1. Introduction
A lump wave is a kind of rational localized wave degenerating in all directions of space. The first report about the lump wave appeared about forty years ago during the study of the rational solutions of the Kadomtsev–Petviashvili equation [1, 2]. As we all know, the long wave limit method developed by Ablowitz et al is an effective method to derive the multiple lump waves of the integrable systems [2, 3]. Then many scholars focused on finding the lump waves and interaction waves between lump waves and other types of localized waves [4–12]. It is a consensus in past studies that there is no phase shift during the interactions between the lump waves. Recently, the research indicates that the Kadomtsev–Petviashvili I (KPI) equation possesses the multiple lump wave with phase shift [13]. In addition, the bound state of the multiple lump waves, i.e. the phenomenon of lump molecules, has been found [13, 14]. The nonlinear superposition of the lump waves, breather waves and solitons is investigated by means of the velocity resonant mechanism, in which all the waves never collide with each other [15, 16]. However, few studies paid attention to studying the dynamical behaviors of the high-order lump molecules.
The KPI equation2 ) [28]. Inspired by the work of Stepanyants et al, this paper mainly considers the high-order lump molecules and interaction solutions of the KPI equation.
$\begin{eqnarray}{\left({u}_{t}+6{{uu}}_{x}+{u}_{{xxx}}\right)}_{x}-3{u}_{{yy}}=0,\end{eqnarray}$
is a generalization of the KdV equation, which was firstly reported by Kadomtsev and Petviashvili [17]. The KPI equation is widely applied in many fields of sciences, such as nonlinear optics, fluid mechanics, Bose–Einstein condensates, ocean waves and solitons [18–20]. Many integrable properties of the KPI equation including Darboux transformation, Bäcklund transformation, conservation laws and analytical solutions have been studied systematically [21–26]. Based on the theory of Hirota's bilinear method and the formula of Wronskian, one can derive the solutions of the KPI equation defined by the Gramian formula $\begin{eqnarray}\begin{array}{l}u\left(x,y,t\right)=2{\left(\mathrm{ln}\tau \right)}_{{xx}},\\ \quad \tau \left(x,y,t\right)=\det \left[{\vartheta }_{{jk}}+{\displaystyle \int }_{-\infty }^{x}{\varphi }_{j}\right.\left.\left(x^{\prime} ,y,t\right){\bar{\varphi }}_{k}\left(x^{\prime} ,y,t\right){\rm{d}}x^{\prime} \Space{0ex}{2.55ex}{0ex}\right],\end{array}\end{eqnarray}$
where all the components of the vector $\left({\varphi }_{1},\cdots ,{\varphi }_{M}\right)$ are the solutions of the linear system $\begin{eqnarray}{\varphi }_{y}-{\rm{i}}{\varphi }_{{xx}}=0,\quad \quad {\varphi }_{t}+4{\varphi }_{{xxx}}=0,\end{eqnarray}$
and ϑjk is a constant M × M matrix. The lump chains of the KPI equation have been constructed by means of the reduction version of the Gramian form of the τ-function [27]. Stepanyants et al investigated the interactions between lumps and plane solitons with the aid of the solution (The organization of the paper is as follows. First of all, we briefly present the procedure for deriving the Gramian determinant solution of the KPI equation. Secondly, the three two-lump molecules in terms of a third-order determinant are constructed. A unified scheme is proposed to derive multiple lump molecules consisting of M N-lump molecules. Furthermore, we investigate the interaction solutions of P line solitons radiating P of the M N-lump molecules. Finally some conclusions are given in the last section.
2. Gramian determinant solution of the KPI equation
In order to construct the multiple lump molecule solutions of the KPI equation, we introduce a complex polynomial function3 ) for any value of λ. More generally, the polynomial function in the degree of p can be given as follows3 ) by substituting this expression into equation (3 ). If one takes the function φj in equation (2 ) as the linear combinations of the functions ${\rho }_{p}\left(x,y,t\right){{\rm{e}}}^{\sigma \left(x,y,t,\lambda \right)}$, a lot of rational solutions of equation (1 ) can be obtained.
$\begin{eqnarray}\sigma \left(x,y,t,\lambda \right)=\lambda x+{\rm{i}}{\lambda }^{2}y-4{\lambda }^{3}t.\end{eqnarray}$
It can be proved that ${{\rm{e}}}^{\sigma \left(x,y,t,\lambda \right)}$ is a solution of the linear system ( $\begin{eqnarray}{\rho }_{p}\left(x,y,t,\lambda \right)={{\rm{e}}}^{-\sigma \left(x,y,t,\lambda \right)}\displaystyle \frac{{\partial }^{p}}{\partial {\lambda }^{p}}{{\rm{e}}}^{\sigma \left(x,y,t,\lambda \right)},\end{eqnarray}$
$\begin{eqnarray*}\begin{array}{l}{\rho }_{0}\left(x,y,t,\lambda \right)=1,\\ \quad {\rho }_{1}\left(x,y,t,\lambda \right)=x+2{\rm{i}}\lambda y-12{\lambda }^{2}t,\\ \quad {\rho }_{2}\left(x,y,t,\lambda \right)=2{\rm{i}}y-24\lambda t+{\left(x+2{\rm{i}}\lambda y-12{\lambda }^{2}t\right)}^{2},\\ \quad {\rho }_{3}\left(x,y,t,\lambda \right)=-24t+3\left(2{\rm{i}}y-24\lambda t\right)\\ \quad \times \left(x+2{\rm{i}}\lambda y-12{\lambda }^{2}t\right)+{\left(x+2{\rm{i}}\lambda y-12{\lambda }^{2}t\right)}^{3},\\ \quad \cdots .\end{array}\end{eqnarray*}$
Then we can conclude that ${\partial }_{\lambda }^{p}{{\rm{e}}}^{\sigma }={\rho }_{p}{{\rm{e}}}^{\sigma }$ is a solution of the linear system (To write the exposition in a simple way, we let λj = aj > 0, aj and b1, ⋯ ,bM are all the real parameters. In addition, the p1, ⋯ ,pM are the non-negative integers. Then function φj can be chosen as2 ) can be transformed into7 ) into equation (2 ) and factoring a common exponential function, a modified version of the Gramian determinant solution of the KPI equation can be obtained as
$\begin{eqnarray}{\varphi }_{j}\left(x,y,t\right)={b}_{j}{\rho }_{{p}_{j}}\left(x,y,t,{a}_{j}\right){{\rm{e}}}^{\sigma \left(x,y,t,{a}_{j}\right)}.\end{eqnarray}$
Thus expression in equation ( $\begin{eqnarray}\begin{array}{l}\quad {\displaystyle \int }_{-\infty }^{x}{\varphi }_{j}\left(x^{\prime} ,y,t\right){\bar{\varphi }}_{k}\left(x^{\prime} ,y,t\right){\rm{d}}x^{\prime} \\ \quad =\displaystyle \frac{{b}_{j}{b}_{k}}{{a}_{j}+{a}_{k}}{\rho }_{{p}_{j}{p}_{k}}\left(x,y,t,{a}_{j},{a}_{k}\right)\\ \quad \times {{\rm{e}}}^{\sigma \left(x,y,t,{a}_{j}\right)+\bar{\sigma }\left(x,y,t,{a}_{k}\right)},\end{array}\end{eqnarray}$
where ${\rho }_{{jk}}\left(x,y,t,s,s^{\prime} \right)$ is a non-homogeneous polynomial in the degree j + k which is determined by the formula $\begin{eqnarray}\begin{array}{l}{\rho }_{{jk}}\left(x,y,t,s,s^{\prime} \right)=\sum _{h=0}^{j+k}{\left(\displaystyle \frac{-1}{s+s^{\prime} }\right)}^{h}\\ \quad \times \displaystyle \frac{{\partial }^{h}}{\partial {x}^{h}}\left[{\rho }_{j}\left(x,y,t,s\right){\bar{\rho }}_{k}\left(x,y,t,s^{\prime} \right)\right].\end{array}\end{eqnarray}$
Thus, a series of function expressions about ${\rho }_{{jk}}\left(x,y,t,s,s^{\prime} \right)$ can be expanded as $\begin{eqnarray}\begin{array}{l}{\rho }_{00}\left(x,y,t,s,s^{\prime} \right)=1,\\ \quad {\rho }_{01}\left(x,y,t,s,s^{\prime} \right)=x-2{\rm{i}}s^{\prime} y-12s{{\prime} }^{2}t-\displaystyle \frac{1}{s+s^{\prime} },\\ \quad {\rho }_{10}\left(x,y,t,s,s^{\prime} \right)=x+2{\rm{i}}{sy}-12{s}^{2}t-\displaystyle \frac{1}{s+s^{\prime} },\\ \quad {\rho }_{11}\left(x,y,t,s,s^{\prime} \right)\\ \quad =\,\left(x+2{\rm{i}}{sy}-12{s}^{2}t\right)\left(x-2{\rm{i}}s^{\prime} y-12s{{\prime} }^{2}t\right)\\ \quad -\displaystyle \frac{2x-2{\rm{i}}s^{\prime} y-12s{{\prime} }^{2}t+2{\rm{i}}{sy}-12{s}^{2}t}{s+s^{\prime} }+\displaystyle \frac{2}{{\left(s+s^{\prime} \right)}^{2}},\\ \quad \cdots .\end{array}\end{eqnarray}$
Substituting the formula ( $\begin{eqnarray}\begin{array}{l}u\left(x,y,t\right)=2\displaystyle \frac{{\partial }^{2}}{\partial {x}^{2}}\,\mathrm{ln}\,\det \,\left[\Space{0ex}{3.59ex}{0ex}{\vartheta }_{{jk}}{{\rm{e}}}^{-\sigma \left(x,y,t,{a}_{j}\right)-\bar{\sigma }\left(x,y,t,{a}_{k}\right)}\right.\\ \quad \left.+\displaystyle \frac{{b}_{j}{b}_{k}}{{a}_{j}+{a}_{k}}{\rho }_{{p}_{j}{p}_{k}}\left(x,y,t,{a}_{j},{a}_{k}\right)\right].\end{array}\end{eqnarray}$
3. Multiple lump molecules and interaction solutions
To construct the multiple lump molecule solution consisting of three two-lump molecules, the parameters among the solution (10 ) are chosen as M = 3, a1 ≠ a2 ≠ a3, p1 = p2 = p3 = 2 and ϑjk is a 3 × 3 zero matrix. Then the three two-lump molecule solutions can be written in the following determinant form8 ) by means of the following set of conjugate functions
$\begin{eqnarray}u\left(x,y,t\right)=2\displaystyle \frac{{\partial }^{2}}{\partial {x}^{2}}\mathrm{ln}\left(\left|\begin{array}{ccc}{\tau }_{11} & {\tau }_{12} & {\tau }_{13}\\ {\tau }_{21} & {\tau }_{22} & {\tau }_{23}\\ {\tau }_{31} & {\tau }_{32} & {\tau }_{33}\end{array}\right|\right),\end{eqnarray}$
where $\begin{eqnarray*}{\tau }_{{jk}}=\displaystyle \frac{{b}_{j}{b}_{k}}{{a}_{j}+{a}_{k}}{\rho }_{22}\left(x,y,t,{a}_{j},{a}_{k}\right),\,j,k=1,2,3,\end{eqnarray*}$
and the function ${\rho }_{22}\left(x,y,t,{a}_{j},{a}_{k}\right)$ is determined by the formula ( $\begin{eqnarray*}\begin{array}{l}{\rho }_{2}\left(x,y,t,{a}_{j}\right)=2y{\rm{i}}-24{a}_{j}t+{x}^{2}+4{a}_{j}{xy}{\rm{i}}\\ \quad -24{a}_{j}^{2}{xt}-4{a}_{j}^{2}{y}^{2}-48{a}_{j}^{3}{yt}{\rm{i}}+144{a}_{j}^{4}{t}^{2},\\ \quad {\bar{\rho }}_{2}\left(x,y,t,{a}_{k}\right)=-2y{\rm{i}}-24{a}_{k}t+{x}^{2}-4{a}_{k}{xy}{\rm{i}}\\ \quad -24{a}_{k}^{2}{xt}-4{a}_{k}^{2}{y}^{2}+48{a}_{k}^{3}{yt}{\rm{i}}+144{a}_{k}^{4}{t}^{2}.\end{array}\end{eqnarray*}$
Figure 1 is plotted to show the evolutionary dynamical behaviors of the three two-lump molecules (11 ). It can be observed from figure 1(a) that the three lump molecules consist of six single lumps, where every two of them form a molecular structure, i.e. the corresponding value of the x-coordinate and the velocity in the x-direction of the two lumps are equal, as t tends to negative large value. Then each pair of lump molecules moves along an independent curve (see figure 1(b)). When t tends to 0, all the lumps gather into a big lump with more concentrated energy. The lump molecules gradually form a triangular structure with the increase of time t, which can be seen in the panels t = 10 and t = 25. Finally, the center points of all the lump waves are located on the x-axis, in which every two lumps form a pair of molecules (see the panel t = 90). Based on the above analysis, we can conclude that the multiple lump molecules maintain the structure of molecules as $\left|t\right|\to \infty $.
Figure 1. Three two-lump molecules ( |
Figure 2 is shown to illustrate the effect of the positive and negative parameters ${a}_{i},\left(i=1,2,3\right)$ on the structure of the multiple lump molecules. Figure 2(a) shows that there are three two-lump molecules on the plane of $\left(x,y\right)$. Since the sign of a1 is negative, the pair of the lump molecule III lies on the straight line y = 0 when t is equal to −30. This structure is similar to the lump molecules presented in figure 1(f). Then the lump molecule III is longitudinally oriented as t gradually increases, which indicates that the motion of the corresponding molecule is reversed if the sign of ai is opposite. By analyzing figures 2(a) and (c), it can be seen that the absolute value of ai plays a decisive role in the velocity and amplitude of lump molecules. With the increase of $\left|{a}_{i}\right|$, the amplitude and velocity of the corresponding lump molecule increase.
Figure 2. Three two-lump molecules ( |
Based on the analysis of the three two-lump molecules, one can establish the following proposition for the multiple lump molecules.
The multiple molecule solutions consisting of M two-lump molecules can be constructed by means of an Mth-order determinant
$\begin{eqnarray}u\left(x,y,t\right)=2\displaystyle \frac{{\partial }^{2}}{\partial {x}^{2}}\mathrm{ln}\left(\left|\begin{array}{cccc}{\tau }_{11} & {\tau }_{12} & \cdots & {\tau }_{1M}\\ {\tau }_{21} & {\tau }_{22} & \cdots & {\tau }_{2M}\\ \vdots & \vdots & \ddots & \vdots \\ {\tau }_{M1} & {\tau }_{M2} & \cdots & {\tau }_{{MM}}\end{array}\right|\right),\end{eqnarray}$
where ${\tau }_{{jk}}=\tfrac{{b}_{j}{b}_{k}}{{a}_{j}+{a}_{k}}{\rho }_{22}\left(x,y,t,{a}_{j},{a}_{k}\right),\,j,k\,=\,1,2,\cdots ,M.$ The dynamical behavior of every two lump molecule among the M two-lump molecules is similar to molecule I or II or III. If the element ${\tau }_{{jk}}$ in the determinant is replaced by the function $\tfrac{{b}_{j}{b}_{k}}{{a}_{j}+{a}_{k}}{\rho }_{33}\left(x,y,t,{a}_{j},{a}_{k}\right)$, the solution (Furthermore, we can consider the interaction solutions between multiple lump molecules and line solitons by making the matrix ϑjk nonzero. If we choose ϑjk = δjk, where δjk is the Kronecker symbol, equation (10 ) can lead to the interaction solutions of M line solitons radiating M two-lump molecules13 ) can derive the interaction solution between M two-lump molecules and Q line solitons in which Q of M two-lump molecules are emitted by the Q line solitons. Furthermore, we can establish a generalized proposition to describe the interaction between M N-lump molecules and P line solitons as follows.
$\begin{eqnarray}u\left(x,y,t\right)=2\displaystyle \frac{{\partial }^{2}}{\partial {x}^{2}}\mathrm{ln}\left(\left|\begin{array}{cccc}{\tilde{\tau }}_{11} & {\tau }_{12} & \cdots & {\tau }_{1M}\\ {\tau }_{21} & {\tilde{\tau }}_{22} & \cdots & {\tau }_{2M}\\ \vdots & \vdots & \ddots & \vdots \\ {\tau }_{M1} & {\tau }_{M2} & \cdots & {\tilde{\tau }}_{{MM}}\end{array}\right|\right),\end{eqnarray}$
where ${\tau }_{{jk}}=\tfrac{{b}_{j}{b}_{k}}{{a}_{j}+{a}_{k}}$ ${\rho }_{22}\left(x,y,t,{a}_{j},{a}_{k}\right)$, j, k = 1, 2, ⋯ ,M, j ≠ k, aj ≠ ak and ${\tilde{\tau }}_{{jj}}={{\rm{e}}}^{-\sigma \left(x,y,t,{a}_{j}\right)-\bar{\sigma }\left(x,y,t,{a}_{j}\right)}$ $+\tfrac{{b}_{j}^{2}}{2{a}_{j}}{\rho }_{22}\left(x,y,t,{a}_{j},{a}_{j}\right),j=1,2,\cdots ,M$. Figure 3 devotes to describing the process of the three line solitons radiating three two-lump molecules. There are three line solitons on the plane $\left(x,y\right)$ as t tends to negative large value, which can be observed in figure 3(a). Then each line soliton gradually emits one longitudinal two-lump molecule, which is shown in the panel t = −10. The three two-lump molecules gradually peel off from the line solitons to form a triangular structure (see figures 3(c), (d) and (e)). Finally, the structure of three line solitons radiating three two-lump molecules is presented in figure 3(f) as t tends to 90. Especially, if the ${\tilde{\tau }}_{{jj}}={\mu }_{j}{{\rm{e}}}^{-\sigma \left(x,y,t,{a}_{j}\right)-\bar{\sigma }\left(x,y,t,{a}_{j}\right)}+\tfrac{{b}_{j}^{2}}{2{a}_{j}}{\rho }_{22}\left(x,y,t,{a}_{j},{a}_{j}\right),$ ${\mu }_{j}=0,1,j=1,2,\cdots ,M,$ and the number of nonzero μj is Q, equation (
Figure 3. Three line solitons radiate three two-lump molecules ( |
The interaction solution of the KPI equation describing P line solitons radiating P of the M N-lump molecules ($P\leqslant M$) can be given as follows
$\begin{eqnarray}u\left(x,y,t\right)=2\displaystyle \frac{{\partial }^{2}}{\partial {x}^{2}}\mathrm{ln}\left(\left|\begin{array}{cccc}{\tilde{\tau }}_{11} & {\tau }_{12} & \cdots & {\tau }_{1M}\\ {\tau }_{21} & {\tilde{\tau }}_{22} & \cdots & {\tau }_{2M}\\ \vdots & \vdots & \ddots & \vdots \\ {\tau }_{M1} & {\tau }_{M2} & \cdots & {\tilde{\tau }}_{{MM}}\end{array}\right|\right),\end{eqnarray}$
where ${\tau }_{{jk}}=\tfrac{{b}_{j}{b}_{k}}{{a}_{j}+{a}_{k}}$ ${\rho }_{{NN}}\left(x,y,t,{a}_{j},{a}_{k}\right)$, $j,k=1,2,\cdots ,M,\,j\ne k,$ ${a}_{j}\ne {a}_{k}$, ${\tilde{\tau }}_{{jj}}={\mu }_{j}{{\rm{e}}}^{-\sigma \left(x,y,t,{a}_{j}\right)-\bar{\sigma }\left(x,y,t,{a}_{j}\right)}$ + $\tfrac{{b}_{j}^{2}}{2{a}_{j}}{\rho }_{{NN}}$ $\left(x,,,y,,,t,,,{a}_{j},,,{a}_{j}\right)$, ${\mu }_{j}=0,1,j=1,2,\cdots ,M,$ and the number of nonzero ${\mu }_{j}$ is P.Proposition 2 gives a unified scheme for constructing the interaction solutions between P line solitons and M N-lump molecules, in which P N-lump molecules are emitted by the P line solitons, respectively. The appearance of multiple lump molecules is similar to the generation of rogue waves [29]. The difference is that it only experiences the process from nonexistence to passing into existence. equation (
4. Conclusions
This paper mainly investigates the multiple lump molecules including three two-lump molecules, high-order M two-lump molecules and M N-lump molecules by means of the modified version of the solution in form of a Gramian for the KPI equation. The interaction dynamical behaviors between the multiple lump molecules and line solitons are studied. We have constructed the interaction solution of three line solitons radiating three two-lump molecules. The process of the interactions has been analyzed with the aid of numerical simulation. Furthermore, the interaction solution of the KPI equation describing P line solitons radiating P of the M N-lump molecules is presented. All obtained new types of solutions will enrich our understanding of the multiple lump waves of the KPI equation. We hope the results are helpful to explain the related nonlinear phenomena in the fields of nonlinear optics, fluid mechanics, ocean waves and solitons.