. y and t, and we get$\begin{eqnarray}{S}_{y}^{\left(i,j\right)}={S}^{(i,j+2)}-{S}^{(i+2,j)}+{S}^{(1,j)}{S}^{(i,0)}-{S}^{(0,j)}{S}^{(i,1)},\end{eqnarray}$$\begin{eqnarray}\begin{array}{rcl}{S}_{t}^{\left(i,j\right)} & = & 4({S}^{(i+3,j)}+{S}^{(i,j+3)}-{S}^{(i,0)}{S}^{(2,j)}\\ & & +{S}^{(i,1)}{S}^{(1,j)}-{S}^{(i,2)}{S}^{(0,j)})\\ & & -\delta {{\boldsymbol{w}}}^{(j)}[\beta (t)]{{\boldsymbol{u}}}^{(i)}.\end{array}\end{eqnarray}$Higher-order x-derivatives can be obtained by repeatedly using relation (2.10a):$\begin{eqnarray}\begin{array}{rcl}{S}_{{xx}}^{\left(i,j\right)} & = & {S}^{(i,j+2)}+{S}^{(i+2,j)}-{S}^{(1,j)}{S}^{(i,0)}-{S}^{(0,j)}{S}^{(i,1)}\\ & & +2\left({S}^{(i+1,j+1)}-{S}^{(0,j)}{S}^{(i+\mathrm{1,0})}\right.\\ & & \left.-{S}^{(0,j+1)}{S}^{(i,0)}+{S}^{(i,0)}{S}^{(\mathrm{0,0})}{S}^{(0,j)}\right),\end{array}\end{eqnarray}$$\begin{eqnarray}\begin{array}{rcl}{S}_{{xxx}}^{\left(i,j\right)} & = & {S}^{(i+3,j)}+{S}^{(i,j+3)}+3{S}^{(i+2,j+1)}+3{S}^{(i+1,j+2)}\\ & & -3{S}^{(i+\mathrm{2,0})}{S}^{(0,j)}-3{S}^{(i,0)}{S}^{(0,j+2)}\\ & & -6{S}^{(i+\mathrm{1,0})}{S}^{(0,j+1)}-3{S}^{(i+\mathrm{1,0})}{S}^{(1,j)}\\ & & -3{S}^{(i,1)}{S}^{(0,j+1)}-{S}^{(i,2)}{S}^{(0,j)}-{S}^{(i,0)}{S}^{(2,j)}\\ & & -3{S}^{(i+\mathrm{1,1})}{S}^{(0,j)}-3{S}^{(i,0)}{S}^{(1,j+1)}\\ & & +6{S}^{(i+\mathrm{1,0})}{S}^{(\mathrm{0,0})}{S}^{(0,j)}+6{S}^{(i,0)}{S}^{(\mathrm{0,0})}{S}^{(0,j+1)}\\ & & -2{S}^{(i,1)}{S}^{(1,j)}+3{S}^{(i,0)}{S}^{(\mathrm{0,0})}{S}^{(1,j)}\\ & & +3{S}^{(i,0)}{S}^{(\mathrm{1,0})}{S}^{(0,j)}+3{S}^{(i,0)}{S}^{(\mathrm{0,1})}{S}^{(0,j)}\\ & & +3{S}^{(i,1)}{S}^{(\mathrm{0,0})}{S}^{(0,j)}-6{S}^{(i,0)}{{S}^{(\mathrm{0,0})}}^{2}{S}^{(0,j)}.\end{array}\end{eqnarray}$To derive ${S}_{{yy}}^{\left(i,j\right)}$, we first look at (2.10a), from which we have$\begin{eqnarray}{S}^{(i,j+2)}={S}_{x}^{(i,j+1)}-{S}^{(i+1,j+1)}+{S}^{(0,j+1)}{S}^{(i,0)},\end{eqnarray}$$\begin{eqnarray}{S}^{(i+2,j)}={S}_{x}^{(i+1,j)}-{S}^{(i+1,j+1)}+{S}^{(0,j)}{S}^{(i+\mathrm{1,0})},\end{eqnarray}$which lead to the following relation$\begin{eqnarray}\begin{array}{l}{S}^{(i+2,j)}-{S}^{(i,j+2)}={\partial }_{x}({S}^{(i+1,j)}\\ \quad -\,{S}^{(i,j+1)})+{S}^{(0,j)}{S}^{(i+\mathrm{1,0})}-{S}^{(0,j+1)}{S}^{(i,0)}.\end{array}\end{eqnarray}$Meanwhile, from (2.10a) we also have$\begin{eqnarray}{S}^{(0,j+1)}={S}_{x}^{(0,j)}-{S}^{(1,j)}+{S}^{(0,j)}{S}^{(\mathrm{0,0})},\end{eqnarray}$$\begin{eqnarray}{S}^{(i+\mathrm{1,0})}={S}_{x}^{(i,0)}-{S}^{(i,1)}+{S}^{(\mathrm{0,0})}{S}^{(i,0)}.\end{eqnarray}$Substituting these relations into (2.10b) yields$\begin{eqnarray*}{\partial }^{-1}{S}_{y}^{\left(i,j\right)}={S}^{(i+1,j)}-{S}^{(i,j+1)}+{\partial }^{-1}({S}_{x}^{(0,j)}{S}^{(i,0)}-{S}_{x}^{(i,0)}{S}^{(0,j)}),\end{eqnarray*}$where ${\partial }^{-1}{\partial }_{x}={\partial }_{x}{\partial }^{-1}=1$ and we always suppose ${S}^{(i,j)}\to 0$ as $x\to \pm \infty $. Next, taking the derivative of ${\partial }^{-1}{S}_{y}^{\left(i,j\right)}$ w.r.t. y and making use of (2.10b) and (2.14), we get$\begin{eqnarray}\begin{array}{rcl}{\partial }^{-1}{S}_{{yy}}^{\left(i,j\right)} & = & {S}^{(i+3,j)}+{S}^{(i,j+3)}-{S}^{(i+2,j+1)}\\ & & -{S}^{(i+1,j+2)}+{S}^{(0,j)}{S}^{(i+\mathrm{1,1})}\\ & & -{S}^{(0,j+1)}{S}^{(i,1)}-{S}^{(1,j)}{S}^{(i+\mathrm{1,0})}+{S}^{(1,j+1)}{S}^{(i,0)}\\ & & +{\partial }^{-1}{\partial }_{y}({S}_{x}^{(0,j)}{S}^{(i,0)}-{S}_{x}^{(i,0)}{S}^{(0,j)}).\end{array}\end{eqnarray}$
Now it is time to derive the KPSCS (1.2). Defining$\begin{eqnarray}v={S}^{(\mathrm{0,0})}\end{eqnarray}$and taking $i=j=0$ in the above derivatives (2.10a), (2.10c), (2.11b) and (2.15), we have$\begin{eqnarray}{v}_{x}={S}^{(\mathrm{0,1})}+{S}^{(\mathrm{0,1})}-{v}^{2},\end{eqnarray}$$\begin{eqnarray}\begin{array}{rcl}{v}_{t} & = & 4({S}^{(\mathrm{3,0})}+{S}^{(\mathrm{0,3})}-{{vS}}^{(\mathrm{2,0})}-{{vS}}^{(\mathrm{0,2})}+{S}^{(\mathrm{0,1})}{S}^{(\mathrm{1,0})})\\ & & -\delta {{\boldsymbol{w}}}^{(0)}\beta (t){{\boldsymbol{u}}}^{(0)},\end{array}\end{eqnarray}$$\begin{eqnarray}\begin{array}{rcl}{v}_{{xxx}} & = & {S}^{(\mathrm{0,3})}+{S}^{(\mathrm{3,0})}+3({S}^{(\mathrm{1,2})}+{S}^{(\mathrm{2,1})})\\ & & -4u({S}^{(\mathrm{2,0})}+{S}^{(\mathrm{0,2})})-8{S}^{(\mathrm{1,0})}{S}^{(\mathrm{0,1})}\\ & & -3({{S}^{(\mathrm{1,0})}}^{2}+{{S}^{(\mathrm{0,1})}}^{2})-6{{vS}}^{(\mathrm{1,1})}\\ & & +12{v}^{2}({S}^{(\mathrm{1,0})}+{S}^{(\mathrm{0,1})})-6{v}^{4},\end{array}\end{eqnarray}$$\begin{eqnarray}\begin{array}{rcl}{\partial }^{-1}{v}_{{yy}} & = & {S}^{(\mathrm{3,0})}+{S}^{(\mathrm{0,3})}-{S}^{(\mathrm{2,1})}-{S}^{(\mathrm{1,2})}\\ & & +2{{vS}}^{(\mathrm{1,1})}-{{S}^{(\mathrm{1,0})}}^{2}-{{S}^{(\mathrm{0,1})}}^{2}.\end{array}\end{eqnarray}$These relations immediately give rise to a potential KP equation with source$\begin{eqnarray}{v}_{t}-{v}_{{xxx}}-6{{v}_{x}}^{2}-3{\partial }^{-1}{v}_{{yy}}=\delta {{\boldsymbol{w}}}^{(0)}\beta (t){{\boldsymbol{u}}}^{(0)}.\end{eqnarray}$Introduce$\begin{eqnarray}\begin{array}{rcl}{\rm{\Phi }} & = & \displaystyle \frac{1}{2}{{\boldsymbol{w}}}^{(0)}\sqrt{\beta (t)}=({\phi }_{1},{\phi }_{2},\cdots ,{\phi }_{N}),\\ {\rm{\Psi }} & = & \displaystyle \frac{1}{2}\sqrt{\beta (t)}{{\boldsymbol{u}}}^{(0)}={\left({\psi }_{1},{\psi }_{2},\cdots ,{\psi }_{N}\right)}^{{\rm{T}}},\end{array}\end{eqnarray}$where$\begin{eqnarray}{\phi }_{j}=\displaystyle \frac{1}{2}{\left({{\boldsymbol{w}}}^{(0)}\sqrt{\beta (t)}\right)}_{j},\,\,{\psi }_{j}=\displaystyle \frac{1}{2}{\left(\sqrt{\beta (t)}{{\boldsymbol{u}}}^{(0)}\right)}_{j},\end{eqnarray}$and ${(\cdot )}_{j}$ stands for the j-th element of the vector $(\cdot )$. With the above notations, we can rewrite (2.18) together with (2.9b) and (2.9e) as$\begin{eqnarray}{v}_{{t}_{3}}-{v}_{{xxx}}-6{{v}_{x}}^{2}-3{\partial }^{-1}{v}_{{yy}}=4\delta {\rm{\Phi }}{\rm{\Psi }},\end{eqnarray}$$\begin{eqnarray}{{\rm{\Phi }}}_{y}={{\rm{\Phi }}}_{{xx}}+2{v}_{x}{\rm{\Phi }},\end{eqnarray}$$\begin{eqnarray}-{{\rm{\Psi }}}_{y}={{\rm{\Psi }}}_{{xx}}+2{v}_{x}{\rm{\Psi }}.\end{eqnarray}$This is a potential version of the KPSCS (1.2). Taking $u=2{v}_{x}$, we get the KPSCS (1.2).
2.2. Explicit solutions to the KPSCS (1.2)
Next, we solve the Sylvester equation (2.1) and dispersion relation (2.3) for given $({\boldsymbol{L}},{\boldsymbol{K}},\beta (t))$, so that we can provide explicit solutions to the KPSCS (1.2).
For convenience, we only consider ${\boldsymbol{L}}$ and ${\boldsymbol{K}}$ in their canonical forms We define the following plane wave factors,$\begin{eqnarray}\begin{array}{rcl}{\rho }_{i} & = & {{\rm{e}}}^{{\eta }_{i}},\,\,{\eta }_{i}={l}_{i}x-{l}_{i}^{2}y+4{l}_{i}^{3}t\\ & & -\displaystyle \frac{1}{2}\delta {\displaystyle \int }_{0}^{t}{\beta }_{i}(z){\rm{d}}z+{\eta }_{i}^{(0)},\,{l}_{i},{\eta }_{i}^{(0)}\in {\mathbb{C}},\end{array}\end{eqnarray}$$\begin{eqnarray}\begin{array}{rcl}{\sigma }_{i} & = & {{\rm{e}}}^{{\xi }_{i}},\,\,{\xi }_{i}={k}_{i}x+{k}_{i}^{2}y+4{k}_{i}^{3}t\\ & & -\displaystyle \frac{1}{2}\delta {\displaystyle \int }_{0}^{t}{\beta }_{i}(z){\rm{d}}z+{\xi }_{i}^{(0)},\,{k}_{i},{\xi }_{i}^{(0)}\in {\mathbb{C}},\end{array}\end{eqnarray}$where ${\beta }_{i}(z)$ is an arbitrary function of z; define an N×N diagonal matrix$\begin{eqnarray}{{\boldsymbol{D}}}^{[N]}({\{{a}_{i}\}}_{1}^{N})=\mathrm{Diag}\{{a}_{1},\cdots ,{a}_{N}\},\end{eqnarray}$and a special N×N skew diagonal matrix$\begin{eqnarray}{{\boldsymbol{H}}}_{0}^{[N]}={\left(\begin{array}{cccc}0 & \cdots & 0 & 1\\ 0 & \cdots & 1 & 0\\ \vdots & \cdots & \vdots & \vdots \\ 1 & \cdots & 0 & 0\end{array}\right)}_{N\times N}.\end{eqnarray}$Introduce more matrix notations$\begin{eqnarray}{{\boldsymbol{G}}}_{{DJ}}^{\left[N;N^{\prime} \right]}({\{{l}_{i}\}}_{1}^{N};b)=-{\left(\displaystyle \frac{{\left(-1\right)}^{j}}{{\left({l}_{i}+b\right)}^{j}}\right)}_{N\times N^{\prime} },\end{eqnarray}$$\begin{eqnarray}{{\boldsymbol{G}}}_{{JD}}^{\left[N;N^{\prime} \right]}(a;{\{{k}_{j}\}}_{1}^{N^{\prime} })=-{\left(\displaystyle \frac{{\left(-1\right)}^{i}}{{\left(a+{k}_{j}\right)}^{i}}\right)}_{N\times N^{\prime} },\end{eqnarray}$$\begin{eqnarray}\begin{array}{rcl}{{\boldsymbol{G}}}_{{JJ}}^{\left[N;N^{\prime} \right]}(a;b) & = & {\left({g}_{i,j}\right)}_{N\times N^{\prime} },\,{g}_{i,j}={{\rm{C}}}_{i+j-2}^{i-1}\displaystyle \frac{{\left(-1\right)}^{i+j}}{{\left(a+b\right)}^{i+j-1}},\\ \,{{\rm{C}}}_{j}^{i} & = & \displaystyle \frac{j!}{i!(j-i)!},\end{array}\end{eqnarray}$the lower triangular Toeplitz matrix$\begin{eqnarray}{{\boldsymbol{T}}}^{[N]}({\{{a}_{i}\}}_{1}^{N})={\left(\begin{array}{cccccc}{a}_{1} & 0 & 0 & \cdots & 0 & 0\\ {a}_{2} & {a}_{1} & 0 & \cdots & 0 & 0\\ \vdots & \vdots & \cdots & \vdots & \vdots & \vdots \\ {a}_{N} & {a}_{N-1} & {a}_{N-2} & \cdots & {a}_{2} & {a}_{1}\end{array}\right)}_{N\times N},\end{eqnarray}$and the skew triangular Toeplitz matrix$\begin{eqnarray}{{\boldsymbol{H}}}^{[N]}({\{{b}_{j}\}}_{1}^{N})={\left(\begin{array}{ccccc}{b}_{1} & \cdots & {b}_{N-2} & {b}_{N-1} & {b}_{N}\\ {b}_{2} & \cdots & {b}_{N-1} & {b}_{N} & 0\\ \vdots & \vdots & \vdots & \vdots & \vdots \\ {b}_{N} & \cdots & 0 & 0 & 0\end{array}\right)}_{N\times N}.\end{eqnarray}$Note that the standard Jordan matrix ${{\boldsymbol{J}}}^{[N]}(a)$ can be expressed by ${{\boldsymbol{T}}}^{[N]}({\{{a}_{i}\}}_{1}^{N})$ with ${a}_{1}=a,{a}_{2}=1,{a}_{3}=\cdots ={a}_{N}=0$.
In the following, we list out three cases of solutions to (2.1) and (2.3).
Case A. When both ${\boldsymbol{L}}$ and ${\boldsymbol{K}}$ are diagonal,$\begin{eqnarray}\begin{array}{rcl}{\boldsymbol{L}} & = & {{\boldsymbol{D}}}^{[N]}({\{{l}_{i}\}}_{1}^{N}),\,\,{\boldsymbol{K}}={{\boldsymbol{D}}}^{[N]}({\{{k}_{i}\}}_{1}^{N}),\\ \beta (t) & = & {{\boldsymbol{D}}}^{[N]}({\{{\beta }_{i}(t)\}}_{1}^{N}),\end{array}\end{eqnarray}$where ${\beta }_{i}(t)$ is an arbitrary function of t, the vectors ${\boldsymbol{r}}$ and ${\boldsymbol{s}}$ in (2.2) are composed by$\begin{eqnarray}{r}_{i}={\rho }_{i},\,\,{s}_{j}={\sigma }_{j},\end{eqnarray}$respectively. Let us denote them as ${\boldsymbol{r}}={{\boldsymbol{r}}}_{D}^{[N]}({\{{l}_{i}\}}_{1}^{N})$ and ${\boldsymbol{s}}={{\boldsymbol{s}}}_{D}^{[N]}({\{{k}_{i}\}}_{1}^{N})$. The dressed Cauchy matrix of this case is given by$\begin{eqnarray}{\boldsymbol{M}}={\left(\displaystyle \frac{{r}_{i}{s}_{j}}{{l}_{i}+{k}_{j}}\right)}_{N\times N}.\end{eqnarray}$
Case B.When both ${\boldsymbol{L}}$ and ${\boldsymbol{K}}$ are Jordan blocks,$\begin{eqnarray}{\boldsymbol{L}}={{\boldsymbol{J}}}^{[N]}({l}_{1}),\,\,{\boldsymbol{K}}={{\boldsymbol{J}}}^{[N]}({k}_{1}),\,\,\beta (t)={\beta }_{1}(t){{\boldsymbol{I}}}_{N},\end{eqnarray}$where ${{\boldsymbol{I}}}_{N}$ is the N-th order unit matrix, ${\boldsymbol{r}}$ and ${\boldsymbol{s}}$ in (2.2) are composed by$\begin{eqnarray}{r}_{i}=\displaystyle \frac{{\partial }_{{l}_{1}}^{i-1}{\rho }_{1}}{(i-1)!},\,\,{s}_{j}=\displaystyle \frac{{\partial }_{{k}_{1}}^{N-j}{\sigma }_{1}}{(N-j)!},\end{eqnarray}$respectively, and we denote them as ${\boldsymbol{r}}={{\boldsymbol{r}}}_{J}^{[N]}({l}_{1})$ and ${\boldsymbol{s}}={{\boldsymbol{s}}}_{J}^{[N]}({k}_{1})$. The dressed Cauchy matrix is given by$\begin{eqnarray}{\boldsymbol{M}}={\boldsymbol{F}}{\boldsymbol{G}}{\boldsymbol{H}},\end{eqnarray}$where$\begin{eqnarray}{\boldsymbol{F}}={{\boldsymbol{T}}}^{[N]}({\{{r}_{i}\}}_{1}^{N}),\,\,{\boldsymbol{G}}={{\boldsymbol{G}}}_{{JJ}}^{\left[N;N\right]}({l}_{1};{k}_{1}),\,{\boldsymbol{H}}={{\boldsymbol{H}}}^{[N]}({\{{s}_{j}\}}_{1}^{N}).\end{eqnarray}$
In addition, one can check case by case that the commutative condition (2.4) holds.
2.3. Reduction to (1.1)
In this subsection we reduce the above solutions of the KPSCS (1.2) to the solutions of the Mel'nikov model (1.1).
First, consider coordinate transformation$\begin{eqnarray*}(x,y,t)\to (x,{\rm{i}}y^{\prime} ,-t^{\prime} ),\end{eqnarray*}$by which the potential KPSCS (2.20) is written as$\begin{eqnarray}3{v}_{y^{\prime} y^{\prime} }-{v}_{t^{\prime} x}-{\left(3{v}_{x}^{2}+{v}_{{xxx}}\right)}_{x}=4\delta {\left({\rm{\Phi }}{\rm{\Psi }}\right)}_{x},\end{eqnarray}$$\begin{eqnarray}{\rm{i}}{{\rm{\Phi }}}_{y^{\prime} }={{\rm{\Phi }}}_{{xx}}+2{v}_{x}{\rm{\Phi }},\,\,{\rm{i}}{{\rm{\Psi }}}_{y^{\prime} }=-{{\rm{\Psi }}}_{{xx}}-2{v}_{x}{\rm{\Psi }}.\end{eqnarray}$Then we impose constraints on Φ, &PSgr; and v by$\begin{eqnarray}{\rm{\Psi }}={\boldsymbol{ \mathcal T }}{{\rm{\Phi }}}^{\ast {\rm{T}}}={\boldsymbol{ \mathcal T }}{{\rm{\Phi }}}^{\dagger },\,\,v={v}^{\ast },\end{eqnarray}$where ∗ stands for complex conjugate and ${\boldsymbol{T}}={{\boldsymbol{T}}}^{\dagger }$ is a constant Hermitian matrix. Under (2.37) we reduce (2.36) to$\begin{eqnarray}3{v}_{y^{\prime} y^{\prime} }-{v}_{t^{\prime} x}-{\left(3{v}_{x}^{2}+{v}_{{xxx}}\right)}_{x}=4\delta {\left({\rm{\Phi }}{\boldsymbol{T}}{{\rm{\Phi }}}^{\dagger }\right)}_{x},\end{eqnarray}$$\begin{eqnarray}{\rm{i}}{{\rm{\Phi }}}_{y^{\prime} }={{\rm{\Phi }}}_{{xx}}+2{v}_{x}{\rm{\Phi }},\end{eqnarray}$which is a generalized potential version of (1.1).
Next, to agree with (2.37), we take$\begin{eqnarray}{l}_{j}={k}_{j}^{* },\,\,{\eta }_{j}^{0}={\xi }_{j}^{0* },\,\,{\beta }_{j}(z)\in {\mathbb{R}}[z],\,\,j=1,2,\cdots ,N,\end{eqnarray}$under which it turns out that ${\boldsymbol{K}}={{\boldsymbol{L}}}^{* }$ and the plain wave factors in (2.21) satisfy$\begin{eqnarray}{\eta }_{j}={\xi }_{j}^{* }\end{eqnarray}$in coordinates $(x,y^{\prime} ,t^{\prime} )$. In the following let us check that, case by case, all the solutions we obtained in the above subsection can agree with the reduction condition (2.37).
Case A.In this case, with (2.39) and (2.40), it is easy to find ${\boldsymbol{s}}={{\boldsymbol{r}}}^{* }$ and ${\boldsymbol{M}}={{\boldsymbol{M}}}^{\dagger }$ from (2.29). Then it follows that$\begin{eqnarray*}\begin{array}{rcl}v & = & {{\boldsymbol{s}}}^{{\rm{T}}}{\left({\boldsymbol{I}}+{\boldsymbol{M}}\right)}^{-1}{\boldsymbol{r}}\\ & = & {{\boldsymbol{r}}}^{\dagger }{\left({\boldsymbol{I}}+{{\boldsymbol{M}}}^{\dagger }\right)}^{-1}{{\boldsymbol{s}}}^{* }\\ & = & {{\boldsymbol{s}}}^{\dagger }{\left({\boldsymbol{I}}+{{\boldsymbol{M}}}^{* }\right)}^{-1}{{\boldsymbol{r}}}^{* }\\ & = & {v}^{* },\end{array}\end{eqnarray*}$and$\begin{eqnarray*}\begin{array}{rcl}{\rm{\Psi }} & = & \displaystyle \frac{1}{2}\sqrt{\beta (t)}{\left({\boldsymbol{I}}+{\boldsymbol{M}}\right)}^{-1}{\boldsymbol{r}}\\ & = & \displaystyle \frac{1}{2}\sqrt{\beta (t)}{\left({\boldsymbol{I}}+{{\boldsymbol{M}}}^{\dagger }\right)}^{-1}{{\boldsymbol{s}}}^{* }\\ & = & \displaystyle \frac{1}{2}{\left[{{\boldsymbol{s}}}^{\dagger }{\left({\boldsymbol{I}}+{{\boldsymbol{M}}}^{* }\right)}^{-1}\right]}^{{\rm{T}}}\\ & = & {{\rm{\Phi }}}^{\dagger }.\end{array}\end{eqnarray*}$Note that in this case ${\boldsymbol{T}}={{\boldsymbol{T}}}_{1}\doteq {{\boldsymbol{I}}}_{N}$ in (2.37).
Case B.In this case, under (2.39) we find sj and rj defined in (2.31) satisfy ${s}_{j}={r}_{N-j+1}^{* }$, which indicates$\begin{eqnarray}{\boldsymbol{r}}={\boldsymbol{T}}{{\boldsymbol{s}}}^{* },\end{eqnarray}$where ${\boldsymbol{T}}={{\boldsymbol{T}}}_{2}\doteq {{\boldsymbol{H}}}_{0}^{[N]}$. Meanwhile, from (2.32b) one has$\begin{eqnarray}{{\boldsymbol{F}}}^{{\rm{T}}}={\boldsymbol{T}}{\boldsymbol{F}}{\boldsymbol{T}},{\boldsymbol{H}}={{\boldsymbol{H}}}^{{\rm{T}}},\ {\boldsymbol{F}}={\boldsymbol{T}}{{\boldsymbol{H}}}^{* },{\boldsymbol{G}}={{\boldsymbol{G}}}^{{\rm{T}}}={{\boldsymbol{G}}}^{* },\end{eqnarray}$which, together with ${{\boldsymbol{T}}}^{2}={{\boldsymbol{I}}}_{N}$, gives rise to$\begin{eqnarray}{{\boldsymbol{M}}}^{{\rm{T}}}={{\boldsymbol{H}}}^{{\rm{T}}}{{\boldsymbol{G}}}^{{\rm{T}}}{{\boldsymbol{F}}}^{{\rm{T}}}={\boldsymbol{H}}{{\boldsymbol{G}}}^{{\rm{T}}}{\boldsymbol{T}}{\boldsymbol{F}}{\boldsymbol{T}}={\boldsymbol{T}}{{\boldsymbol{F}}}^{* }{{\boldsymbol{G}}}^{* }{{\boldsymbol{H}}}^{* }{\boldsymbol{T}}={\boldsymbol{T}}{{\boldsymbol{M}}}^{* }{\boldsymbol{T}}.\end{eqnarray}$Then we have$\begin{eqnarray}\begin{array}{rcl}v & = & {{\boldsymbol{s}}}^{{\rm{T}}}{\left({\boldsymbol{I}}+{\boldsymbol{M}}\right)}^{-1}{\boldsymbol{r}}\\ & = & {{\boldsymbol{r}}}^{\dagger }{\boldsymbol{T}}{\left({\boldsymbol{I}}+{\boldsymbol{T}}{{\boldsymbol{M}}}^{\dagger }{\boldsymbol{T}}\right)}^{-1}{\boldsymbol{T}}{{\boldsymbol{s}}}^{* }\\ & = & {{\boldsymbol{s}}}^{\dagger }{\left({\boldsymbol{I}}+{{\boldsymbol{M}}}^{* }\right)}^{-1}{{\boldsymbol{r}}}^{* }\\ & = & {v}^{* },\end{array}\end{eqnarray}$and$\begin{eqnarray}\begin{array}{rcl}{\rm{\Psi }} & = & \displaystyle \frac{1}{2}\sqrt{{\beta }_{1}(t^{\prime} )}{\left({\boldsymbol{I}}+{\boldsymbol{M}}\right)}^{-1}{\boldsymbol{r}}\\ & = & \displaystyle \frac{1}{2}\sqrt{{\beta }_{1}(t^{\prime} )}{\left({\boldsymbol{I}}+{\boldsymbol{T}}{{\boldsymbol{M}}}^{\dagger }{\boldsymbol{T}}\right)}^{-1}{\boldsymbol{T}}{{\boldsymbol{s}}}^{* }\\ & = & \displaystyle \frac{1}{2}\sqrt{{\beta }_{1}(t^{\prime} )}{\boldsymbol{T}}{\left({\boldsymbol{I}}+{{\boldsymbol{M}}}^{* }\right)}^{-1}{{\boldsymbol{s}}}^{* }\\ & = & \displaystyle \frac{1}{2}{\boldsymbol{T}}{[{{\boldsymbol{s}}}^{{\rm{T}}}{\left({\boldsymbol{I}}+{\boldsymbol{M}}\right)}^{-1}]}^{* }\sqrt{{\beta }_{1}(t^{\prime} )}\\ & = & {\boldsymbol{T}}{{\rm{\Phi }}}^{\dagger }.\end{array}\end{eqnarray}$
Case C.This mixed case can be viewed as a formal ‘linear combination' of the first two cases in terms of block matrix. One can check that in this case we have the same expressions as (2.41), (2.42), (2.43), (2.44) and (2.45), but with$\begin{eqnarray}{\boldsymbol{T}}={{\boldsymbol{T}}}_{3}\doteq \mathrm{Diag}\{{{\boldsymbol{I}}}_{{N}_{1}},{{\boldsymbol{H}}}_{0}^{[{N}_{2}]},\cdots ,{{\boldsymbol{H}}}_{0}^{[{N}_{s}]}\}.\end{eqnarray}$
Let us summarize the above three cases.
The generalized Mel'nikov-type equation with self-consistent sources (2.38), i.e. ($u=2{v}_{x}$)$\begin{eqnarray}3{u}_{y^{\prime} y^{\prime} }-{u}_{t^{\prime} x}-{\left(3{u}^{2}+{u}_{{xx}}+8\delta {\rm{\Phi }}{\boldsymbol{T}}{{\rm{\Phi }}}^{\dagger }\right)}_{{xx}}=0,\end{eqnarray}$$\begin{eqnarray}{\rm{i}}{{\rm{\Phi }}}_{y^{\prime} }=u{\rm{\Phi }}+{{\rm{\Phi }}}_{{xx}},\end{eqnarray}$has the following solutions:$\begin{eqnarray}u=2{[{{\boldsymbol{r}}}^{\ast {\rm{T}}}{\boldsymbol{ \mathcal T }}{\left({\boldsymbol{ \mathcal I }}+{\boldsymbol{ \mathcal M }}\right)}^{-1}{\boldsymbol{r}}]}_{x},\,\,{\rm{\Phi }}=\displaystyle \frac{1}{2}{{\boldsymbol{r}}}^{\ast {\rm{T}}}{\boldsymbol{ \mathcal T }}{\left({\boldsymbol{ \mathcal I }}+{\boldsymbol{ \mathcal M }}\right)}^{-1}\sqrt{\beta ({t}^{{\rm{{\prime} }}})},\end{eqnarray}$where ${\boldsymbol{T}}$ is taken as one of the following three forms:$\begin{eqnarray}{{\boldsymbol{T}}}_{1}={{\boldsymbol{I}}}_{N},\end{eqnarray}$$\begin{eqnarray}{{\boldsymbol{T}}}_{2}={{\boldsymbol{H}}}_{0}^{[N]},\end{eqnarray}$$\begin{eqnarray}{{\boldsymbol{T}}}_{3}=\mathrm{Diag}\{{{\boldsymbol{I}}}_{{N}_{1}},{{\boldsymbol{H}}}_{0}^{[{N}_{2}]},\cdots ,{{\boldsymbol{H}}}_{0}^{[{N}_{s}]}\},\end{eqnarray}$and ${\boldsymbol{r}}$, $\beta (t)$ and ${\boldsymbol{M}}$ are respectively given in cases A, B and C in section 2.2, with (2.39).
Let us see more details about the equation (2.47). Case A ${\boldsymbol{T}}={{\boldsymbol{T}}}_{1}$ yields$\begin{eqnarray}3{u}_{y^{\prime} y^{\prime} }-{u}_{t^{\prime} x}-{\left(3{u}^{2}+{u}_{{xx}}+8\delta \displaystyle \sum _{j=1}^{N}| {\phi }_{j}{| }^{2}\right)}_{{xx}}=0,\end{eqnarray}$$\begin{eqnarray}{\rm{i}}{\phi }_{j,y^{\prime} }=u{\phi }_{j}+{\phi }_{j,{xx}},\,\,j=1,2,\cdots ,N,\end{eqnarray}$which can be viewed as a generalized Mel'nikov model to describe interactions of the long wave u and N different short-wave packets $\{| {\phi }_{j}| \}$1(1 In case A, both ${\boldsymbol{L}}$ and ${\boldsymbol{K}}$ are diagonal and $\{{l}_{j}\}$ are distinct. From (3.1) one can see that, as 1-soliton, u and $| {\phi }_{1}{| }^{2}$ are only different from the amplitude ${\beta }_{1}(t^{\prime} )$. This indicates that the short-wave packets $\{| {\phi }_{j}{| }^{2}\}$ are distinguished not only by the eigenvalues $\{{l}_{j}\}$ (as usual soliton solutions) but also by ${\beta }_{j}(t^{\prime} )$.), while the Mel'nikov model (1.1) is of N=1. For case B, in which ${\boldsymbol{T}}={{\boldsymbol{T}}}_{2}$ is given in (2.50), we have$\begin{eqnarray}3{u}_{y^{\prime} y^{\prime} }-{u}_{t^{\prime} x}-{\left(3{u}^{2}+{u}_{{xx}}+8\delta \displaystyle \sum _{j=1}^{N}{\phi }_{j}{\phi }_{N-j+1}^{* }\right)}_{{xx}}=0,\end{eqnarray}$$\begin{eqnarray}{\rm{i}}{\phi }_{j,y^{\prime} }=u{\phi }_{j}+{\phi }_{j,{xx}},\,\,j=1,2,\cdots ,N.\end{eqnarray}$In particular, when N=2, (2.53) has the following form$\begin{eqnarray}3{u}_{y^{\prime} y^{\prime} }-{u}_{t^{\prime} x}-{[3{u}^{2}+{u}_{{xx}}+8\delta ({\phi }_{1}{\phi }_{2}^{* }+{\phi }_{2}{\phi }_{1}^{* })]}_{{xx}}=0,\end{eqnarray}$$\begin{eqnarray}{\rm{i}}{\phi }_{j,y^{\prime} }=u{\phi }_{j}+{\phi }_{j,{xx}},\,\,j=1,2.\end{eqnarray}$
For the general case C, noting the fact that any Hermitian matrix ${\boldsymbol{T}}$ can be diagonalized via$\begin{eqnarray*}{\boldsymbol{T}}={{\boldsymbol{V}}}^{{\rm{T}}}{\boldsymbol{E}}{{\boldsymbol{V}}}^{* },\,\mathrm{with}\,{\boldsymbol{E}}=\mathrm{Diag}\{{\epsilon }_{1},\cdots ,{\epsilon }_{N}\},\end{eqnarray*}$where ${\epsilon }_{j}=\pm 1$, we introduce a new vector $\bar{{\rm{\Phi }}}={\rm{\Phi }}{{\boldsymbol{V}}}^{{\rm{T}}}\,=({\bar{\phi }}_{1},{\bar{\phi }}_{2},\cdots ,{\bar{\phi }}_{N})$, then (2.47) has an alternative form:$\begin{eqnarray}3{u}_{y^{\prime} y^{\prime} }-{u}_{t^{\prime} x}-{\left(3{u}^{2}+{u}_{{xx}}+8\delta \displaystyle \sum _{j=1}^{N}{\epsilon }_{j}| {\bar{\phi }}_{j}{| }^{2}\right)}_{{xx}}=0,\end{eqnarray}$$\begin{eqnarray}{\rm{i}}{\bar{\phi }}_{j,y^{\prime} }=u{\bar{\phi }}_{j}+{\bar{\phi }}_{j,{xx}},\,\,j=1,2,\cdots ,N.\end{eqnarray}$Note that, considering the diagonal form ${\boldsymbol{E}}$ of the Hermitian matrix, for the standard form of cases B and C, there must be some $\{{\epsilon }_{j}\}$ to take −1, i.e. ${\displaystyle \prod }_{j=1}^{N}{\epsilon }_{j}=-1$.
3. Dynamics
In the following we discuss dynamics of the obtained solutions and investigate effects of short-wave packets as well as the arbitrary function ${\beta }_{j}(t)$. We take δ=1 and for convenience we omit ' from t and y to avoid confusion.
3.1. 1-soliton and short-wave packet effect
From (2.48), we have the 1-soliton solution of (2.52) with N=1,$\begin{eqnarray}u=\displaystyle \frac{{\left({l}_{1}^{* }+{l}_{1}\right)}^{2}}{2}{{\rm{sech}} }^{2}\displaystyle \frac{{\eta }_{1}+{\eta }_{1}^{* }-\mathrm{ln}| {l}_{1}^{* }+{l}_{1}| }{2},\end{eqnarray}$$\begin{eqnarray}{\phi }_{1}=\displaystyle \frac{\sqrt{{l}_{1}^{* }+{l}_{1}}}{4}\sqrt{{\beta }_{1}(t)}{{\rm{e}}}^{\displaystyle \frac{{\eta }_{1}^{* }-{\eta }_{1}}{2}}{\rm{{\rm{sech}} }}\displaystyle \frac{{\eta }_{1}+{\eta }_{1}^{* }-\mathrm{ln}| {l}_{1}^{* }+{l}_{1}| }{2},\end{eqnarray}$where$\begin{eqnarray}{\eta }_{1}={l}_{1}x-{\rm{i}}{l}_{1}^{2}y-4{l}_{1}^{3}t+\displaystyle \frac{1}{2}{\int }_{0}^{t}{\beta }_{1}(z){\rm{d}}z+{\eta }_{1}^{(0)}.\end{eqnarray}$Next, let us assume$\begin{eqnarray}{l}_{j}={a}_{j}+{\rm{i}}{b}_{j},\,\,{\eta }_{j}^{(0)}={c}_{j}+{\rm{i}}{d}_{j},\,\,{a}_{j},{b}_{j},{c}_{j},{d}_{j}\in {\mathbb{R}}.\end{eqnarray}$Then, the long wave u and short-wave packet $| {\phi }_{1}| $ are written as$\begin{eqnarray}u=2{a}_{1}^{2}{{\rm{sech}} }^{2}{\xi }_{1},\end{eqnarray}$$\begin{eqnarray}| {\phi }_{1}{| }^{2}=\displaystyle \frac{1}{8}| {\beta }_{1}(t)| | {a}_{1}| {{\rm{sech}} }^{2}{\xi }_{1},\end{eqnarray}$where$\begin{eqnarray}\begin{array}{rcl}{\xi }_{1} & = & {a}_{1}[x+2{b}_{1}y-4({a}_{1}^{2}-3{b}_{1}^{2})t]\\ & & +\displaystyle \frac{1}{2}{\displaystyle \int }_{0}^{t}{\beta }_{1}(t){\rm{d}}t+{c}_{1}-\displaystyle \frac{1}{2}\mathrm{ln}(2{a}_{1}).\end{array}\end{eqnarray}$
As depicted in figure 1(a), (3.3a) is a line soliton with a constant amplitude $2{a}_{1}^{2}$. This line on the (x, y) plane is ξ1=0 and it moves with velocity$\begin{eqnarray}\begin{array}{l}(x^{\prime} (t),y^{\prime} (t))\\ =\,\left\{\begin{array}{ll}-\left[4({a}_{1}^{2}-3{b}_{1}^{2})+\tfrac{1}{2{a}_{1}}{\beta }_{1}(t)\right]\left(1,\tfrac{1}{2{b}_{1}}\right),\, & {b}_{1}\ne 0,\\ -\left(4{a}_{1}^{2}+\tfrac{1}{2{a}_{1}}{\beta }_{1}(t)\right)(1,0), & {b}_{1}=0.\end{array}\right.\end{array}\end{eqnarray}$A stationary line soliton appears when ${\beta }_{1}(t)=-8{a}_{1}({a}_{1}^{2}\,-3{b}_{1}^{2})$. The short-wave packet, depicted in figure 1(b), is also a line soliton like u but its amplitude is time-dependent and governed by a1 and β1(t). When β1(t)=0, the short wave disappears and the system (1.1) degenerates to the KP-II equation that has usual line solitons, each of which has a constant amplitude and a constant velocity. If ${\beta }_{1}(t)\ne 0$, the short wave exists, and this results in a line soliton u whose amplitude is still independent of time but velocity changes with β1(t). This is the effect of the short-wave packet $| {\phi }_{1}| $2(2 In [2-4] Mel'nikov derived solutions for (1.1) using a direct method, but at that time the arbitrary β1(t) was not introduced. Now one can realize that the effect of the short-wave packet is actually governed by this function.). To illustrate how the arbitrary β1(t) affects motions of u and $| {\phi }_{1}| $, we employ a vertical line soliton ($y^{\prime} (t)\equiv 0$, i.e. b1=0), which can be represented by the cross-section at any y, and we record positions of the cross-section at y=0 and observe how the cross-section changes with time. These are depicted in figure 2. One can see from figure 2(c) that the amplitude of the vertical line soliton u does not change with time but its velocity changes, while from figure 2(d) we can see that both the amplitude and velocity of the packet $| {\phi }_{1}{| }^{2}$ change with time.
Figure 1.
New window|Download| PPT slide Figure 1.1-soliton solution of the equation (2.52). (a) Shape of u given in (3.3a) with ${l}_{1}=0.5+0.2{\rm{i}},{\beta }_{1}(z)=\tfrac{1}{1+{z}^{2}},{\eta }_{1}^{(0)}=0,\,t=0$. (b) Shape of $| {\phi }_{1}{| }^{2}$ given in (3.3b) with ${l}_{1}=0.5+0.2{\rm{i}},{\beta }_{1}(z)=\tfrac{1}{1+{z}^{2}},{\eta }_{1}^{(0)}=0,\,t=0$.
Figure 2.
New window|Download| PPT slide Figure 2.Vertical line soliton solution of equation (2.52). (a) Shape of u given in (3.3a) at t=0 with ${l}_{1}=1,\ {\beta }_{1}(t)=8\sin t+8a({a}^{2}\,-3{b}^{2}),{\eta }_{1}^{(0)}=0$. (b) Shape of $| {\phi }_{1}{| }^{2}$ given in (3.3b) at t=0 with same l1, β1(t) and ${\eta }_{1}^{(0)}$. (c) 2D plot of (a) at y=0 and t=−2 (red solid curve), t=0 (blue dot-dashed curve) and t=1 (black dashed curve). (d) 2D plot of (b) at y=0 and t=−2 (red solid curve), t=0 (blue dot-dashed curve) and t=1 (black dashed curve).
3.2. 2-soliton interaction
Consider equation (2.52) with N=23(3 If we take β2(t)≡0 then we will have a 2-soliton solution for the Mel'nikov equation (1.1).). In case A when ${\boldsymbol{L}}={{\boldsymbol{D}}}^{[2]}({\{{l}_{i}\}}_{1}^{2})$, we get a 2-soliton solution for the equation (2.52),$\begin{eqnarray}u=2{\left(\mathrm{ln}\tau \right)}_{{xx}},\end{eqnarray}$$\begin{eqnarray}{\phi }_{1}=\displaystyle \frac{\sqrt{{\beta }_{1}(t)}}{2}\displaystyle \frac{{{\rm{e}}}^{{\eta }_{1}^{* }}(1+{M}_{22})-{{\rm{e}}}^{{\eta }_{2}^{* }}{M}_{21}}{\tau },\end{eqnarray}$$\begin{eqnarray}{\phi }_{2}=\displaystyle \frac{\sqrt{{\beta }_{2}(t)}}{2}\displaystyle \frac{{{\rm{e}}}^{{\eta }_{2}^{* }}(1+{M}_{11})-{{\rm{e}}}^{{\eta }_{1}^{* }}{M}_{12}}{\tau },\end{eqnarray}$where$\begin{eqnarray}\tau =| {\boldsymbol{I}}+{\boldsymbol{M}}| =(1+{M}_{11})(1+{M}_{22})-{M}_{12}{M}_{21},\end{eqnarray}$$\begin{eqnarray}\begin{array}{rcl}{M}_{11} & = & \displaystyle \frac{{{\rm{e}}}^{{\eta }_{1}+{\eta }_{1}^{* }}}{{l}_{1}+{l}_{1}^{* }},\,\,{M}_{12}=\displaystyle \frac{{{\rm{e}}}^{{\eta }_{1}+{\eta }_{2}^{* }}}{{l}_{1}+{l}_{2}^{* }},\\ {M}_{21} & = & \displaystyle \frac{{{\rm{e}}}^{{\eta }_{2}+{\eta }_{1}^{* }}}{{l}_{2}+{l}_{1}^{* }},\,\,{M}_{22}=\displaystyle \frac{{{\rm{e}}}^{{\eta }_{2}+{\eta }_{2}^{* }}}{{l}_{2}+{l}_{2}^{* }},\end{array}\end{eqnarray}$$\begin{eqnarray}{\eta }_{j}={l}_{j}x-{\rm{i}}{l}_{j}^{2}y-4{l}_{j}^{3}t+\displaystyle \frac{1}{2}{\int }_{0}^{t}{\beta }_{j}(z){\rm{d}}z+{\eta }_{j}^{(0)}.\end{eqnarray}$
For convenience, we employ notations (3.2) and introduce ${\gamma }_{j}=\mathrm{Re}[{\eta }_{j}]$ and ${\theta }_{j}=\mathrm{Im}[{\eta }_{j}]$, i.e.$\begin{eqnarray}\begin{array}{rcl}{\gamma }_{j} & = & {a}_{j}[x+2{b}_{j}y-4({a}_{j}^{2}-3{b}_{j}^{2})t)]\\ & & +\displaystyle \frac{1}{2}{\displaystyle \int }_{0}^{t}{\beta }_{j}(z){\rm{d}}z+{c}_{j},\end{array}\end{eqnarray}$$\begin{eqnarray}{\theta }_{j}={b}_{j}x-({a}_{j}^{2}-{b}_{j}^{2})y-4{b}_{j}(3{a}_{j}^{2}-{b}_{j}^{2})t+{d}_{j}.\end{eqnarray}$With these notations one can rewrite τ as$\begin{eqnarray}\begin{array}{rcl}\tau & = & 1+\displaystyle \frac{1}{2{a}_{1}}{{\rm{e}}}^{2{\gamma }_{1}}+\displaystyle \frac{1}{2{a}_{2}}{{\rm{e}}}^{2{\gamma }_{2}}\\ & & +\displaystyle \frac{1}{4{a}_{1}{a}_{2}}\displaystyle \frac{{\left({a}_{1}-{a}_{2}\right)}^{2}+{\left({b}_{1}-{b}_{2}\right)}^{2}}{{\left({a}_{1}+{a}_{2}\right)}^{2}+{\left({b}_{1}-{b}_{2}\right)}^{2}}{{\rm{e}}}^{2({\gamma }_{1}+{\gamma }_{2})},\end{array}\end{eqnarray}$and the long wave u and short-wave packets $| {\phi }_{j}{| }^{2}$ as$\begin{eqnarray}u=2{\left(\mathrm{ln}\tau \right)}_{{xx}},\end{eqnarray}$$\begin{eqnarray}| {\phi }_{1}{| }^{2}=| {\beta }_{1}(t)| \displaystyle \frac{{A}_{1}}{4{\tau }^{2}},\end{eqnarray}$$\begin{eqnarray}| {\phi }_{2}{| }^{2}=| {\beta }_{2}(t)| \displaystyle \frac{{A}_{2}}{4{\tau }^{2}},\end{eqnarray}$where$\begin{eqnarray*}\begin{array}{rcl}{A}_{1} & = & {{\rm{e}}}^{2{\gamma }_{1}}\left[1+\displaystyle \frac{{a}_{1}^{2}-{a}_{2}^{2}+{\left({b}_{1}-{b}_{2}\right)}^{2}}{{a}_{2}[{\left({a}_{1}+{a}_{2}\right)}^{2}+{\left({b}_{1}-{b}_{2}\right)}^{2}]}{{\rm{e}}}^{2{\gamma }_{2}}\right.\\ & & \left.+\displaystyle \frac{{\left({a}_{1}-{a}_{2}\right)}^{2}+{\left({b}_{1}-{b}_{2}\right)}^{2}-8{a}_{2}^{2}({a}_{1}-{a}_{2})}{4{a}_{2}^{2}[{\left({a}_{1}+{a}_{2}\right)}^{2}+{\left({b}_{1}-{b}_{2}\right)}^{2}]}{{\rm{e}}}^{4{\gamma }_{2}^{2}}\right],\\ {A}_{2} & = & {{\rm{e}}}^{2{\gamma }_{2}}\left[1+\displaystyle \frac{{a}_{2}^{2}-{a}_{1}^{2}-{\left({b}_{1}-{b}_{2}\right)}^{2}}{{a}_{1}[{\left({a}_{1}+{a}_{2}\right)}^{2}+{\left({b}_{1}-{b}_{2}\right)}^{2}]}{{\rm{e}}}^{2{\gamma }_{1}}\right.\\ & & \left.+\displaystyle \frac{{\left({a}_{1}-{a}_{2}\right)}^{2}+{\left({b}_{1}-{b}_{2}\right)}^{2}+8{a}_{1}^{2}({a}_{1}-{a}_{2})}{4{a}_{1}^{2}[{\left({a}_{1}+{a}_{2}\right)}^{2}+{\left({b}_{1}-{b}_{2}\right)}^{2}]}{{\rm{e}}}^{4{\gamma }_{1}^{2}}\right].\end{array}\end{eqnarray*}$
Note that one can redefine cj in γj and then rewrite the τ (3.7) as$\begin{eqnarray}\tau =1+{{\rm{e}}}^{2{\gamma }_{1}}+{{\rm{e}}}^{2{\gamma }_{2}}+\displaystyle \frac{{\left({a}_{1}-{a}_{2}\right)}^{2}+{\left({b}_{1}-{b}_{2}\right)}^{2}}{{\left({a}_{1}+{a}_{2}\right)}^{2}+{\left({b}_{1}-{b}_{2}\right)}^{2}}{{\rm{e}}}^{2({\gamma }_{1}+{\gamma }_{2})}.\end{eqnarray}$This is a standard Hirota's form for 2-soliton solutions of the KdV-type equations. The phase factor $\tfrac{{\left({a}_{1}-{a}_{2}\right)}^{2}+{\left({b}_{1}-{b}_{2}\right)}^{2}}{{\left({a}_{1}+{a}_{2}\right)}^{2}+{\left({b}_{1}-{b}_{2}\right)}^{2}}$ cannot be zero unless ${l}_{1}={l}_{2}$, which means there is no Y-shape resonance as the KP-II equation (cf [19]).
Scattering of two line solitons described by (3.8a) is depicted in figure 3(a), in which the velocity of each soliton is actually governed by βj(t). The short-wave packets $| {\phi }_{j}{| }^{2}$ exhibit ‘ghost' soliton behavior, i.e. one can see a line soliton interacted with an invisible line soliton. See figures 3(b) and (c).
Figure 3.
New window|Download| PPT slide Figure 3.2-soliton solutions (3.8) of the equation (2.52), with ${l}_{1}=0.5+0.1{\rm{i}},{l}_{2}=0.8+0.5{\rm{i}}$, ${\beta }_{1}(t)=\sin t,\ {\beta }_{2}(t)=1$, ${\eta }_{1}^{(0)}={\eta }_{2}^{(0)}=0$ and fixed t=2. (a) Shape of u. (b) Shape of $| {\phi }_{1}{| }^{2}$. (c) Shape of $| {\phi }_{2}{| }^{2}$.
To see the effects of the arbitrary βj(t), we take vertical line solitons of u as an example, in which one line soliton is stationary by taking ${\beta }_{2}(t)=8{a}_{2}({a}_{2}^{2}-3{b}_{2}^{2})$ while the other runs with velocity $(1-8\sin t)/2{a}_{1}$ by taking ${\beta }_{1}(t)\,=8\sin t+8{a}_{1}({a}_{1}^{2}-3{b}_{1}^{2})-1$. Figure 4(b) records the positions of two line solitons by the cross-section at y=0 with different times, which shows how two line solitons interact under the effect of βj(t). One can see that at first the two line solitons overlap completely at t=0, then the higher line soliton moves left while the lower one stays stationary, then the higher one moves from left to right and when it ‘passes' the lower one, the latter has a phase shift. Such an oscillation cannot happen in the usual KP-II case.
Figure 4.
New window|Download| PPT slide Figure 4.Vertical two line soliton solution (3.8a) of equation (2.52), with ${l}_{1}=1,{l}_{2}=0.5,{\beta }_{1}(t)=8\sin t+8{a}_{1}({a}_{1}^{2}-3{b}_{1}^{2})-1$, ${\beta }_{2}(t)=8{a}_{2}({a}_{2}^{2}-3{b}_{2}^{2}),{\eta }_{1}^{(0)}={\eta }_{2}^{(0)}=0$ and fixed t=π/2. (a) Shape of u. (b) 2D plot of (a) at y=0 and t=0 (red solid curve), t=π/2 (blue solid curve), t=π (blue dot-dashed curve), t=5.3 (red dashed curve) and t=6 (black dashed curve).
3.3. Resonant short-wave packets
This is case B, of which the equation is (2.53). If we only consider two short-wave packets, (2.53) degenerates to (2.54). Resonance in this case all βj(t) that govern the short-wave packets are the same, we can say that this case provides solutions with resonant short-wave packets4(4 Resonance results from a limit procedure where, among two soliton parameters (e.g. l1, l2), one approaches another. This is also called the double-pole case (or in general multiple-pole case), cf. [20, 21].).
For equation (2.54), when taking ${\boldsymbol{L}}={{\boldsymbol{J}}}^{[2]}({l}_{1})$ in case B, we have$\begin{eqnarray}{r}_{1}={{\rm{e}}}^{{\eta }_{1}}={{\rm{e}}}^{{\gamma }_{1}+{\rm{i}}{\theta }_{1}},\,{r}_{2}={\partial }_{{l}_{1}}{{\rm{e}}}^{{\eta }_{1}},\,{s}_{1}={r}_{2}^{* },\,{s}_{2}={r}_{1}^{* },\end{eqnarray}$$\begin{eqnarray}{\boldsymbol{M}}={\boldsymbol{F}}{\boldsymbol{G}}{\boldsymbol{H}}=\left(\begin{array}{cc}{M}_{11} & {M}_{12}\\ {M}_{21} & {M}_{22}\end{array}\right),\end{eqnarray}$$\begin{eqnarray}{\boldsymbol{F}}=\left(\begin{array}{cc}{r}_{1} & 0\\ {r}_{2} & {r}_{1}\end{array}\right),\,{\boldsymbol{G}}=\left(\begin{array}{cc}\tfrac{1}{2{a}_{1}} & \tfrac{-1}{{\left(2{a}_{1}\right)}^{2}}\\ \tfrac{-1}{{\left(2{a}_{1}\right)}^{2}} & \tfrac{2}{{\left(2{a}_{1}\right)}^{3}}\end{array}\right),\,{\boldsymbol{H}}=\left(\begin{array}{cc}{r}_{2}^{* } & {r}_{1}^{* }\\ {r}_{1}^{* } & 0\end{array}\right),\end{eqnarray}$where η1 is defined as in (3.5f) with j=1, γ1=Re[η1] and θ1=Im[η1], and the explicit forms of Mij are$\begin{eqnarray*}\begin{array}{rcl}{M}_{11} & = & \displaystyle \frac{(x+{\rm{i}}2{l}_{1}^{* }y-12{l}_{1}^{* 2}t){{\rm{e}}}^{2{\gamma }_{1}}}{2{a}_{1}}\\ & & -\displaystyle \frac{{{\rm{e}}}^{2{\gamma }_{1}}}{{\left(2{a}_{1}\right)}^{2}},\,\,{M}_{12}=\displaystyle \frac{{{\rm{e}}}^{2{\gamma }_{1}}}{2{a}_{1}},\\ {M}_{21} & = & \displaystyle \frac{{\left(x-2{\rm{i}}{l}_{1}y-12{l}_{1}^{2}t\right)}^{2}{{\rm{e}}}^{2{\gamma }_{1}}}{2{a}_{1}}\\ & & -\displaystyle \frac{[4{b}_{1}y+2({a}_{1}^{2}-{b}_{1}^{2})]{{\rm{e}}}^{2{\gamma }_{1}}}{{\left(2{a}_{1}\right)}^{2}}+\displaystyle \frac{2{{\rm{e}}}^{2{\gamma }_{1}}}{{\left(2{a}_{1}\right)}^{3}},\\ {M}_{22} & = & \displaystyle \frac{(x-2{\rm{i}}{l}_{1}y-12{l}_{1}^{2}t){{\rm{e}}}^{2{\gamma }_{1}}}{2{a}_{1}}-\displaystyle \frac{{{\rm{e}}}^{2{\gamma }_{1}}}{{\left(2{a}_{1}\right)}^{2}}.\end{array}\end{eqnarray*}$Thus, solutions to equation (2.54) are$\begin{eqnarray}u=2{\left(\mathrm{ln}\tau \right)}_{{xx}},\end{eqnarray}$$\begin{eqnarray}{\phi }_{1}=\displaystyle \frac{\sqrt{{\beta }_{1}(t)}}{2}\displaystyle \frac{{r}_{2}^{* }(1+{M}_{22})-{r}_{1}{M}_{21}}{\tau },\end{eqnarray}$$\begin{eqnarray}{\phi }_{2}=\displaystyle \frac{\sqrt{{\beta }_{1}(t)}}{2}\displaystyle \frac{{r}_{1}(1+{M}_{11})-{r}_{2}^{* }{M}_{12}}{\tau },\end{eqnarray}$where$\begin{eqnarray}\begin{array}{rcl}\tau & = & | {\boldsymbol{I}}+{\boldsymbol{M}}| =1\\ & & +\left[\displaystyle \frac{x-2{b}_{1}y-6({a}_{1}^{2}-{b}_{1}^{2})t}{{a}_{1}}-\displaystyle \frac{1}{2{a}_{1}^{2}}\right]{{\rm{e}}}^{2{\gamma }_{1}}-\displaystyle \frac{1}{{\left(2{a}_{1}\right)}^{4}}{{\rm{e}}}^{4{\gamma }_{1}}.\end{array}\end{eqnarray}$It is easy to write out short-wave packets $| {\phi }_{1}{| }^{2}$ and $| {\phi }_{2}{| }^{2}$.
However, one can prove that no matter what b1 is, there are always zeros of τ in ${{\mathbb{R}}}^{3}$. For this reason we only present below the picture of u and its density plot in figure 5.
Figure 5.
New window|Download| PPT slide Figure 5.Double-pole solution (3.11a) of the equation (2.53), with ${l}_{1}=1+0.5{\rm{i}},\,{\beta }_{1}(t)=8\sin t,{\eta }_{1}^{(0)}=1$ and t=0. (a) Shape of u. (b) Density plot of (a) where the bright stripes stand for singularities.
4. Concluding remarks
We investigated the Mel'nikov model (1.1) with one short-wave packet and its extended version (2.47) with N short-wave packets. Both models can be viewed as complex reductions of the KPSCS (1.2). We have found that each short-wave packet $| {\phi }_{j}{| }^{2}$ is actually characterized by an arbitrary function βj(t), which was never reported in the literatures on the research of the Mel'nikov model (1.1) and its extensions [1-4, 22-26]. This ${\beta }_{j}(t)$ governs the effect of the short-wave packet and can bring more variety of interactions of line solitons. If we switched the roles of y and t, like in [23], we would have more interesting interactions as in the (1+1)-dimensional case (cf. [27, 28]).
The method we employed is the Cauchy matrix approach, where we ‘insert' βj(t) in the time evolution of plain wave factors. First, we derived the KPSCS (1.2) and then we implemented complex reductions to get the extended Mel'nikov model (2.47). Note that the Cauchy matrix approach is a method that starts from solutions and then equations follow. This enables us to view the Mel'nikov-type models from a new angle and form greater understanding: the Mel'nikov model (2.55) is in fact complicated. The model parameters {εj} are related to the type of solutions. For example, the resonance of short-wave packets can only happen when ${\displaystyle \prod }_{j=1}^{N}{\epsilon }_{j}=-1$. We will leave this part for future research.
Acknowledgments
This project is supported by the NSF of China (Nos. 11875040 and 11631007).
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,Department of Mathematics, 
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