⇦Chinese Physics Letters, 2020, Vol. 37, No. 3, Article code 030501 Soliton Molecules and Some Hybrid Solutions for the Nonlinear Schrödinger Equation * Bao Wang (汪保)1, Zhao Zhang (张钊)2, Biao Li (李彪)2** Affiliations 1Robotics Institute, Ningbo University of Technology, Ningbo 315211 2School of Mathematics and Statistics, Ningbo University, Ningbo 315211 Received 29 December 2019, online 22 February 2020 *Supported by the National Natural Science Foundation of China under Grant Nos. 11775121, Department of Education of Zhejiang Province under Grant No. Y201839043 and the K.C. Wong Magna Fund in Ningbo University.
**Corresponding author. Email: libiao@nbu.edu.cn Citation Text: Wang B, Zhang Z and Li B 2020 Chin. Phys. Lett. 37 030501    Abstract Based on velocity resonance and Darboux transformation, soliton molecules and hybrid solutions consisting of soliton molecules and smooth positons are derived. Two new interesting results are obtained: the first is that the relationship between soliton molecules and smooth positons is clearly pointed out, and the second is that we find two different interactions between smooth positons called strong interaction and weak interaction, respectively. The strong interaction will only disappear when $t \to \infty$. This strong interaction can also excite some periodic phenomena. DOI:10.1088/0256-307X/37/3/030501 PACS:05.45.Yv, 02.30.Ik, 47.20.Ky, 52.35.Mw © 2020 Chinese Physics Society Article Text The famous nonlinear Schrödinger (NLS) equation is a very important system in many technological fields. Many studies started with the NLS equation and then extended to other systems. In Ref. [1], authors took the NLS equation as an example to illustrate the generation mechanism of higher-order rouge waves. Higher-order rouge waves for a complex modified KdV system are also obtained in a similar way.[2] In a very important recent research about the NLS equation,[3] Wang et al. not only derived breather positons but also clearly pointed out the relationship between higher-order breather positons and higher-order rouge waves. Under some conditions, the NLS equation $$ \begin{aligned} iq_t + q_{xx} + 2q^*q^2 = 0 \end{aligned}~~ \tag {1} $$ can be derived from an Ablowitz–Kaup–Newell–Segur system.[4] Here $q=q(x,t)$ denotes a complex function respect to $\{x,t\}$. The Lax pair and $n$-fold Darboux transformation of Eq. (1) are shown in Refs. [3,4]. The new solution $q^{[n]}$ can be generated by $n$-fold Darboux transformation from a seed solution $q$:[3,4] $$ \begin{aligned} q^{[n]} = q - 2i\frac{{N_{2n} }}{{D_{2n} }}, \end{aligned}~~ \tag {2} $$ with $$ N_{2n} = \left| {\begin{array}{*{20}c} {\phi _{11} } & {\phi _{12} } & {\lambda _1 \phi _{11} } & {\lambda _1 \phi _{12} } & \cdots & {\lambda _1^{n - 1} \phi _{11} } & {\lambda _1^n \phi _{11} } \\ {\phi _{21} } & {\phi _{22} } & {\lambda _2 \phi _{21} } & {\lambda _2 \phi _{22} } & \cdots & {\lambda _2^{n - 1} \phi _{21} } & {\lambda _2^n \phi _{21} } \\ {\phi _{31} } & {\phi _{32} } & {\lambda _3 \phi _{31} } & {\lambda _3 \phi _{32} } & \cdots & {\lambda _3^{n - 1} \phi _{31} } & {\lambda _3^n \phi _{31} } \\ {\phi _{41} } & {\phi _{42} } & {\lambda _4 \phi _{41} } & {\lambda _4 \phi _{42} } & \cdots & {\lambda _4^{n - 1} \phi _{41} } & {\lambda _4^n \phi _{41} } \\ \vdots & \vdots & \vdots & \vdots & \ddots & \vdots & \vdots \\ {\phi _{2n1} } & {\phi _{2n2} } & {\lambda _{2n} \phi _{2n1} } & {\lambda _{2n} \phi _{2n2} } & \cdots & {\lambda _{2n}^{n - 1} \phi _{2n1} } & {\lambda _{2n}^n \phi _{2n1} } \\ \end{array}} \right| , $$ $$ D_{2n} = \left| {\begin{array}{*{20}c} {\phi _{11} } & {\phi _{12} } & {\lambda _1 \phi _{11} } & {\lambda _1 \phi _{12} } & \cdots & {\lambda _1^{n - 1} \phi _{11} } & {\lambda _1^{n - 1} \phi _{12} } \\ {\phi _{21} } & {\phi _{22} } & {\lambda _2 \phi _{21} } & {\lambda _2 \phi _{22} } & \cdots & {\lambda _2^{n - 1} \phi _{21} } & {\lambda _2^{n - 1} \phi _{22} } \\ {\phi _{31} } & {\phi _{32} } & {\lambda _3 \phi _{31} } & {\lambda _3 \phi _{32} } & \cdots & {\lambda _3^{n - 1} \phi _{31} } & {\lambda _3^{n - 1} \phi _{32} } \\ {\phi _{41} } & {\phi _{42} } & {\lambda _4 \phi _{41} } & {\lambda _4 \phi _{42} } & \cdots & {\lambda _4^{n - 1} \phi _{41} } & {\lambda _4^{n - 1} \phi _{42} } \\ \vdots & \vdots & \vdots & \vdots & \ddots & \vdots & \vdots \\ {\phi _{2n1} } & {\phi _{2n2} } & {\lambda _{2n} \phi _{2n1} } & {\lambda _{2n} \phi _{2n2} } & \cdots & {\lambda _{2n}^{n - 1} \phi _{2n1} } & {\lambda _{2n}^{n - 1} \phi _{2n2} } \\ \end{array}} \right|, $$ and ${\it\Psi}_j=[\phi_{j1} \, \phi_{j2} ]^{\rm T}$ $\left (j=1, 2,\ldots, 2n \right)$ is a set of eigenfunctions with $\lambda=\lambda_j$. Equation (2) satisfies some constraints as follows: $$\begin{aligned} &\lambda _{2j} = \lambda _{2j - 1}^*, ~~\phi _{2j,1} = - \phi _{2j - 1,2}^* (\lambda _{2j - 1}), \\ & \phi _{2j,2} = \phi _{2j - 1,1}^* (\lambda _{2j - 1}),~~ j=1, 2,\ldots, n. \end{aligned}~~ \tag {3} $$ In recent years, soliton molecules have attracted increasing attention from scholars.[5–8] Since Lou[8] proposed the velocity resonance mechanism, researchers have theoretically discovered a variety of soliton molecules.[9,10] As far as we know, the soliton molecules of Eq. (1) have not been discovered by other researchers, so we mainly study the soliton molecules and hybrid solutions consisting of soliton molecules and other form solutions for the NLS system. First, molecules containing multiple solitons are constructed through velocity resonance. We obtain breather solutions with zero background by controlling the distance between solitons in the molecule. Furthermore, the relationship between smooth positon solutions and breather solutions is explained in detail. Finally, we mainly discuss the elastic interactions between smooth positons and soliton molecules. Two types of interactions between smooth positons are shown in this study. Eigenfunctions corresponding to $\lambda_j$ can be derived as the following form when the seed solution $q=0$: $$ {\it\Psi}_j=\left[ \begin{array}{c} \phi_{{{\it j1}}}\\ \phi_{{{\it j2}}} \end{array} \right] =\left[ \begin{array}{c} {{\rm e}^{-i\lambda_{{j}} \left(2\,{\lambda_{{j}}}t+x \right) -\alpha}}\\ {{\rm e}^{i\lambda_{{j}} \left(2\,{\lambda_{{j}}}t+x \right) +\alpha}} \end{array} \right].~~ \tag {4} $$ Here $\lambda_j=\xi_j+i\eta_j$, and $\alpha$ is a real number. Similar to the method in Ref. [10], if the $n$-soliton solution (2) with eigenfunctions (4) satisfies the velocity resonance condition $$ \begin{aligned} &{\rm Re}\left(\lambda_1 \right)={\rm Re}\left(\lambda_2 \right)=\cdots={\rm Re}\left(\lambda_n \right), \end{aligned}~~ \tag {5} $$ then a molecule consisting of $n$ solitons can be derived.
cpl-37-3-030501-fig1.png
Fig. 1. (a) A molecule consisting of four solitons decried by Eq. (2) with parameters $n=4, ~\alpha=10, ~\lambda_1=-\frac{1}{4}+\frac{i}{4},~ \lambda_3=-\frac{1}{4}+\frac{i}{5}, ~\lambda_5=-\frac{1}{4}+\frac{i}{3}, ~\lambda_7=-\frac{1}{4}+\frac{i}{2}$. (b) A molecule consisting of two solitons decried by Eq. (2) with parameters $n=2, \alpha=2, \lambda_1=-\frac{i}{8}, \lambda_3=\frac{i}{6}$. (c) The breather solution has the same parameters as (b) but $\alpha=0$.
Although there are molecules consisting of multiple solitons in the NLS system, we find that molecules containing two identical solitons cannot be obtained by comparing Eq. (1) with the mKdV equation.[10] From Figs. 1(b) and 1(c), we find that the soliton molecule is converted to a breather solution with zero background when the two solitons in a molecule are close enough to each other. In other words, two solitons with the same velocity ($\lambda_1=\xi_1+i\eta_1, \lambda_2=\xi_1+i\eta_2$) will be converted to a breather solution $q_b$ when $\alpha=0$: $$ \begin{aligned} &q_{b} = \frac{N_b}{D_b}, \end{aligned}~~ \tag {6} $$ with $$\begin{align} N_b =\,& 4\, \left(\eta_{{1}}-\eta_{{2}} \right) \\ &\cdot\bigg [ \eta_{{1}}{{\rm e}^{ \left(4\,i{\eta_{{1}}}^{2}\!-4\,i{\xi_{{1}}}^{2}+16\,\xi_{{1}}\eta_{{1 }}+8\,\eta_{{2}}\xi_{{1}} \right) t-2\, \left(i\xi_{{1}}\!-2\,\eta_{{1} }-\eta_{{2}} \right) x}}\\ &+\!\eta_{{1}}{{\rm e}^{ \left(4\,i{\eta_{{1}}}^ {2}-4\,i{\xi_{{1}}}^{2}+16\,\xi_{{1}}\eta_{{1}}-8\,\eta_{{2}}\xi_{{1}} \right) t-2\, \left(i\xi_{{1}}-2\,\eta_{{1}}+\eta_{{2}} \right) x}}\\ &-\!\eta_{{2}} \Big({{\rm e}^{ \left(4\,i{\eta_{{2}}}^{2}-4\,i{\xi_{{1} }}^{2}+8\,\xi_{{1}}\eta_{{1}} \right) t-2\, \left(i\xi_{{1}}-\eta_{{1 }} \right) x}}\\ &+\!{{\rm e}^{ \left(4\,i{\eta_{{2}}}^{2}\!-4\,i{\xi_{{1}}}^ {2}\!+24\,\xi_{{1}}\eta_{{1}} \right) t-2\, \left(i\xi_{{1}}-3\,\eta_{{ 1}} \right) x}} \Big) \bigg ] \left(\eta_{{1}}\!+\!\eta_{{2}} \right),\\ D_b=\,&-4\,\eta_{{1}}\eta_{{2}}{{\rm e}^{ \left(-4\,i{\eta_{{1}}}^{2}+4\,i{ \eta_{{2}}}^{2}+16\,\xi_{{1}}\eta_{{1}} \right) t+4\,x\eta_{{1}}}}\\ &-4\,\eta_{{1}}\eta_{{2}}{{\rm e}^{ \left(4\,i{\eta_{{1}}}^{2}-4\,i{\eta_{ {2}}}^{2}+16\,\xi_{{1}}\eta_{{1}} \right) t+4\,x\eta_{{1}}}}\\ &+\left( \eta_{{1}}-\eta_{{2}} \right) ^{2}{{\rm e}^{8\, \left(\eta_{{1}}-\eta _{{2}} \right) \left(t\xi_{{1}}+x/4 \right) }}\\ &+ \left(\eta_{{1}}+ \eta_{{2}} \right) ^{2}{{\rm e}^{2\, \left(4\,t\xi_{{1}}+x \right) \left(3\,\eta_{{1}}-\eta_{{2}} \right) }}\\ &+ \left(\eta_{{1}}+\eta_{{ 2}} \right) ^{2}{{\rm e}^{8\, \left(t\xi_{{1}}+x/4 \right) \left( \eta_{{1}}+\eta_{{2}} \right) }}\\ &+{{\rm e}^{2\, \left(4\,t\xi_{{1}}+x \right) \left(3\,\eta_{{1}}+\eta_{{2}} \right) }} \left(\eta_{{1}} -\eta_{{2}} \right) ^{2}. \end{align} $$ By a tedious calculation, we find that $\left| q_{b1} \right|=0$ when $x \to \infty, t \to \infty$. This means that breather solutions obtained through module resonance has the zero background. However, breather solutions obtained by Darboux transformation with a nonzero seed solution do not have this characteristic.[11,12] Crests of first-order breather solution $\left| q_{b} \right|$ satisfy the linear equation $x=4\,\xi_1t$. When $\eta_1\cdot\eta_2>0$, $\left\{ t=\,{\frac {\pi\,j}{2({\eta_{{1}}}^{2}-{\eta_{{2}}}^{2})}},~x= \,{\frac {-2\xi\,\pi\,j}{{\eta_{{1}}}^{2}-{\eta_{{2}}}^{2}}} \right\}$, $~j \in \mathbb{Z}$ corresponds to the crest of the breather solution $\left| q_{b} \right|$. The height of $\left| q_{b} \right|$ is $2\left|\eta_1+\eta_2\right|$. When $\eta_1\cdot\eta_2 < 0$, $\!\left\{ t={\frac {\pi\, \left(2j+1 \right) }{4({\eta_{{1}}}^{2}-{\eta_{{2}}}^{2})}}, ~x=\!-{\frac {\xi\,\pi\, \left(2j+1 \right) }{{ \eta_{{1}}}^{2}\!-{\eta_{{2}}}^{2}}} \right\}$, $~j \in \mathbb{Z}$ corresponds to the crest of the breather solution $\left| q_{b} \right|$. The height of $\left| q_{b} \right|$ is $2\left|\eta_1-\eta_2\right|$. Inspired by Ref. [3], we find that the second-order smooth positon solution $q_{2-p}$ is the limit of the breather solution $q_b$ when its period goes to infinity. According to the description in the previous two paragraphs, this means $\eta_1 \to \eta_2 $. The mathematical expression of $q_{2-b}$ will be represented by the determinant of eigenfunctions in the following part. So far, we have clearly pointed out the relationship among soliton molecules, breather solutions and smooth positons.
cpl-37-3-030501-fig2.png
Fig. 2. The evolution from a breather solution $\left| q_{b} \right| $ to a second-order smooth positon $\left| q_{2-p} \right| $: (a) a breather $\left| q_{b} \right| $ described by Eq. (6) with parameter selections $\xi_1=0, \eta_1=\frac{1}{2},~ \eta_2=\frac{1}{3} $; (b) a breather $\left| q_{b} \right| $ described by Eq. (6) with parameter selections $\xi_1=0,~ \eta_1=\frac{1}{2},~ \eta_2=\frac{3}{7} $; and (c) a second-order smooth positon $\left| q_{2-p} \right| $ described by Eq. (6) with parameter selections $\xi_1=0, ~\eta_1=\frac{1}{2}, ~\eta_2 \to \frac{1}{2} $.
Figure 2 clearly shows the evolution process from a breather solution to a second-order smooth positon. The maximum value $4 \left| \eta_1 \right|$ of $\left| q_{2-p} \right| $ is obtained at the origin. Similar to the conclusion in Refs. [13,14], the dynamics properties of second-order smooth positon $\left| q_{2-p} \right| $ for the NLS system are shown in the following. Proposition 1: A more precise approximate trajectory of second-order smooth positons are some curves defined by $$\begin{alignat}{1} &x=-4\,\xi_{{1}}t \pm {\frac {\ln (4\,\sqrt {{\eta_{{1}}}^{2}t} ) }{\eta_{{1}}}} +\frac{\alpha}{\eta_1}, ~~ t \to \infty,~~ \tag {7} \end{alignat} $$ $$\begin{alignat}{1} &x=-4\,\xi_{{1}}t \pm {\frac {\ln (4\,\sqrt {-{\eta_{{1}}}^{2}t} ) }{\eta_{{1}}}} +\frac{\alpha}{\eta_1}, ~t \to -\infty.~~ \tag {8} \end{alignat} $$ Only along the trajectories of Eqs. (7) and (8), the height $2 \left| \eta_1 \right|$ of second-order smooth positons can be derived. Next, we will introduce two kinds of interactions: the elastic interactions between smooth positons and soliton molecules, and the interactions between smooth positons. Proposition 2: A hybrid of an $m$th-order smooth positon and a molecule containing $l$ solitons has the following form on the basics of partial velocity resonance and semi-degenerate Darboux transformation: $$ \begin{aligned} &q_{\rm m-l-hyb} = - 2i\frac{{N_{2n}' }}{{D_{2n}' }}, \end{aligned}~~ \tag {9} $$ with $$\begin{align} &N_{2n}' = \left({\left. {\frac{{\partial ^{h(i)} }}{{\partial \epsilon ^{h(i)} }}} \right|_{\epsilon = 0} (N_{2n})_{ij} (\lambda _j + \epsilon)} \right)_{2n \times 2n}, \\ &D_{2n}' = \left({\left. {\frac{{\partial ^{h(i)} }}{{\partial \epsilon ^{h(i)} }}} \right|_{\epsilon = 0} (D_{2n})_{ij} (\lambda _j + \epsilon)} \right)_{2n \times 2n}, \end{align} $$ $$ \begin{aligned} h(x)=\begin{cases} \left[ {\frac{{x - 1}}{2}} \right], &x\leq 2m,\\ 0, &x > 2\,m, \end{cases} \end{aligned} $$ where $n = m + l,~ \lambda_1 =\lambda_3 = \cdots= \lambda_{2 m-1} $. Note $\lambda_{2m+1},\cdots, \lambda_{2m+2l-1}$ satisfies Eq. (5), $[x]$ denotes the floor function of $x$ and $D_{2n}, N_{2n}$ are shown by Eq. (2). In proposition 2, if $l = 0$, then hybrid solutions $q_{\rm m-l-hyb} $ will be converted to higher-order smooth positons $q_{m-p}$. From Fig. 3, we can roughly observe that there is no change except the phase shift before and after the collision between soliton molecules and higher-order smooth positons. According to the method in Ref. [15], in Fig. 3(a), the trajectories of second-order smooth positons before and after the collision are $x_{{-\inf}}=-t_{{-\inf}} \pm 2\,\ln \left(2\,\sqrt {-t_{{-\inf}}} \right) +8+\theta$ and $x_{{\inf}}=-t_{{\inf}} \pm 2\,\ln \left(2\,\sqrt {t_{{\inf}}} \right) +8-\theta$, $\theta=\frac{\ln \left(17 \right)}{2} +\frac{\ln \left(5 \right)}{2} $, respectively. Before and after the interaction, the height of second-order smooth positon $\left| q_{2-p} \right| $ is $1$. Proposition 3: Based on the degenerate Darboux transformation, a hybrid of an $m$th-order smooth positon and an $l$th-order smooth positon is provided by $$ \begin{aligned} &q_{\rm m - l-pos} = - 2i\frac{{N_{2n}' }}{{D_{2n}' }}, \end{aligned}~~ \tag {10} $$ with $$\begin{align} &N_{2n}' = \left({\left. {\frac{{\partial ^{g(i)} }}{{\partial \epsilon ^{g(i)} }}} \right|_{\epsilon = 0} (N_{2n})_{ij} (\lambda _j + \epsilon)} \right)_{2n \times 2n},\\ &D_{2n}' = \left({\left. {\frac{{\partial ^{g(i)} }}{{\partial \epsilon ^{g(i)} }}} \right|_{\epsilon = 0} (D_{2n})_{ij} (\lambda _j + \epsilon)} \right)_{2n \times 2n}, \end{align} $$ where $ n=m+l, ~\lambda_1=\lambda_3=\cdots= \lambda_{2\,m-1},~ \lambda_{2m+1}=\lambda_{2m+3}=\cdots=\lambda_{2m+2l-1}, ~\lambda_1 \neq \lambda_{2m+1}, $ $$ \begin{aligned} g(x)=\begin{cases} \left[ {\frac{{x-1}}{2}} \right], &x \leq i \leq 2m,\\ \left[ \frac{x-1}{2} \right]-m, &2m+1 \leq i \leq 2m+2l. \end{cases} \end{aligned} $$ Here $D_{2n},~ N_{2n}, ~{\it\Psi}_j$ are shown by Eqs. (2) and (4). Note that $D_{2n}'$ and $N_{2n}'$ in proposition 3 are different from those in proposition 2.
cpl-37-3-030501-fig3.png
Fig. 3. Elastic interaction $\left| q_{\rm m-l-hyb} \right| $ between soliton molecules and higher-order smooth positons: (a) $\left| q_{2-2-hyb} \right| $ described by Eq. (9) with $\lambda_1=\lambda_3=\frac{1}{4}+\frac{i}{2}, \lambda_5=\frac{i}{2}, \lambda_7=\frac{i}{4}, ~\alpha=4 $; (b) $\left| q_{3-2-hyb} \right| $ described by Eq. (9) with $\lambda_1=\lambda_3=\lambda_5=\frac{i}{2}, ~\lambda_7=-\frac{1}{4}+\frac{i}{4}, ~\lambda_9=-\frac{1}{4}+\frac{i}{2}, ~\alpha=4 $.
cpl-37-3-030501-fig4.png
Fig. 4. (a) $\left| q_{2-2-{\rm pos}} \right| $ described by Eq. (10) with $\lambda_1=\lambda_3=-\frac{1}{4}+\frac{i}{2},~ \lambda_5=\lambda_7=\frac{1}{4}+\frac{i}{2}$; (b) $\left| q_{2-2-{\rm pos}} \right| $ described by Eq. (10) with $\lambda_1=\lambda_3=\frac{i}{2},~ \lambda_5=\lambda_7=\frac{3i}{4}$.
Figure 4 clearly displays two different types of interactions between higher-order smooth positons. Combining propositions 1 and 3, we have the following conclusions: (i) When ${\rm Re}(\lambda_1) \neq {\rm Re}(\lambda_{2m+1}) $, the phenomenon shown in Fig. 4(a) will be obtained, which is called the weak interaction. This weak interaction is characterized by its short duration. Before and after this weak interaction, nothing changes except the phase. (ii) When ${\rm Re}(\lambda_1) = {\rm Re}(\lambda_{2m+1}) $, the result described in Fig. 4(b) can be generated, which is called the strong interaction. The salient feature of this strong interaction is its long duration. In other words, the strong interaction will only disappear when $|t| \to \infty $. This strong interaction can also trigger periodic phenomena. Compared with Ref. [10], in addition to the soliton molecules and hybrid solutions consisting of soliton molecules and smooth positons, two interesting and novel results are also obtained: the connection between soliton molecules and smooth positons is obtained; two different types of interactions between smooth positons are generated. The link between soliton molecules and smooth positons has been described clearly. Finally, we elaborate on two completely different types of interactions called the strong interaction and the weak interaction. Although these pictures are very beautiful, we have some problems because our professional ability is not very strong. By constraining the eigenvalues, we obtain breather solutions with zero background in this study. Furthermore, can higher-order rouge waves sitting on zero background be derived? How to obtain the dynamic properties of hybrid solutions described by proposition 3 accurately? We believe that these questions will be answered soon thanks to the efforts of many scholars. The authors would like to express their sincere thanks to Professor S.Y. Lou for his guidance and encouragement. The authors sincerely thank Dr. J.C. Chen and L.H. Wang for their discussion. References Generating mechanism for higher-order rogue wavesFew-cycle optical rogue waves: Complex modified Korteweg–de Vries equationGeneration of higher-order rogue waves from multibreathers by double degeneracy in an optical fiberDeterminant representation of Darboux transformation for the AKNS systemSpontaneous crystallization and filamentation of solitons in dipolar condensatesUltrafast measurements of optical spectral coherence by single-shot time-stretch interferometryReal-Time Observation of the Buildup of Soliton MoleculesSoliton molecules and asymmetric solitons in fluid systems via velocity resonanceSoliton Molecules, Asymmetric Solitons and Hybrid Solutions for (2+1)-Dimensional Fifth-Order KdV EquationSoliton molecules and novel smooth positons for the complex modified KdV equationDarboux Transformation of the Second-Type Derivative Nonlinear Schrödinger EquationThe n th-order degenerate breather solution for the Kundu–Eckhaus equationDynamics of the smooth positons of the complex modified KdV equationGenerating mechanism and dynamic of the smooth positons for the derivative nonlinear Schrödinger equationTrajectory equation of a lump before and after collision with line, lump, and breather waves for (2+1)-dimensional Kadomtsev–Petviashvili equation
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