On fixed proximal pairs of $E_r$-mappings

1.   Introduction

"A rogue wave" is a single wave that comes out in the ocean "from nowhere" with a much higher amplitude than the average crests around it ^[1]. It was thought to be mystical until someone had a similar experience ^[2]. Those who are involved in such a disaster surely will focus on saving lives rather than recording evidence ^[3]. For such reason, there does not exist a clear understanding of this occurrence until now ^[4]. The superposition principle based on linear theory is inadequate to explain large amplitude, but nonlinear theory would be more helpful ^[5]. There are two main characteristics of rogue waves: "appear from nowhere" and "disappear without a trace." The first characteristic lies in modulation instability as the solution encounters abnormal initial input that will increase exponentially ^[6]. On the other hand, Bespalov-Talanov instability does not last long and will return the solution to its initial state ^[7]. As a result, in a conservative nonlinear system, each wave that "appears from nowhere" must "disappear without a trace" ^[8]. In contrast to soliton solution, lump solution is a kind of rational function solution that is localized in each direction ^[9]. A general rational solution was presented for an integrable system such as the KdV equation, the Boussinesq equation, the Toda lattice equation via the Wronskian and Casoratian determinant method.

Lately, Tian derived lump solution, multi-kink soliton and discussed the interaction between tripe and lump soliton from a (3+1)-dimensional Kadomtsev-Petviashvili equation with the help of Hirota's bilinear method and the Bäcklund transformation ^[10]. Moreover, the mechanism and transition of the solution have been investigated for the (2+1)-dimensional Sawada-Kotera equation using phase shift analysis and characteristic line ^[11]. It is worth noting that the $\bar{\partial}$-dressing method is proposed by Tian to study the three-component couple Hirota(tcCH) equation ^[12]. The coupled Maxwell-Bloch equations and the AKNS type integrable shallow water wave equation are studied by Darboux transformation ^[13,14,15]. Other work that involves Darboux transformation and symmetry condition can be seen in ^[16]. Recently, the consistent Riccati expansion method and the consistent tanh expansion method are employed to investigate KdV equation ^[17,18]. Bäcklund-transformation and symbolic computation are also effective in dealing with nonlinear models ^{[19,20,21,22,23,24]}.

In 1993, Camassa and Holm derived a completely integrable dispersive shallow water wave equation via Hamiltonian methods, i.e.,

where $u = u(x, t)$ is the height of the water's free surface over a flat bottom, related to the spatial coordinate $x$ and time $t$; $k$ is a constant that has to do with the shallow water wave speed; the subscripts stand for the partial derivatives ^[25]. As a small amplitude expansion of incompressible Euler's equations for unidirectional wave motion at the free surface under the influence of gravity, the higher order terms (the right-hand side) are usually neglected in the small amplitude, shallow water limit ^[26]. Leaving those terms brings about the Benjamin-Bona-Mahony equation or the Korteweg-de Vries equation in the same order ^[27]. Therefore, the Camassa-Holm(CH) equation can be treated as an extension of the Benjamin-Bona-Mahony(BBM) equation ^[28,29]. This extension renders solitons whose limiting form($k\rightarrow 0$) is peak as their first derivative is discontinued(compared with the smooth isolated waves) ^[30]. Such "peakons" dominate the solutions of the initial value problems under the condition of $k = 0$, i.e.,

where $c$ is the wave speed, which means $u$ has slope discontinuities ^[31]. If a derivative discontinuity occurs, we can limit a verticality at each inflection point to break a smooth initial condition into a series of peakons. Moreover, multisolitons can be obtained by superimposing the single solitons and solving as a completely integrable finite dimensional bi-Hamiltonian system, i.e., it can be written in two different kinds of Hamiltonian forms ^[32]. It is this property modifies the CH equation as a compatibility condition for a linear isospectral problem, such that the inverse scattering transformation can solve the initial value problem. Boyd pointed out why CH equation is important: it is a shallow water model like the KdV equation; two is it gives the peaked periodic waves with discontinuous first derivatives ^[33]. The higher order terms will come small and disposable under a slowly varying condition $\xi = x-ct$. Therefore, the soliton is given to the lowest order by the solutions of

According to Eq (1.2), Wazwaz proposed two modified forms of CH equation

and

where $a > 0, k\in R$, $n$ is the strength of the nonlinearity ^[34,35]. Under this sense, Wazwaz further presented two variants of (2+1)-dimensional Camassa-Holm-KP(CH-KP) equations

and

Since Eqs (1.5) and (1.6) are derived from the modified CH equations (1.3) and (1.4), which are similar to the deduction of the Kadomtsev-Petviashvili equation, thus they are called the "Camassa-Holm-KP equation" ^[36].

With the help of simulation method and dynamical system theory, Xie derived loops, periodic cusp waves, kinks, and peakons of a generalized (2+1)-dimensional CH-KP equation ^[37]. Biswas obtained analytic 1-soliton solution of two generalized CH-KP equations ^[38]. Tian investigated breathers, rogue waves and other kinds of solitary waves of a generalized CH-KP equation, through bilinear formalism and homoclinic breather limit approach^[39]. Lai studied solitary patterns, periodic and algebraic traveling wave solutions for two generalized (2+1)-dimensional CH-KP equations by direct integration ^[40]. As for (3+1)-dimensional cases, a generalized (3+1)-dimensional time fractional CH-KP equation was educed in the sense of Riemann-Liouville fractional derivatives with the help of Agrawal's method, Euler-Lagrange equation and semi-inverse method ^[41].

On the other hand, by adding $u_{tt}$ to the KP equation, Wazwaz introduced a generalized KP-Boussinesq equation

which makes a significant impact on the phase shift and the dispersion relation ^[42]. In this paper, following the same manners of Wazwaz, we propose an extension to the CH-KP equation:

where $u = u(x, y, t)$ is the height of the water's free surface over a flat bottom; $\alpha, \beta, \gamma, \lambda, \delta$ are nonzero constants. Equation (1.7) simulates the dispersion's role in the development of patterns in a liquid drop, and describes left and right traveling waves like the Boussinesq equation.

To our knowledge, the rational and generalized rational solution has not been reported for Eq (1.7) yet. Therefore, in this paper, we intend to find rational and generalized rational solutions of Eq (1.7) with the help of bilinear form and symbolic computation. In Section 2, we derive the second order, third order and fourth order rational solutions and explore the inner connections between the bilinear equation and rational solution by discussing the relevance between the complex roots and the formation of the waves. In Section 3, we obtain generalized rational solutions with two arbitrary parameters, that can be used to modulate the evolution progress. The detailed dynamical behaviors for the generalized rational solutions are discussed in Section 4, including the scatter behavior, exact locations where lump waves space, and the moving path. In Section 5, we analyze the multiple dark wave solitons. Section 6 contains the summary of this paper.

2.   Rational solutions of Eq (1.7)

In this section, we will construct rational solutions of Eq (1.7) and assess the inner link between the bilinear equation and rational solution. Taking $\xi = x+t$, Eq (1.7) will be translated to

where $\alpha, \lambda, \beta, \gamma, \delta$ are real parameters. Using the transform $u = \frac{6 \gamma}{\beta}(\ln f)_{\xi\xi}$, (2.1) becomes

which is equal to

As we can see, the bilinear equation (2.2) has three parts, and $\alpha, \lambda$ serve as the same effect in the second part. Therefore, in order to facilitate the following computation, we treat $\alpha, \lambda$ as the same parameter(setting $\alpha+\lambda = \alpha$). Of course, one can retrieves the original result by replacing $\alpha$ with $\alpha+\lambda$. With the help of bilinear form (2.2), the rational solution of Eq (1.7) can be obtained by the following theorem.

Theorem 2.1. The generalized CH-KP equation (1.7) has rational solution

with

where $a_{i, j}$ is real parameter, $\beta, F_n\neq 0$ in order to make $u_n$ analytic ^[43].

Using upon procedure one has

Remark 2.1. The solutions plotted in Figure 1 can be thought of as bound states of individual solitons, namely, two-, three- and four-humped multisolitons ^[44].

We can derive two kinds of waves by choosing different coordinates. For instance, choosing $(\xi, y)$, we can derive multi-lump soliton and $(x, t)$ for multi-wave soliton. Firstly, let us consider the multi-lump soliton. The second order, third order and fourth order rational solutions are illustrated in Figure 1, showing two, three, four lumps under the conditions: $\alpha = \delta = -\gamma = 1, \beta = -6$ for Figure 1(a) and (b); $\alpha = -2, \delta = -1, \gamma = 1, \beta = 6$ for Figure 1(c). Second order rational solution $u_2$ has two separated peaks while there is one shaper peak in the middle surrounded by two shorter peaks for $u_3$. As for $u_4$, it has four divided peaks, two in the middle and two guarding around. Apparently, we can conclude that the $n$th order rational solution $u_n$ has $n$ separated peaks and the maximum value locates at $y = 0$ with $n$ maximum values of $u_n$. On the other hand, multi-wave solitons are shown in Figures 2–4. According to the evolutional plots, these waves stem from a plain wave background and hit the maximum amplitude at $y = 0$. Then, they disappear with time. Interestingly, we find that higher order multi-wave solitons may contain lower ones. For example, the two-wave soliton does exist in the evolution progress of the three-wave soliton (see Figure 3(a)) and the three-wave soliton does exist in the evolution progress of the four-wave soliton (see Figure 4(a)).

Remark 2.2. Here, the solitons are plotted in ($\xi, t$)-plane and ($x, t$)-plane, and they are evolutionary of variable $y$. For some other models that transformed by $\xi = x+y$ in (2.1), the corresponding solutions will be evolutionary of $t$.

In this part, we will look into the connection between polynomial $F_n$ and rational solution $u_n$, through discussing the relevance between the complex roots and the formation of the waves. In Figure 5, each picture gives the complex root of $F_n = 0$ on the complex field, for $y = 0, y = 3n(n = 2, 3, 4)$ of $x$, i.e.,

As shown in Figure 5, the red points are closer to the real axis than the blue points, which means the complex roots of $F_n = 0$ will leave the real axis when $y$ grows. This result also accords with Figure 6. Above pictures demonstrate a "triangular pattern" for $y = 0, y = 3n$. For instance, the complex roots of $F_2(\xi, 0) = 0$ construct two adjacent triangles, and the complex roots of $F_2(\xi, 6) = 0$ form two divided triangles.

We overlay the sectional view of $u_4$ on the map of the complex roots of $F_4(\xi, y) = 0$ in Figure 7. Unlike Figures 5 and 6, we choose six different values of $y$, that is $y = -15, -2, 0, 1, 4, 15.$ As shown in Figure 7, the roots move to the real axis as $y$ increases and $y = 0$ is the moment that those roots cluster together near the base point most closely. Next, they move away from the base point and form as isolated triangles like the initial condition. Together with the sectional view of $u_4$, we come to the conclusion that more complex roots around the base point, higher amplitude the wave has.

3.   Generalized rational solutions of Eq (1.7)

Above, we have investigated the rational solutions of Eq (1.7) and assessed the inner link between the bilinear equation and the rational solution. Next, we are going to construct the generalized rational solutions of Eq (1.7) by introducing two free parameters $\mu, \nu$, which possess more complex mechanisms than their counterpart. In this case, the generalized rational solution also has two kinds of waves, multi-lump soliton and dark-multi-wave soliton (compare with the bright-multi-wave soliton in Section 2). With the help of bilinear form (2.2), the generalized rational solution of Eq (1.7) can be obtained by the following theorem.

Theorem 3.1. The extended CH-KP equation (1.7) has generalized rational solution

with

and

where $\bar{F}_{n}\neq 0$ and $\mu, \nu, b_{i, j}, c_{i, j}$ are real parameters ^[43].

3.1.   Second order generalized rational solution

Substituting $\bar{F_2}$ into (2.2) and equating the coefficients of all powers of $\xi, y$ to zero, we have

where $a_{3.3}, b_{1.1}, c_{1.1}$ are arbitrary constants. Substituting into (3.1) and using $u = \frac{6 \gamma}{\beta}(\ln f)_{\xi\xi}$, the solution of Eq (1.7) can be written as

Here, $\alpha = -\gamma = \sigma = a_{3.3} = b_{1.1} = c_{1.1} = 1, \beta = -6$. Starting from the rational soliton $u_2(\xi, y)$ (see Figure 1(a)), the evolution plots of the generalized rational soliton $\bar{u}_2(\xi, y, \mu, \nu)$ are displayed in Figure 8. Numerical simulation indicates that there are two peaks for $\bar{u}_2(\xi, y, \mu, \nu)$ when $\mu, \nu$ are small. One of the peaks shrinks, and divides into two small hills as $\mu, \nu$ increase. After that, those tiny hills will rise up if $\mu, \nu$ continue increase. Eventually $\bar{u}_2(\xi, y, \mu, \nu)$ contains three separated lump waves and organizes a triangle (see Figure 9).

3.2.   Third order generalized rational solution

Substituting $\bar{F_3}$ into (2.2) and equating the coefficients of all powers of $\xi, y$ to zero, we have

Substituting into (3.1) and using $u = \frac{6 \gamma}{\beta}(\ln f)_{\xi\xi}$, the solution of Eq (1.7) can be written as

Here, $\alpha = -\gamma = \sigma = 1, \beta = -6$. From Figure 1(b), we know that the rational solution $u_3(\xi, y)$ only has three peaks, with one in the middle and two guarding around. As $\mu, \nu$ increases, two bilateral peaks start to split. To be specific, the left side peak divides into three, while the right side peak divides into two(see Figure 10). For sufficient large $\mu, \nu$, $\bar{u}_3(\xi, y, \mu, \nu)$ organizes a pentagon with one peak in the middle and five around it(see Figure 11).

3.3.   Fourth order generalized rational solution

Substituting $\bar{F_4}$ into (2.2) and equating the coefficients of all powers of $\xi, y$ to zero, we have

where $c_{4, 6}$ is an arbitrary constant. Substituting into (3.1) and using $u = \frac{6 \gamma}{\beta}(\ln f)_{\xi\xi}$, the solution of Eq (1.7) can be written as

Here, $\alpha = -2, \beta = 6, \gamma = 1, \sigma = -1$. The evolution of $\bar{u}_4(\xi, y, \mu, \nu)$ is a bit more complicated than the lower order ones. In Figure 12, two guarding peaks divide into four separated peaks, as another three tiny hills rise nearby the central peaks. For sufficient large $\mu, \nu$, $\bar{u}_4(\xi, y, \mu, \nu)$ possesses one double-peak wave in the middle and seven in a ring around it, composing a heptagram(see Figure 13).

Remark 3.1. The multi-breather soliton that has similar structure such as forming a triangle, a pentagon and a heptagram can be seen in a generalized KP-BBM equation ^[45].

4.   Scatter of the lumps

Next, we will study the scatter behavior of the multi-lump soliton by selecting different $\mu, \nu$. Based on previous discussion, we know that $\bar{u}_n$ is capable of scattering to three-, six- and eight-lump waves as $\mu, \nu$ increases. It is natural to wonder, what will happen if these parameters continue to increase? Therefore, we choose three different values of $\mu, \nu$ for each solution to analyze their behavior. According to Figures 14–16, $\bar{u}_2, \bar{u}_3, \bar{u}_4$ own homologous character: if $\mu, \nu$ continue to rise, the lumps will maintain their shape and expand the clearance to a larger picture.

According to above analysis, we know that if $\mu, \nu$ continues to increase, these separated lumps will keep their formation and extend over a wider space. Obviously, the propagation observes a specific pattern. For this purpose, we will calculate the exact locations where extremum are and discuss their features. Under the same parametric conditions in Section 3, solve $\partial \bar{u}_n/\partial \xi = 0, \partial \bar{u}_n/\partial y = 0$ and omit extra point, we can obtain the location of extremum. Consequently, one has the following three circumcircles (see Figure 17)

where the lump waves locate, for $\bar{u}_2(\xi, y, 10^4, 10^4), \bar{u}_3(\xi, y, 10^7, 10^7), \bar{u}_4(\xi, y, 10^{10}, 10^{10})$, severally. It is noteworthy that during our calculation, the selection of the point is processed in a random order, and we find that other extremum accords with the same path.

We select several alternative values of $\mu, \nu$ for each circumstance and overlay them in one picture. In Figure 18(a), we set $\mu = \nu = 6\times 10^4, 2\times 10^5, 5\times 10^5, 1\times 10^6$. In Figure 18(b), we set $\mu = \nu = 1\times 10^7, 1\times 10^8, 5\times 10^8, 1.6\times 10^9.$ In Figure 18(c), we set $\mu = \nu = 5\times 10^{10}, 4\times 10^{11}, 2\times 10^{12}, 7\times 10^{12}.$ Based on above superimposed contour plots, as $\mu, \nu$ grows, the space between each other turns bigger, and the lump waves all move along a straight line in three, five, or seven different paths (red lines), respectively. Furthermore, except for $\bar{u}_2$, the higher order generalized rational solution has one stable solitary wave in the middle of the picture. Therefore, we have answered the query proposed at the beginning of Section 4: "what will happen if these parameters continue to increase?".

5.   Dark wave dynamics

For the generalized rational solution $\bar{u}_n$, it also has two appearances: multi-lump and multi-wave. Above, we have established the multi-lump soliton and studied its mechanism. Further, we will analyze the dark-multi-waves. Different from the bright-multi-waves obtained in Section 2, the dark-multi-waves do not come from a plain wave background and do not disappear with time. In Figure 19, there is one stripe soliton hidden in the valley at the initial input $y = -0.2$. It starts to shrink at first, then rises up and forms two gullies. As for the second order generalized rational solution $\bar{u}_2$, there are three stripe solitons in the valley at the initial datum. Then, these stripe solitons converge into one and rise up above the surface (see Figure 20). The evolution of the third order generalized rational solution $\bar{u}_3$ is more complicated than the others. As is shown in Figure 21, there are three hidden stripe solitons in the middle at the beginning. Then, they converge into two and divide into three and finally leave us four gullies.

6.   Conclusions

This paper proposes a new extension to the Camassa-Holm-Kadomtsev-Petviashvili (CH-KP) equation by adding an additional term $u_{tt}$, which simulates the dispersion's role in the development of patterns in a liquid drop and describes left and right traveling waves like the Boussinesq equation. We have derived the rational and generalized rational solutions for this equation through its bilinear form and symbolic computation. By plotting them in different planes, two waveforms exist: multi-lump and multi-wave. Two-, three- and four-lump solitons are demonstrated in Figure 1. Except for the multi-lump soliton, we have plotted the multi-wave solitons in Figures 2–4, showing as multiple order line rogue waves. According to our numerical simulations, these waves stem from a plain wave background and hit the maximum amplitude at $y = 0$. Then, they disappear with time. Furthermore, we have explored the inner link between $F_n$ and $u_n$ by discussing the relevance between complex roots of $F_n = 0$ and the formation of waves. As for the generalized rational solution $\bar{u}_n$ with two free parameters $\mu, \nu$, we have systematically analyzed its scatter behavior by setting different $\mu, \nu$. The multiple order dark waves are introduced in Section 5, and they are different from the multi-wave solitons in Section 2. We believe these results may aid in explaining the progress of rogue waves, as they not only provide a complete picture of how a rogue wave can "come out of nowhere and disappear without a trace", but also makes it realistic in the actual world. Moreover, multi-component and higher order rational solutions demonstrating diverse interesting phenomena, particularly in fully discrete and (3+1)-dimensional cases, would be very interesting topics in future research.

Conflict of interest

The authors declare that they have no conflict of interest.