1 Introduction
The study of nonlinear science has emerged as a powerful tool to understanding of many natural phenomena. In the past few decades, the soliton solutions have attracted more and more scholars' attention due to their crucial role in many branches of physics and engineering. Especially in Bose-Einstein condensations (BECs), nonlinear control, fluid dynamics and so on.[1-7] In recent years, the solitons and other related issues of nonlinear evolution equations (NLEEs) have become a hot topic.[8-11] It is worth noting that lump waves have been found by many researchers. Many methods to obtain soliton solutions of NLEEs are proposed with the deepening of research,[12-13] such as Hirota bilinear method,[14] inverse scattering transformation,[15] Darboux transformation (DT).[16] Lump waves can be observed in many fields, among which oceanics and nonlinear optics are the most common.[17-19] Numerous theoretical and experimental studies of lump waves are mentioned.[20-24]
In this paper, we consider the following (2+1)-dimen-sional generalized Konopelchenko-Dubrovsky-Kaup-Ku-pershmidt (gKDKK) equation[25]
$u_{t}+h_{1}u_{xxx}+h_{2}uu_{x}+h_{3}u_{xxxxx}+h_{4}v_{y}+h_{5}u_{xxy}\,, \\ \quad +h_{6}(u_{x}v+uv_{x})+h_{7}(u_{x}u_{xx}+uu_{xxx})+h_{8}u^{2}u_{x}=0\,, \\ u_{y}=v_{x}\,,$
where the coefficients $h_{i}$ $(i=1,2,\ldots,8)$ are the real parameters. When different special coefficient of $h_{i}$ are chose, the Bogoyavlensky-Konoplechenko equation,[26] the isospectral BKP equation[27] and the (2+1)-dimensional Sawada-Kotera equation[28] can be obtained, respectively. It is no exaggeration to say that many physical phenomena can be described by Eq. (1). The (2+1)-dimensional gKDKK equation was investigated as long ago as 2016 by Feng,[25] and it is pointed out that Eq. (1) has periodic wave solutions and asymptotic behaviors, which can be used to describe certain situations from the fluid mechanics, ocean dynamics and plasma physics. Recently, Man and Lou put forward a new way of thinking to get the lump and lumpoff solutions of the NLEEs in Ref. [29]. This result is very helpful for us to study some physical phenomena in engineering. The main aim of this paper is to investigate the lumps, lumpoff and the rogue waves with predictability of the gKDKK equation.
The rest of this paper is structured as follows. In Sec. 2, the general lump solutions for (2+1)-dimensional gKDKK equation are obtained with the help of the dependent variable transformation $u=12h_{1}h_{2}^{-1}(\ln f)_{xx}$ and the moving path of the lump waves is also described. In Sec. 3, we discussed the lumpoff waves based on the assumed equation Eq. (23). In Sec. 4, the special rogue waves of the Eq. (1) and the time and place of its occurrence are provided. Finally, conclusions and discussions are provided in Sec. 5.
2 General Lump Solutions for gKDKK Equation
In the present paper we consider a variable transformation
$ u=12h_{1}h_{2}^{-1}(\ln f)_{xx}\,.$
Substitution of Eq. (2) into Eq. (1), the Hirota bilinear form for the gKDKK equation can be expressed as[25]
${\rm Eq. (1)}=(D_{x}D_{t}\!+\!h_{1}D_{x}^{4}\!+\!h_{3}D_{x}^{6}+h_{4}D_{y}^{2}\!+\!h_{5}D_{x}^{3}D_{y})f \cdot f \\ = 2[(ff_{xt}-f_{x}f_{t})+h_{1}(3f_{xx}^{2}-4f_{x}f_{xxx}+ff_{xxxx}) \\ + \!h_{3}(ff_{xxxxxx}\!-\!6f_{x}f_{xxxxx}\!+\!15f_{xx}f_{4x} \!-\!10f_{xxx}^{2}) \\ +h_{4}(ff_{yy}-f_{y}^{2})+h_{5}(3f_{xx}f_{xy}-3f_{x}f_{xxy} \\ +ff_{xxxy}-f_{xxx}f_{y})]\,.$
$f=x^{T}Ax+f_{0}\,,$
with
$A=\left\{\begin{matrix} a_{00}&a_{01}&a_{02}&a_{03} \\ a_{10}&a_{11}&a_{12}&a_{13} \\ a_{20}&a_{21}&a_{22}&a_{23} \\ a_{30}&a_{31}&a_{32}&a_{33} \\\end{matrix} \right\} \\ x^{T}=(x_{0},x_{1},x_{2},x_{3})\,,$
in which $A\in R^{4\times4}$ is a symmetric matrix, $f_{0}$ is a positive constant. In particular, putting $x_{0}=1$, $x_{1}=x$, $x_{2}=y$, and $x_{3}=t$, then $f$ can be written as follows
$f=\sum_{i\leq j=0}^{3}a_{ij}x_{i}x_{j}+f_{0} \\ \hphantom{f}=a_{11}x^{2}+a_{22}y^{2}+a_{33}t^{2}+2a_{12}xy+2a_{13}xt \\ \hphantom{f=} +2a_{23}yt+2a_{01}x+2a_{02}y+2a_{03}t+a_{00}+f_{0}\,.$
From the properties of logarithmic function, it is easy to find that $f$ must be positive. Therefore, we suppose that $a_{ij}$ can be represented as
$ a_{ij}=\vec{A_{i}}\cdot\vec{A_{j}}=\sum_{k=1}^{p}A_{ik}A_{jk}\,,$
where
$\vec{A_{1}}=\vec{l}=(l_{1},l_{2},\ldots,l_{p})\,, \\ \vec{A_{2}}=\vec{m}=(m_{1},m_{2},\ldots,m_{p})\,, \\ \vec{A_{3}}=\vec{n}=(n_{1},n_{2},\ldots,n_{p})\,, \\ \vec{A_{0}}=\vec{\chi}=(\chi_{1},\chi_{2},\ldots,\chi_{p})$
are $p$ dimension vectors and $l_{k}$, $m_{k}$, $n_{k}$, $\chi_{k}$ are real constants to be determined later. Moreover, if take $\vec{\rho}=\sum_{i=0}^{3}x_{i}\vec{A_{i}}$, one can get
$\sum_{i\leq j=0}^{3}a_{ij}x_{i}x_{j}=\vec{\rho}\cdot\vec{\rho}=\sum_{k=1}^{p}\rho_{k}^{2}\geq0\,.$
Thus $f$ is always positive with $a_{ij}$ defined by Eq. (7). Substituting Eq. (6) into Eq. (3) and collecting all the coefficients of the same exponent of $x$, $y$, $t$, we can get a set of algebraic equations. By solving these objective equations, we find that these equations need only five solutions as follows
$a_{03}=\frac{h_{4}(a_{01}a_{22}-2a_{02}a_{12})}{a_{11}}\,, \quad a_{13}=\frac{h_{4}(a_{11}a_{22}-2a_{12}^{2})}{a_{11}}\,, \quad a_{23}=-\frac{h_{4}a_{12}a_{22}}{a_{11}}\,, \quad a_{33}=\frac{h_{4}^{2}a_{22}^{2}}{a_{11}}\,,$
$ f_{0}=-a_{00}+\frac{h_{4}a_{11}a_{02}^{2}-2h_{4}a_{01}a_{02}a_{12}+h_{4}a_{22}a_{01}^{2}-3h_{5}a_{11}^{2}a_{12}}{h_{4}(a_{11}a_{22}-a_{12}^{2})}-\frac{3h_{1}a_{11}^{3}}{h_{4}(a_{11}a_{22}-a_{12}^{2})}\,,$
where $a_{00}$, $a_{01}$, $a_{02}$, $a_{11}$, $a_{12}$, and $a_{22}$ are all arbitrary constants. Furthermore, inserting Eqs. (6)-(8) into the bilinear form Eq. (3), we can also find the following relationships
$n_{k}=\frac{h_{4}(a_{22}l_{k}-2a_{12}m_{k})}{a_{11}}\,,$
$ f_{0}=-a_{00}+\frac{h_{4}a_{11}a_{02}^{2} -2h_{4}a_{01}a_{02}a_{12}+h_{4}a_{22}a_{01}^{2} -3h_{5}a_{11}^{2}a_{12}}{h_{4}(a_{11}a_{22}-a_{12}^{2})} -\frac{3h_{1}a_{11}^{3}}{h_{4}(a_{11}a_{22}-a_{12}^{2})}\,.$
It means that these objective equations need only two constraint conditions under the constraint of Eqs. (10) and (11).
Then, according to the results in Ref. [29], taking $p=3$ for $\vec{A_{i}}$, the lump solutions of Eq. (1) will be more generalized than other values of $p$. That being said, the two constraint conditions can be written as
$n_{1}=-\frac{h_{4}[l_{1}(m_{1}^{2}-m_{2}^{2}-m_{3}^{2})+2m_{1}(l_{2}m_{2}+l_{3}m_{3})]}{l_{1}^{2}+l_{2}^{2}+l_{3}^{2}}\,, \quad n_{2}=-\frac{h_{4}[l_{2}(m_{2}^{2}-m_{1}^{2}-m_{3}^{2})+2m_{2}(l_{1}m_{1}+l_{3}m_{3})]}{l_{1}^{2}+l_{2}^{2}+l_{3}^{2}}\,, \\ n_{3}=-\frac{h_{4}[k_{3}(m_{3}^{2}-m_{1}^{2}-m_{2}^{2})+2m_{3}(l_{1}m_{1}+l_{2}m_{2})]}{l_{1}^{2}+l_{2}^{2}+l_{3}^{2}}\,,$
$f_{0}=-\frac{h_{4}[\Phi_{1}-\Phi_{2}]}{h_{4}[(l_{1}m_{2}-l_{2}m_{1})^{2}+(l_{2}m_{3}-l_{3}m_{2})^{2} +(l_{1}m_{3}-l_{3}m_{1})^{2}]} \\ \hphantom{f_0=} -\frac{3h_{1}(l_{1}^{2}+l_{2}^{2}+l_{3}^{2})^{3}}{h_{4}[(l_{1}m_{2}-l_{2}m_{1})^{2}+(l_{2}m_{3}-l_{3}m_{2})^{2}+(l_{1}m_{3}-l_{3}m_{1})^{2}]}\,,$
where $l_{k}$, $m_{k}$, $\chi_{k}$ $(k=1,2,3)$ are all arbitrary parameters, and
$\Phi_{1}=\chi_{1}(l_{2}m_{3}-l_{3}m_{2})+\chi_{2}(l_{3}m_{1}-l_{1}m_{3})+\chi_{3}(l_{1}m_{2}-l_{2}m_{1})]^{2}\,, \\ \Phi_{2}=3h_{5}[(l_{1}^{2}+l_{2}^{2}+l_{3}^{2})^{2}(l_{1}m_{1}+l_{2}m_{2}+l_{3}m_{3})]\,.$
By applying the transformation $u=12h_{1}h_{2}^{-1}(\ln f)_{xx}$, the general lump solution of the (2+1)-dimensional gKDKK equation has the following forms
$u_{\rm lump}=12h_{1}h_{2}^{-1}\bigg[\frac{2a_{12}}{f_{\rm lump}}-{4(a_{11}x+a_{12}y-[{h_{4}(a_{11}a_{22}-2a_{12}^{2})}/{a_{11}}]t+a_{01})^{2}}{f_{\rm lump}^{2}}\bigg]\,,$
in which
$f_{\rm lump}=a_{11}x^{2}+a_{22}y^{2}+\frac{h_{4}^{2}a_{22}^{2}}{a_{11}}t^{2}+2a_{12}xy+\frac{2h_{4}(a_{11}a_{22}-2a_{12}^{2})}{a_{11}}xt+\frac{2h_{4}a_{12}a_{22}}{a_{11}}yt+2a_{01}x+2a_{02}y \\ +\frac{2h_{4}(a_{01}a_{22}-2a_{02}a_{12})}{a_{11}}t+\frac{h_{4}a_{11}a_{02}^{2} -h_{4}a_{01}a_{02}a_{12}+h_{4}a_{22}a_{01}^{2}-3h_{5}a_{11}^{2}a_{12}}{h_{4}(a_{11}a_{22} -a_{12}^{2})}-\frac{3h_{1}a_{11}^{3}}{h_{4}(a_{11}a_{22}-a_{12}^{2})}\,,$
where $a_{ij}$ consists of $l_{k}$, $m_{k}$, $n_{k}$, $\chi_{k}$ in Eq. (7) with Eq. (8), and $n_{k}$ is defined by $l_{k}$ and $m_{k}$.
Especially, if we can find the critical point of the lump waves, the moving path of the lump waves can be described. Consider the case of $f_{x}=f_{y}=0$, we have
$x=-\frac{h_{4} a_{22}}{a_{11}}t-\frac{a_{01}a_{22}-a_{12}a_{02}}{a_{11}a_{22}-a_{12}^{2}}\,, \\ y=\frac{2h_{4} a_{12}}{a_{11}}t+\frac{a_{01}a_{12}-a_{11}a_{02}}{a_{11}a_{22}-a_{12}^{2}}\,.$
That is to say, the lump wave move along the straight line
$y=-\frac{2a_{12}}{a_{22}}x-\frac{a_{01}a_{12}a_{22}+a_{11}a_{02}a_{22}-2a_{12}^{2}a_{02}}{a_{22}(a_{11}a_{22}-a_{12}^{2})}\,.$
The graphical representation of lump solution Eq. (17) is described in Fig.1 with the following special parameters:
$l_{1}=-1\,, \quad l_{2}=-1\,, \quad l_{3}=\frac{1}{2}\,, \quad m_{1}=1\,, \\ m_{2}=1\,,\quad m_{3}=1\,, \quad \chi_{1}=1\,, \quad \chi_{2}=1\,, \quad \chi_{3}=1\,, \\ h_{1}=1\,, \quad h_{2}=6, \quad h_{4}=-5, \quad h_{5}=-5\,.$
Moreover, the moving path of lump waves is given by calculating the expression Eq. (19), one has
$ y=x-1\,.$
This observation can be clearly seen in Fig.1, the lump wave has the localized characteristic when $t=0$, and will propagate along a straight line as time changes.
Fig.1 (Color online) Space diagrams (a)-(c) and density plots (d)-(f) of lump solution Eq. (17) for Eq. (1) with the parameters Eq. (21). (a), (d) $t=-8$; (b), (e) $t=0$; (c), (f) $t=8$. |
Fig.2 (Color online) Space diagrams (a)-(c) and density plots (d)-(f) of lumpoff waves Eq. (27) for Eq. (1) with the parameters Eq. (28). (a), (d) $t=-2$; (b), (e) $t=4$; (c), (f) $t=12$. |
3 Lumpoff Solutions for gKDKK Equation
The so-called lumpoff solution is the interaction between lump wave solutions and stripe soliton wave solutions. At one point in time, the two are separated from each other and exist alone. But as time goes on, the lump waves will be swallowed by the solitary soliton waves. Before the beginning of the structural lump solutions, we assume $f_{\rm lumpoff}$ can be expressed as
$ f_{\rm lumpoff}=f_{\rm lump}+k\exp(l_{0}x+m_{0}y+n_{0}t+\chi_{0})\,,$
where $f_{\rm lump}$ is derived in Eq. (18), and $k$, $l_{0}$, $m_{0}$, $n_{0}$, $\chi_{0}$ are undetermined. The lump solutions and exponential solutions constitute the lumpoff waves . It is not hard to find that the exponentiation part is dominant when
$ l_{0}x+m_{0}y+n_{0}t+\chi_{0}>0\,. $
Otherwise, the lump solution only appears (that is $l_{0}x+m_{0}y+n_{0}t+\chi_{0}<0$).
Substituting Eq. (23) into Eq. (1), we get
$l_{0}^{2}=\frac{2h_{4}a_{11}(h_{1}a_{11}+h_{5}a_{12}+\sqrt{h_{1}^{2}a_{11}^{2}+2h_{1}h_{5}a_{11}a_{12}+h_{5}^{2}a_{11}a_{22}})}{3h_{5}^{2}a_{11}^{2}}\,,$
$m_{0}=-\frac{h_{5}}{2h_{4}}l_{0}^{3}+\frac{a_{12}}{a_{11}}l_{0}\,,$
$n_{0}=-\frac{h_{3}l_{0}^{6}+h_{1}l_{0}^{4}+h_{5}l_{0}^{3}m_{0}+h_{4}m_{0}^{2}}{l_{0}}\,,$
where $a_{11}$, $a_{12}$, $a_{22}$ are defined by Eq. (7) with Eq. (8), and $k$, $\chi_{0}$ are free constants.
The above results make us understand that $l_{0}$ and $m_{0}$ are completely determined by
Eqs. (24)-(25). However, the $n_{0}$ is related to $l_{0}$ and $m_{0}$. So what does that tell us the soliton waves are produced by lump waves. The existence of such lump waves in the soliton wave also exists.
Based on the condition above, substituting the $f_{\rm lumpoff}$ into Eq. (2), the lumpoff solutions can be written as
$u_{\rm lumpoff}= \frac{12h_{1}h_{2}^{-1}}{f_{\rm lumpoff}}[2a_{11}\!+kl_{0}^{2}\exp(l_{0}x\!+m_{0}y+n_{0}t\!+\chi_{0})] \\ -\frac{12h_{1}h_{2}^{-1}}{f_{\rm lumpoff}^{2}}[2a_{11}x+2a_{12}y+2a_{13}t+2a_{01} +kl_{0}\exp(l_{0}x+m_{0}y+n_{0}t+\chi_{0})]^{2}\,,$
where $l_{0}$, $m_{0}$, $n_{0}$ are given by Eqs. (24)-(26), and $k$, $\chi_{0}$ are arbitrary constants.
The corresponding dynamic characteristics of the lumpoff waves are plotted in Fig.2 with the following special parameters:
$l_{1}=-1\,, \quad l_{2}=1\,, \quad l_{3}=\frac{1}{2}\,, \quad m_{1}=1\,, \quad m_{2}=\frac{1}{2}\,, \\ m_{3}=\frac{1}{2}\,, \quad \chi_{1}=1\,, \quad \chi_{2}=1\,, \quad \chi_{3}=0\,, \quad h_{1}=-5\,, \\ h_{2}=-6\,, \quad h_{3}=1\,, \quad h_{4}=5\,, \quad h_{5}=45\,.$
Observation Eq. (23) is easy to find that the generation of lumpoff waves is based on the premise that the lump part is unchanged. The moving path of lumpoff solution is given by calculating the expression Eq. (19) and has the following forms
$y=\frac{1}{3}x-\frac{39}{53}\,.$
Figure 2 shows the process of evolution for different selections of parameter. Obviously, the lump wave is cut by the soliton. We also notice that the lump waves appear when $l_{0}x+m_{0}y+n_{0}t+\chi_{0}<0$ and covered by soliton in the end.
4 Rogue Waves with Predictability for gKDKK Equation
In the section, the special rogue wave solutions of Eq. (1) are considered. Its particularity lies in that the arising time and space can be predicted. In fact, the lump waves can be regarded as a special rogue waves. Next, we construct the rogue wave solutions for the gKDKK equation as follows
$f_{\rm rogue}=f_{\rm lump}+\frac{\mu}{2}\exp(l_{0}x+m_{0}y+n_{0}t+\chi_{0}) \\ \hphantom{f_{\rm rogue}=} +\frac{\mu}{2}\exp(-l_{0}x-m_{0}y-n_{0}t-\chi_{0})+\lambda_{0} \\ \hphantom{f_{\rm rogue}} =f_{\rm lump}+\mu\cosh(l_{0}x+m_{0}y+n_{0}t+\chi_{0})+\lambda_{0}\,,$
where $f_{\rm lump}$ is shown in Eq. (18), and the specific expression of $l_{0}$, $m_{0}$, $n_{0}$ are provided by Eqs. (24)-(26), $\mu$ and $\lambda_{0}$ being two arbitrary constants to be determined.
By observing Eq. (30), it is easy to find that the rogue wave $f_{\rm rogue}$ is composed of two parts of lump wave and exponential part. In Eq. (30), the cosh part is obviously dominant. In other words, if and only if the following conditions are satisfied
$ l_{0}x+m_{0}y+n_{0}t+\chi_{0}\sim0\,,$
the lump wave is emerge. That is to say, only soliton wave appears and lump wave will appear. Substituting Eq. (30) into Eq. (3) and collecting all relevant coefficients of $x$, $y$, $t$, cosh, sinh, a series of equations have been obtained. Based on the previous computational results, we have
$ \lambda_{0}=\frac{\mu^{2}l_{0}^{2}(3h_{1}a_{11}^{2}l_{0}^{2}+3h_{5}a_{11}a_{12}l_{0}^{2}-2a_{12}^{2}+2a_{11}a_{22})}{4a_{11}(a_{11}a_{22}-a_{12}^{2})}\,,$
where $a_{11}$, $a_{12}$, $a_{22}$ are given by Eq. (7) with Eq. (8), and $\mu$ is a free constant.
Via expressions (2), the rogue solution of gKDKK equation can be written as
$u_{\rm rogue}= 12h_{1}h_{2}^{-1}\Big[\frac{2a_{11}+\mu l_{0}^{2}\cosh(l_{0}x+m_{0}y+n_{0}t+\chi_{0})}{f_{\rm rogue}} \\ -\frac{(2a_{11}x+2a_{12}y+2a_{13}t+2a_{01}+\mu l_{0}^{2}\sinh(l_{0}x+m_{0}y+n_{0}t+\chi_{0}))^{2}}{f_{\rm rogue}^{2}}\Big]\,,$
where $l_{0}$, $m_{0}$, $n_{0}$ are defined by Eqs. (24)-(26), and $a_{01}$, $a_{11}$, $a_{12}$, $a_{13}$ are defined by Eqs. (7)-(8). $\mu$ and $\chi_{0}$ are all free constants.
In addition, we can see clearly from our results that the path and the emerge time of the rouge wave may be predict. Because the rogue wave will disappear with the loss of dominance, and it will appear only when $l_{0}x+m_{0}y+n_{0}t+\chi_{0}\sim0$. Therefore, on the basis of the moving path of lump waves Eq. (17), we can predict the appearance time and place of the special rogue waves by utilizing the center line $l_{0}x+m_{0}y+n_{0}t+\chi_{0}=0$ of a pair of resonance stripe soliton waves. The time $t$ reads
$ t=\frac{2a_{11}(h_{5}l_{0}^{3}a_{01}a_{11}a_{12}-h_{5}l_{0}^{3}a_{02}a_{11}^{2}+2h_{4}l_{0}a_{01}a_{11}a_{22}-2h_{4}l_{0}a_{01}a_{12}^{2}-2h_{4}\chi_{0}a_{11}^{2}a_{22}+2h_{4}\chi_{0}a_{11}a_{12}^{2})}{l_{0}(a_{11}a_{22}-a_{12}^{2})(2h_{5}^{2}l_{0}^{4}a_{11}^{2}-4h_{3}h_{4}l_{0}^{4}a_{11}^{2}-h_{5}^{2}l_{0}^{4}a_{11}^{2}-4h_{1}h_{4}l_{0}^{2}a_{11}^{2}-12h_{4}h_{5}l_{0}^{2}a_{11}a_{12}-4h_{4}^{2}a_{11}a_{12}+4h_{4}^{2}a_{11}^{2})}\,,$
and the place $x$, $y$ read
$x=-\frac{a_{01}a_{22}-a_{12}a_{02}}{a_{11}a_{22}-a_{12}^{2}} \\ \hphantom{y=} -\frac{2h_{4}a_{22}(h_{5}l_{0}^{3}a_{01}a_{11}a_{12}-h_{5}l_{0}^{3}a_{02}a_{11}^{2}+2h_{4}l_{0}a_{01}a_{11}a_{22}-2h_{4}l_{0}a_{01}a_{12}^{2}-2h_{4}\chi_{0}a_{11}^{2}a_{22}+2h_{4}\chi_{0}a_{11}a_{12}^{2})}{l_{0}(a_{11}a_{22}-a_{12}^{2})(2h_{5}^{2}l_{0}^{4}a_{11}^{2}-4h_{3}h_{4}l_{0}^{4}a_{11}^{2}-h_{5}^{2}l_{0}^{4}a_{11}^{2}-4h_{1}h_{4}l_{0}^{2}a_{11}^{2}-12h_{4}h_{5}l_{0}^{2}a_{11}a_{12}-4h_{4}^{2}a_{11}a_{12}+4h_{4}^{2}a_{11}^{2})}\,, \\ y=\frac{a_{01}a_{12}-a_{11}a_{02}}{a_{11}a_{22}-a_{12}^{2}} \\ \hphantom{y=} +\frac{4h_{4}a_{12}(h_{5}l_{0}^{3}a_{01}a_{11}a_{12}-h_{5}l_{0}^{3}a_{02}a_{11}^{2}+2h_{4}l_{0}a_{01}a_{11}a_{22}-2h_{4}l_{0}a_{01}a_{12}^{2}-2h_{4}\chi_{0}a_{11}^{2}a_{22}+2h_{4}\chi_{0}a_{11}a_{12}^{2})}{l_{0}(a_{11}a_{22}-a_{12}^{2})(2h_{5}^{2}l_{0}^{4}a_{11}^{2}-4h_{3}h_{4}l_{0}^{4}a_{11}^{2}-h_{5}^{2}l_{0}^{4}a_{11}^{2}-4h_{1}h_{4}l_{0}^{2}a_{11}^{2}-12h_{4}h_{5}l_{0}^{2}a_{11}a_{12}-4h_{4}^{2}a_{11}a_{12}+4h_{4}^{2}a_{11}^{2})}\,,$
where $\chi_{0}$ is free parameter, and $l_{0}$ is given in Eq. (24).
In order to analyze the propagation characteristics of the rogue wave in detail, we choose the following appropriate parameters to plot Fig.3:
$l_{1}=-\frac{1}{4}\,, \quad l_{2}=1\,, \quad l_{3}=-\frac{1}{4}\,, \quad m_{1}=1\,, \quad m_{2}=-\frac{1}{2}\,, m_{3}=1\,, \quad \chi_{1}=1\,, \\ \chi_{2}=0\,, \quad \chi_{3}=\frac{1}{2}\,, \quad h_{1}=1\,, \quad h_{2}=6\,, \quad h_{3}=5\,, \quad h_{4}=5\,, \quad h_{5}=15\,.$
Figure 3 describes the rogue wave will appear when $t$ is at a special value. But with the change of time $t$, the rogue waves will eventually be covered by the solitary waves.
Fig.3 (Color online) Space diagrams (a)-(c) and density plots (d)-(f) of rogue waves Eq. (33) for Eq. (1) with the parameters Eq. (36). (a), (d) $t=-1$; (b), (e) $t=0$; (c) (f) $t=1$. |
4 Conclusions and Discussions
In this paper, we mainly investigated the lump waves, lumpoff and rogue waves of the (2+1)-dimensional generalized Konopelchenko-Dubrovsky-Kaup-Kupershmidt equation. First, by constructing the special quadratic function Eq. (6) with a symmetric matrix, we have obtained the general lump solutions based on transform Eq. (2). It is worth emphasizing that the moving path of the lump waves Eq. (15) are given. Second, with the help of the ansatz Eq. (23), the lumpoff solution is also considered. Besides, the soliton is induced by the lumps, and so we say that the existence of the lump waves determines the existence of soliton. Furthermore, the rogue waves with predictability are derived when double solitons are induced by the lumps, we display the appearance time and place of the special rogue waves in Eqs. (34)-(35). Finally, the dynamic properties of these solutions are discussed by some 3-dimensional plots and contour plots with choices some special parameters. The lump waves and lumpoff are expected to play an increasingly important role in mathematical physics and engineering.