Dan-Dan Zhang^1,4, Lei Wang^,1,4,∗, Lei Liu², Tai-Xing Liu¹, Wen-Rong Sun³¹School of Mathematics and Physics, North China Electric Power University, Beijing 102206, China
²Beijing Computational Science Research Center, Beijing 100193, China
³School of Mathematics and Physics, University of Science and Technology Beijing, Beijing 100083, China

First author contact: ⁴Co-first authors.^∗Author to whom any correspondence should be addressed.
Received:2021-03-17Revised:2021-06-16Accepted:2021-06-16Online:2021-07-16


Abstract
In this article, we consider the (3+1)-dimensional generalized Kadomtsev–Petviashvili (GKP) equation in fluids. We show that a variety of nonlinear localized waves can be produced by the breath wave of the GKP model, such as the (oscillating-) W- and M-shaped waves, rational W-shaped waves, multi-peak solitary waves, (quasi-) Bell-shaped and W-shaped waves and (quasi-) periodic waves. Based on the characteristic line analysis and nonlinear superposition principle, we give the transition conditions analytically. We find the interesting dynamic behavior of the converted nonlinear waves, which is known as the time-varying feature. We further offer explanations for such phenomenon. We then discuss the classification of the converted solutions. We finally investigate the interactions of the converted waves including the semi-elastic collision, perfectly elastic collision, inelastic collision and one-off collision. And the mechanisms of the collisions are analyzed in detail. The results could enrich the dynamic features of the high-dimensional nonlinear waves in fluids.
Keywords： state transition;time-varying feature;nonlinear superposition principle;interaction;classification


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Dan-Dan Zhang, Lei Wang, Lei Liu, Tai-Xing Liu, Wen-Rong Sun. Shape-changed propagations and interactions for the (3+1)-dimensional generalized Kadomtsev–Petviashvili equation in fluids. Communications in Theoretical Physics, 2021, 73(9): 095001- doi:10.1088/1572-9494/ac0ba5

1. Introduction

It is common knowledge that the exact solutions of nonlinear partial differential equations are of great significance in the fields of fluids, optics, Bose–Einstein condensate and plasmas, since they can provide valuable physical information and give deep understandings for some nonlinear wave phenomena [1–9]. Types of nonlinear wave include solitons [10], rogue waves [11], Akhmediev breathers [12], Kuznetsov–Ma breathers [13–15], superregular breathers [16, 17] and lump waves [18–28], to name a few. Solitons are local waves generated by the interaction of nonlinear and dispersive effects [10]. Rogue waves ‘appear from nowhere and disappear without traces’ [11]. Akhmediev breathers [12] and Kuznetsov–Ma breathers [13–15] are periodic breathers in space and time, respectively. The superregular breathers are composed of quasi-Akhmediev breather pairs and develop from the localized perturbations [16, 17, 29, 30]. The lump solutions, which are reported in certain high-dimensional models, are a type of the rational solutions and localized in space in all directions [18], and have aroused widespread concerns in recent years [18–28]. The breath-wave solution shows the periodic feature along a specific direction [31], the limiting case (the period of the breath wave approaches infinity) of which becomes the lump wave [18–28].

Under certain conditions, rogue waves and breathers can be converted into some other kinds of nonlinear waves [32–39]. This phenomenon is called the state transition of nonlinear waves [32–39]. The breather can be seen as the nonlinear combination of two wave components [periodic wave component (PWC) and solitary wave component (SWC)] [32–39]. When the group velocities of the two wave components are identical, the breather can be evolved into a sequence of converted waves, for instance, the Bell-shaped waves, W- and M-shaped waves, periodic waves and multi-peak solitary waves [32–39]. In the low-dimensional models, some results have been achieved [32–39]. For example, Akmediev et al have found that the breathers can be converted into solitons in the nonlinear Schrödinger equations with higher-order terms [32, 33]. Wang et al have discussed the state transitions of breathers and superregular breathers in some higher-order nonlinear Schrödinger equations [30, 34–37]. Liu et al have reported the state transitions between the W-shaped waves and rogue waves in the Hirota systems [38, 39].Zhao et al have investigated the relationship between the fundamental excitation and modulation instability in the Lakshmanan–Porsezian–Daniel model which contains higher-order terms [40]. Zhao et al have further found that two W-typed solitary waves are produced on the finite wave background from the weak modulation signal [41]. And two mechanisms, both modulation stability and modulation instability, are involved in the process of the generation for these W-typed waves [41].

Although many scholars have researched the state transitions of breathers in some low dimensional models [32–39], such mechanisms in the high dimensional cases have remained largely unexplored. Very recently, Wang et al have found the transitions of the lump-typed and breath waves in the (2+1)-dimensional Korteweg–de Vries (KdV) model [42]. They have discussed the transition mechanisms between the lump and W-typed waves, and the state transitions of the breath wave have also been presented [42]. Wang et al have studied the state transition of the high-dimensional Ito model [43]. The transition mechanisms of the single and two-breath waves are presented analytically, and the gradient relationship of the converted solutions has been obtained [43]. However, to the best of our knowledge, there are few relevant studies on the (3+1)-dimensional models.

In this article, we will consider the (3+1)-dimensional generalized Kadomtsev–Petviashvili equation as follows [44–47](1)$\begin{eqnarray}{u}_{{xt}}+\beta {u}_{x}^{2}+\alpha {u}_{{xxxx}}+\beta {{uu}}_{{xx}}+\gamma {u}_{{yy}}+\gamma {u}_{{zz}}=0,\end{eqnarray}$which can describe the propagations of nonlinear waves in fluid dynamics. In this equation, α, β and γ are some arbitrary constants, x, y, z are some spatial variables, t is a time variable, and u is a function of these four variables. Equation (1) contains a lot of classical equations. For example, when α=1, β=6, γ=0, it can be simplified to the KdV equation [48](2)$\begin{eqnarray}{u}_{t}+{u}_{{xxx}}+6{{uu}}_{x}=0,\end{eqnarray}$which can account for nonlinear wave phenomena in engineering and physics [49]. When α=1, β=6, γ=±1 (γ=−1 and γ=1 correspond to the KPI and KPII equations, respectively), z=0, equation (1) is converted into the (2+1)-dimensional KP equations [50](3)$\begin{eqnarray}{u}_{{xt}}+{u}_{{xxxx}}+6({u}_{x}^{2}+{{uu}}_{{xx}})\pm {u}_{{yy}}=0.\end{eqnarray}$In fluids, the (2+1)-dimensional KP equation characterizes the small amplitude and weakly dispersive water wave evolution in the shallow water regime [51–55]. When α=1, β=6, γ=±3, equation (1) can be reduced to the (3+1)-dimensional KP equations [56–58](4)$\begin{eqnarray}{({u}_{t}+6{{uu}}_{x}+{u}_{{xxx}})}_{x}\pm 3{u}_{{yy}}\pm 3{u}_{{zz}}=0,\end{eqnarray}$which have been found to characterize the three-dimensional solitary waves in weakly dispersive media [59, 58]. Ma has studied the rational solutions and periodic traveling wave solutions of equation (4) [58]. When choosing γ=−3 in equation (4), Khalfallah has acquired the periodic- and soliton-like solutions [60], and Cui et al have discussed the multiple breath-wave solutions [61]. Moreover, Tian et al have studied the lump and semi-rational solutions of equation (4) with γ=−3, and discussed the interactions between the lumps and solitons [62, 63]. When α=1, β=6, γ=−1 in equation (1), Ma et al have obtained two categories of lump solutions [64]. When α=−1, β=6, γ=−1 in equation (1), Wu has acquired the one- and two-periodic wave solutions [65]. Wang et al have presented the rogue wave solution and solitary wave solution of equation (1) by using the Bell’s polynomials, and the interactions between them have been studied [44]. Mao et al have obtained the lump solutions of equation (1) by Hirota’s bilinear method, and they have investigated the interaction between a lump and a stripe soliton [45].

In this article, we use the wave component analysis (nonlinear superposition principles) [43] to find the conditions of the state transition of equation (1). We show the interesting time-varying feature of converted waves of equation (1), which does not appear in the (1+1)-dimensional models, and illustrate the essence by using the phase shift and characteristic line analyses. Furthermore, we show that the phase shift difference between the SWC and PWC can result in the inelastic collision of the converted waves of equation (1). The classification for different kinds of converted waves is presented based on the ratio of wave numbers of wave components.

The arrangement of this article is structured as follows: in section 2, we will show the transition condition analytically and study the dynamics of time-varying feature of the first-order converted waves by analysing the phase shifts and characteristic lines. The classification for different kinds of converted waves of equation (1) will be investigated in section 3. The nonlinear interactions will be discussed in section 4. In the end, section 5 will give the conclusion of this article.

2. Time-varying dynamics of converted waves

2.1. Analysis of superposition components

Under the following variable transformation [44, 45](5)$\begin{eqnarray}u=\displaystyle \frac{12\alpha }{\beta }{(\mathrm{ln}f)}_{{xx}},\end{eqnarray}$the bilinear form of equation (1) can be written as [44, 45]:(6)$\begin{eqnarray}({D}_{x}{D}_{t}+\alpha {D}_{x}^{4}+\gamma ({D}_{y}^{2}+{D}_{z}^{2}))f\cdot f=0,\end{eqnarray}$in which the bilinear differential operators are defined by [66](7)$\begin{eqnarray}\begin{array}{rcl}{D}_{\zeta }^{p}{D}_{\eta }^{q}a\cdot b & = & {\left(\displaystyle \frac{\partial }{\partial \zeta }-\displaystyle \frac{\partial }{\partial \zeta ^{\prime} }\right)}^{p}\left(\displaystyle \frac{\partial }{\partial \eta }\right.\\ & & {\left.-\displaystyle \frac{\partial }{\partial \eta ^{\prime} }\right)}^{q}a(\zeta ,\eta )\cdot b\left(\zeta ^{\prime} ,\eta ^{\prime} \right){| }_{(\zeta ,\eta )=\left(\zeta ^{\prime} ,\eta ^{\prime} \right)}.\end{array}\end{eqnarray}$To present the the first- and second-order soliton solutions, the functions f₁ and f₂ can be given by(8)$\begin{eqnarray}{f}_{1}=1+{{\rm{e}}}^{{\theta }_{1}},\,\,{\theta }_{1}={k}_{1}x+{l}_{1}y+{m}_{1}z+{\omega }_{1}t+{\delta }_{1},\end{eqnarray}$and(9)$\begin{eqnarray}\begin{array}{l}{f}_{2}=1+{{\rm{e}}}^{{\theta }_{1}}+{{\rm{e}}}^{{\theta }_{2}}+{d}_{12}{{\rm{e}}}^{{\theta }_{1}+{\theta }_{2}},\\ {\theta }_{i}={k}_{i}x+{l}_{i}y+{m}_{i}z+{\omega }_{i}t+{\delta }_{i},\\ {\omega }_{i}=-\displaystyle \frac{\alpha {k}_{i}^{4}+\gamma {l}_{i}^{2}+\gamma {m}_{i}^{2}}{{k}_{i}},\end{array}\end{eqnarray}$(10)$\begin{eqnarray}{d}_{12}=-\displaystyle \frac{\alpha \left({k}_{1}-{k}_{2}\right){}^{4}+\left({k}_{1}-{k}_{2}\right)\left({\omega }_{1}-{\omega }_{2}\right)+\gamma \left({l}_{1}-{l}_{2}\right){}^{2}+\gamma \left({m}_{1}-{m}_{2}\right){}^{2}}{\alpha \left({k}_{1}+{k}_{2}\right){}^{4}+\left({k}_{1}+{k}_{2}\right)\left({\omega }_{1}+{\omega }_{2}\right)+\gamma \left({l}_{1}+{l}_{2}\right){}^{2}+\gamma \left({m}_{1}+{m}_{2}\right){}^{2}},\end{eqnarray}$where k_i, l_i, m_i, δ_i (i=1, 2) are some free parameters. Then we pluralize the corresponding parameters, that is,(11)$\begin{eqnarray}\begin{array}{l}{k}_{1}={a}_{1}+{\rm{i}}{b}_{1}={k}_{2}{}^{* },\,\,{l}_{1}={c}_{1}+{\rm{i}}{d}_{1}={l}_{2}{}^{* },\\ {m}_{1}={g}_{1}+{\rm{i}}{h}_{1}={m}_{2}{}^{* },\\ {\delta }_{1}=\mathrm{ln}\displaystyle \frac{{\lambda }_{1}}{2}+{\beta }_{1}+{\rm{i}}{\eta }_{1}={\delta }_{2}{}^{* },\end{array}\end{eqnarray}$where * stands for conjugate, and the parameters ${a}_{1},\,{b}_{1},\,{c}_{1},\,{d}_{1},\,{g}_{1},\,{h}_{1},\,{\lambda }_{1}\,(\gt 0),{\beta }_{1},{\eta }_{1}$ are all real constants. Substituting (11) into(9), we have(12)$\begin{eqnarray}{f}_{2}\sim 2\sqrt{{\lambda }_{2}}\cosh \left({\xi }_{1}+\displaystyle \frac{\mathrm{ln}{\lambda }_{2}}{2}\right)+{\lambda }_{1}\cos \left({{\rm{\Lambda }}}_{1}\right),\end{eqnarray}$where(13)$\begin{eqnarray}\begin{array}{l}{\xi }_{1}={a}_{1}x+{c}_{1}y+{g}_{1}z+{\omega }_{1R}t+{\beta }_{1},\\ {{\rm{\Lambda }}}_{1}={b}_{1}x+{d}_{1}y+{h}_{1}z+{\omega }_{1I}t+{\eta }_{1},\\ {\lambda }_{2}=\displaystyle \frac{1}{4}{d}_{12}{\lambda }_{1}^{2},\end{array}\end{eqnarray}$(14)$\begin{eqnarray}{\omega }_{1R}=\displaystyle \frac{-\alpha {a}_{1}^{5}+2\alpha {a}_{1}^{3}{b}_{1}^{2}+{a}_{1}\left(3\alpha {b}_{1}^{4}+\gamma \left(-{c}_{1}^{2}+{d}_{1}^{2}-{g}_{1}^{2}+{h}_{1}^{2}\right)\right)+{b}_{1}\left(-2\gamma \left({c}_{1}{d}_{1}+{g}_{1}{h}_{1}\right)\right)}{{a}_{1}^{2}+{b}_{1}^{2}},\end{eqnarray}$(15)$\begin{eqnarray}{\omega }_{1I}=\displaystyle \frac{-3\alpha {a}_{1}^{4}{b}_{1}-2\alpha {a}_{1}^{2}{b}_{1}^{3}-2{a}_{1}\gamma \left({c}_{1}{d}_{1}+{g}_{1}{h}_{1}\right)+{b}_{1}\left(\alpha {b}_{1}^{4}+\gamma \left({c}_{1}^{2}-{d}_{1}^{2}+{g}_{1}^{2}-{h}_{1}^{2}\right)\right)}{{a}_{1}^{2}+{b}_{1}^{2}}.\end{eqnarray}$Substituting equation (12) into (5), the first-order breath-wave solution can be expressed as(16)$\begin{eqnarray}\begin{array}{rcl}{u}_{1} & = & \displaystyle \frac{12\alpha \left(2\sqrt{{\lambda }_{2}}{\lambda }_{1}\left({a}_{1}^{2}-{b}_{1}^{2}\right)\cos \left({{\rm{\Lambda }}}_{1}\right)\cosh \left(\tfrac{\mathrm{ln}{\lambda }_{2}}{2}+{\xi }_{1}\right)+4{a}_{1}^{2}{\lambda }_{2}-{b}_{1}^{2}{\lambda }_{1}^{2}\right)}{\beta \left({\lambda }_{1}\cos \left({{\rm{\Lambda }}}_{1}\right)+2\sqrt{{\lambda }_{2}}\cosh \left(\tfrac{\mathrm{ln}{\lambda }_{2}}{2}+{\xi }_{1}\right)\right){}^{2}}\\ & & +\displaystyle \frac{12\alpha \left(4{a}_{1}{b}_{1}\sqrt{{\lambda }_{2}}{\lambda }_{1}\sin \left({{\rm{\Lambda }}}_{1}\right)\sinh \left(\tfrac{\mathrm{ln}{\lambda }_{2}}{2}+{\xi }_{1}\right)\right)}{\beta \left({\lambda }_{1}\cos \left({{\rm{\Lambda }}}_{1}\right)+2\sqrt{{\lambda }_{2}}\cosh \left(\tfrac{\mathrm{ln}{\lambda }_{2}}{2}+{\xi }_{1}\right)\right){}^{2}}.\end{array}\end{eqnarray}$Notice that when ${d}_{12}\gt 1$, the nonsingular solution of (16) exists. The breath-wave solution (16) is comprised of the hyperbolic functions and trigonometric functions. The hyperbolic functions $\left[\cosh \left(\tfrac{\mathrm{ln}{\lambda }_{2}}{2}+{\xi }_{1}\right),\sinh \left(\tfrac{\mathrm{ln}{\lambda }_{2}}{2}+{\xi }_{1}\right)\right]$ determine the dynamic behavior of the SWC while the trigonometric ones [cos $\left({{\rm{\Lambda }}}_{1}\right)$, sin $\left({{\rm{\Lambda }}}_{1}\right)]$ describe that of the PWC. So the breath wave is formed by the nonlinear combination of such two types of wave components. We can extract the SWC (figure 1(a)) and PWC (figure 1(b)) from equation (16), which are respectively given by(17)$\begin{eqnarray}{u}_{{\rm{s}}}=\displaystyle \frac{12\alpha {a}_{1}^{2}\sqrt{\tfrac{\gamma {h}_{1}^{2}{\lambda }_{1}^{2}}{3\alpha {a}_{1}^{4}+\gamma {h}_{1}^{2}}}\left({\lambda }_{1}\cos (A)\cosh (B)+\sqrt{\tfrac{\gamma {h}_{1}^{2}{\lambda }_{1}^{2}}{3\alpha {a}_{1}^{4}+\gamma {h}_{1}^{2}}}\right)}{\beta \left({\lambda }_{1}\cos (A)+\sqrt{\tfrac{\gamma {h}_{1}^{2}{\lambda }_{1}^{2}}{3\alpha {a}_{1}^{4}+\gamma {h}_{1}^{2}}}\cosh (B)\right){}^{2}},\end{eqnarray}$(18)$\begin{eqnarray}{u}_{{\rm{p}}}=-\displaystyle \frac{12\alpha {b}_{1}^{2}{\lambda }_{1}\left(\sqrt{{\lambda }_{1}^{2}\left(1-\tfrac{3\alpha {b}_{1}^{4}}{\gamma {g}_{1}^{2}}\right)}\cosh (M)\cos (N)+{\lambda }_{1}\right)}{\beta \left(\sqrt{{\lambda }_{1}^{2}\left(1-\tfrac{3\alpha {b}_{1}^{4}}{\gamma {g}_{1}^{2}}\right)}\cosh (M)+{\lambda }_{1}\cos (N)\right){}^{2}},\end{eqnarray}$with(19)$\begin{eqnarray}\,A=-\displaystyle \frac{2\gamma {g}_{1}{h}_{1}t}{{a}_{1}}+{\eta }_{1},\end{eqnarray}$(20)$\begin{eqnarray}\,\begin{array}{r}B\,=\,{a}_{1}x+{c}_{1}y-\alpha {a}_{1}^{3}t-\displaystyle \frac{\gamma t\left({c}_{1}^{2}+{g}_{1}^{2}-{h}_{1}^{2}\right)}{{a}_{1}}\\ +\ \displaystyle \frac{1}{2}\mathrm{ln}\left(\displaystyle \frac{\gamma {h}_{1}^{2}{\lambda }_{1}^{2}}{12\alpha {a}_{1}^{4}+4\gamma {h}_{1}^{2}}\right)+{\beta }_{1},\end{array}\end{eqnarray}$(21)$\begin{eqnarray}\,M=\displaystyle \frac{1}{2}\mathrm{ln}\left(\displaystyle \frac{{\lambda }_{1}^{2}\left(\gamma {g}_{1}^{2}-3\alpha {b}_{1}^{4}\right)}{4\gamma {g}_{1}^{2}}\right)-\displaystyle \frac{2\gamma {g}_{1}{h}_{1}t}{{b}_{1}}+{\beta }_{1},\end{eqnarray}$(22)$\begin{eqnarray}\,N={b}_{1}x+{d}_{1}y+\alpha {b}_{1}^{3}t-\displaystyle \frac{\gamma t\left({d}_{1}^{2}-{g}_{1}^{2}+{h}_{1}^{2}\right)}{{b}_{1}}+{\eta }_{1}.\end{eqnarray}$

Next, we analyze the two types of wave components in depth. The velocities of the SWC along the x and y directions are(23)$\begin{eqnarray}\,{V}_{{s}_{1}-x}=\displaystyle \frac{{\omega }_{1R}}{{a}_{1}},\,\,\,\,{V}_{{s}_{1}-y}=\displaystyle \frac{{\omega }_{1R}}{{c}_{1}},\end{eqnarray}$and those of the PWC read(24)$\begin{eqnarray}\,{V}_{{p}_{1}-x}=\displaystyle \frac{{\omega }_{1I}}{{b}_{1}},\,\,\,\,{V}_{{p}_{1}-y}=\displaystyle \frac{{\omega }_{1I}}{{d}_{1}}.\end{eqnarray}$The phases of the SWC $[{\phi }_{\mathrm{swc}}(t)]$ and PWC $[{\phi }_{\mathrm{pwc}}(t)]$ are expressed as(25)$\begin{eqnarray}\,\begin{array}{r}{\phi }_{\mathrm{swc}}(t)=-\alpha {a}_{1}^{3}t-\displaystyle \frac{\gamma t\left({c}_{1}^{2}+{g}_{1}^{2}-{h}_{1}^{2}\right)}{{a}_{1}}\\ +\ \displaystyle \frac{1}{2}\mathrm{ln}\left(\displaystyle \frac{\gamma {h}_{1}^{2}{\lambda }_{1}^{2}}{12\alpha {a}_{1}^{4}+4\gamma {h}_{1}^{2}}\right)+{\beta }_{1},\end{array}\end{eqnarray}$(26)$\begin{eqnarray}\,{\phi }_{\mathrm{pwc}}(t)=\alpha {b}_{1}^{3}t-\displaystyle \frac{\gamma t\left({d}_{1}^{2}-{g}_{1}^{2}+{h}_{1}^{2}\right)}{{b}_{1}}+{\eta }_{1}.\end{eqnarray}$We can see that the phases of the two wave components are related to t. And their initial phases (t=0) are(27)$\begin{eqnarray}\,\begin{array}{r}{\phi }_{\mathrm{swc}}(0)\,=\,\displaystyle \frac{1}{2}\mathrm{ln}\left(\displaystyle \frac{\gamma {h}_{1}^{2}{\lambda }_{1}^{2}}{12\alpha {a}_{1}^{4}+4\gamma {h}_{1}^{2}}\right)+{\beta }_{1},\\ \,{\phi }_{\mathrm{pwc}}(0)\,=\,{\eta }_{1}.\end{array}\end{eqnarray}$In addition, the equations of characteristic lines $({L}_{{s}_{1}}$, ${L}_{{p}_{1}})$ of the breath-wave solution (16) are expressed as follows:(28)$\begin{eqnarray}\,\begin{array}{l}{L}_{{s}_{1}}:\,\,{a}_{1}x+{c}_{1}y+{g}_{1}z+{\omega }_{1R}t+{\beta }_{1}+\displaystyle \frac{\mathrm{ln}{\lambda }_{2}}{2}=0,\\ {L}_{{p}_{1}}:\,\,{b}_{1}x+{d}_{1}y+{h}_{1}z+{\omega }_{1I}t+{\eta }_{1}=0.\end{array}\end{eqnarray}$On the corresponding characteristic lines, both the hyperbolic functions and trigonometric functions are the constant ones. Thus, the value of the SWC (see equation (17)) (PWC (see equation (18))) remains unchanged on the characteristic line ${L}_{{s}_{1}}$ (${L}_{{p}_{1}}$). In the expression of the SWC (see equation (17)), a trigonometric function $\cos (A)$ appears, which contains the time variable. Similarly, for the PWC (see equation (18)), the hyperbolic function $\cosh (M)$ is also associated with the time variable. Therefore, with the change of time t, the two functions affect the dynamics of the two wave components respectively. In addition, on any of the characteristic lines, the breath-wave solution (16) is not a constant function.

Next, we further analyze the features of characteristic lines for the two classes of nonlinear waves. The parameters (the wave numbers in x and y directions) of the characteristic lines (cross structure) for the breath wave satisfy the following condition(29)$\begin{eqnarray}\,{a}_{1}{d}_{1}-{b}_{1}{c}_{1}\ne 0.\end{eqnarray}$In this case, for these two types of wave components, their characteristic directions are different from each other, as shown in figure 2(a). In other words, their characteristic lines are intersecting. Instead, we consider another situation where the two characteristic lines $({L}_{{s}_{1}}$, ${L}_{{p}_{1}})$ are parallel, i.e.(30)$\begin{eqnarray}{a}_{1}{d}_{1}-{b}_{1}{c}_{1}=0.\end{eqnarray}$Under this circumstance, the characteristic direction of the SWC is parallel to that of the PWC, as displayed in figure 2(b). Then the converted wave will appear. Furthermore, for the converted wave in the (x, z)- and (y, z)-planes, the corresponding transition conditions are respectively given by(31)$\begin{eqnarray}{a}_{1}{h}_{1}-{b}_{1}{g}_{1}=0,\,\,\,\,\,{c}_{1}{h}_{1}-{d}_{1}{g}_{1}=0.\end{eqnarray}$Regarding the analyses of such waves, we will discuss in detail below.

Figure 1.

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Figure 1.(a) The SWC given by equation (17) with (α, β, γ, a₁, b₁, c₁, d₁, g₁, h₁, λ₁, β₁, η₁, z , t)=(−0.1, −4, 0.5, 1, 0, 1, 0, 4, 1, 2, 0, 0, 0, 0). (b) The PWC given by equation (18) with (α, β, γ, a₁, b₁, c₁, d₁, g₁, h₁, λ₁, β₁, η₁, z, t)=(−0.1, −4, 0.5, 0, 2, 0, 2, 4, 1, 2, 0, 0, 0, 0).

Figure 2.

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Figure 2.The schematic diagram of the nonlinear superposition principles of two types of nonlinear waves of equation (1): (a) the breath wave; (b) the converted wave. In (a), the characteristic direction of the SWC is different from that of the PWC, and their characteristic lines are intersecting (${L}_{S}\nparallel {L}_{P}$). In (b), the characteristic direction of the SWC is the same as that of the PWC, and their characteristic lines are parallel to each other (${L}_{S}\parallel {L}_{P}$). ‘L_S’ (red line) is the characteristic line of the SWC and ‘L_P’ (blue line) is that of the PWC. Arrows of different colors represent the directions of characteristic lines for the corresponding wave components (SWC and PWC).

2.2. Time-varying feature

In this section, based on the transition condition (30), we will study the dynamics of time-varying feature of the converted nonlinear waves. We find that the wave numbers a₁ and b₁ influence the shapes of the converted waves. The value of a₁ affects the localization of the SWC while the value of b₁ determines the frequency of the PWC. Specifically, the localization of the SWC strengthens (the width starts to get narrow) as the value of a₁ increases. The frequency of the PWC increases as the value of b₁ increases. The values of c₁ and d₁ influence the characteristic directions of the converted waves in the (x, y)-plane. In addition, we note the phase functions of the two wave components are described by equations (25) and(26). Then the phase difference is(32)$\begin{eqnarray}\begin{array}{l}{\phi }_{\mathrm{swc}}(t)-{\phi }_{\mathrm{pwc}}(t)=\left(-\alpha {a}_{1}^{3}-\alpha {b}_{1}^{3}\right.\\ -\left.\displaystyle \frac{\gamma \left({c}_{1}^{2}+{g}_{1}^{2}-{h}_{1}^{2}\right)}{{a}_{1}}+\displaystyle \frac{\gamma \left({d}_{1}^{2}-{g}_{1}^{2}+{h}_{1}^{2}\right)}{{b}_{1}}\right)t\\ +\displaystyle \frac{1}{2}\mathrm{ln}\left(\displaystyle \frac{\gamma {h}_{1}^{2}{\lambda }_{1}^{2}}{12\alpha {a}_{1}^{4}+4\gamma {h}_{1}^{2}}\right)+{\beta }_{1}-{\eta }_{1}.\end{array}\end{eqnarray}$From the above equation, we can conclude that the superposition region of the two wave components will change over time t. This will cause the changes in the shapes of the converted waves. This phenomenon is called time-varying feature [43]. A detailed analysis will be given below.

A. W- and M-shaped waves

We first study the time-varying feature of the W- and M-shaped waves. Let us consider the case of a₁=b₁=0.5 and c₁=d₁=0.75 in equation (16), which satisfies the transition condition(30). In figure 3(a), we can observe a W-shaped wave with two valleys and one peak at t=−0.85. As time evolves, the W-shaped wave turns into an M-shaped one with a lower amplitude (<0.1) at t=0.025, showing two peaks (figure 3(b)). Notice that the shape of the converted wave has changed significantly over time. As time goes on, the M-shaped wave changes back into the W-shaped one at t=0.85 that is different from the previous one, as displayed in figure 3(c). Compared with the W-shaped wave in figure 3(a), the wave in figure 3(c) has the lower peak and shallower valleys. In fact, as time goes by, the shape of the wave alternates between the W and M shapes. Figure 3(d) shows the sectional views of the wave at three different moments, from which we can observe the obvious shape-changed feature. The evolution of the characteristic lines for the converted wave is displayed in figure 3(e). The distance between the characteristic lines changes over time. Then, with the help of the phase shift and superposition components, we analyze the essence of the time-varying feature of such wave. Since the velocities of the SWC (see equation (23)) and PWC (see equation (24)) are different, which leads to the phase difference between them (see equation (32)). Thus, the superposition region of these two wave components changes with time. For example, when t=t₁, the SWC is superposed with two valleys and a peak of the PWC. When t=t₂, the SWC is superposed with a valley and two peaks of the PWC. Figure 3(f) shows a schematic diagram of the nonlinear superposition principle for such waves.

Figure 3.

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Figure 3.Plots of the W-shaped and M-shaped waves with (α, β, γ, a₁, b₁, c₁, d₁, g₁, h₁, λ₁, β₁, η₁, z )=(−0.1, −4, 0.1, 0.5, 0.5, 0.75, 0.75, 4, 1, 2, 0, 0, 0). (a): t=−0.85. (b): t=0.025. (c): t=0.85. (d): Cut plots at different times. (e): The characteristic lines at different moments. (f): The schematic diagram of analysis of wave components.


B. Oscillating W- and oscillating M-shaped waves

Taking a₁=0.5, b₁=2, c₁=0.75, d₁=3 in equation (16), the breath wave of equation (1) is converted into the wave with the similar time-varying feature, as depicted in figures 4(a)–(c). Compared with the W- and M-shaped waves in figures 3(a)–(c), the waves exhibit certain oscillatory characteristic, which can be clearly observed in figure 4(d). And it can be clearly seen that the wave patterns of figures 4(a) and (c) are nearly the same except for their positions. Figure 4(e) displays the positions of the characteristic lines at three different moments. The superposition principle of the oscillating W-shaped and oscillating M-shaped waves is presented in figure 4(f). We can see that the frequency of the PWC increases, compared with that in figure 3(f). This results in the oscillatory characteristic of the converted wave.

Figure 4.

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Figure 4.Plots of the oscillating W-shaped and oscillating M-shaped waves with (α, β, γ, a₁, b₁, c₁, d₁, g₁, h₁, λ₁, β₁, η₁, z)=(−0.1, −4, 0.1, 0.5, 2, 0.75, 3, 4, 1, 2, 0, 0, 0). (a) t=−6.82. (b) t=0.82. (c) t=8.46. (d) Cut plots at different times. (e) The characteristic lines at different moments. (f): The schematic diagram of analysis of wave components.


C. Multi-peak solitary waves

Now let us keep the value of b₁ unchanged (see Part B) and reduce the value of a₁, i.e. a₁=0.25, b₁=2. In figures 5(a)–(c), we can observe that the localization of the wave weakens and does not change with time, compared with figures 4(a)–(c). As shown in figure 5(d), the widths of the the converted wave at different moments are approximately equal to 40. The wave has more peaks than that in figure 4(d), which is called the multi-peak solitary wave. The distance between these two characteristic lines varies over time for the multi-peak solitary wave, as demonstrated in figure 5(e). In figure 5(f), one can find that the frequency of the PWC has greatly increased, which leads to multiple peaks appearing in the converted wave.

Figure 5.

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Figure 5.Plots of the multi-peak solitary waves with (α, β, γ, a₁, b₁, c₁, d₁, g₁, h₁, λ₁, β₁, η₁, z)=(−0.1, −4, 0.1, 0.25, 2, 0.375, 3, 4, 1, 2, 0, 0, 0). (a) t=−10.67. (b) t=0.97. (c) t=12.62. (d) Cut plots at different times. (e) The characteristic lines at different moments. (f) The schematic diagram of analysis of wave components.


D. Quasi W- and quasi Bell-shaped waves

We take a₁=c₁=1, b₁=d₁=0.01 in equation (16), and a novel converted wave of equation (1) can be obtained. Figure 6(a) is a quasi Bell-shaped wave, which has one peak. As time goes by, the quasi Bell-shaped wave changes into a quasi W-shaped one, as shown in figure 6(b), and it has one peak and two valleys. As time changes further, the quasi W-shaped wave changes back to the quasi Bell-shaped one seen from figure 6(c). The sectional views of the three moments are shown in figure 6(d). Figure 6(e) shows the characteristic lines at three moments. The distance between the characteristic lines changes visibly. Figure 6(f) displays a schematic diagram of superposition principle of the quasi Bell- and quasi W-shaped waves. Quite different from the previous three cases, the frequency of this PWC is extremely slight. Hence the influence of this SWC on the converted wave is much greater than that of the PWC.

Figure 6.

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Figure 6.Plots of the quasi Bell-shaped and quasi W-shaped waves with (α, β, γ, a₁, b₁, c₁, d₁, g₁, h₁, λ₁, β₁, η₁, z )=(−0.1, −4, 0.5, 1, 0.01, 1, 0.01, 4, 1, 2, 0, 0, 0). (a) t=0. (b) t=0.7. (c) t=1.55. (d) Cut plots at different times. (e) The characteristic lines at different moments. (f) The schematic diagram of analysis of wave components.


E. Bell-shaped and W-shaped waves

Let us consider an extreme case, i.e. b₁=d₁=0 in equation (16). In this case, the waveform alternates between the Bell and W shapes with time, as can be seen from figures 7(a)–(c). The time-varying feature of such wave is similar to that of the quasi Bell-shaped wave (see figures 6(a)–(c)). Figure 7(d) is the sectional views of the wave at different moments. The evolution of the characteristic lines over time is shown in figure 7(e). Compared with the characteristic lines of the quasi Bell-shaped and quasi W-shaped waves in figure 6(e), the number of the characteristic lines is reduced from 6 to 3. The PWC disappears completely, so does its characteristic line. In figure 7(f), there is only the SWC. Under this circumstance, the quasi Bell-shaped and quasi W-shaped waves turn into the pure Bell-shaped and pure W-shaped waves, respectively. The pure solutions we discuss here are aimed at the sense of space. It should be noted that since the expression of the solution (17) contains cos(A), the shape of the converted wave varies with time.

Figure 7.

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Figure 7.Plots of the Bell-shaped and W-shaped waves with (α, β, γ, a₁, b₁, c₁, d₁, g₁, h₁, λ₁, β₁, η₁, z )=(−0.1, −4, 0.5, 1, 0, 1, 0, 4, 1, 2, 0, 0, 0). (a) t=0. (b) t=0.7. (c) t=1.55. (d) Cut plots at different times. (e) The characteristic lines at different moments. (f) The schematic diagram of analysis of wave components.


F. (Quasi-) periodic waves

We select the parameters a₁=c₁=0.01, b₁=d₁=2 in equation (16). In this case, the effect of the PWC is significantly stronger than that of the SWC. As displayed in figures 8(a)–(c), the converted wave shows the periodic state along the orthogonal direction of the characteristic direction, which can be called the quasi periodic wave. It is composed of a series of W- or Bell-shaped waves. In particular, it can be found that the wave is a short-lived structure. The amplitude of the wave (see figure 8(a)) gradually increases with time, reaches the maximum value at t=0 (see figure 8(b)), and then decreases gradually (see figure 8(c)). The sectional views in figure 8(d) have shown the distinct changes in the shape of the wave. The distance between the characteristic lines also varies clearly in figure 8(e). Figure 8(f) is a schematic diagram of the superposition principle of the quasi periodic wave. The frequency of the PWC is obviously stronger than that of the preceding parts. The SWC only has a small effect on the converted wave.

Figure 8.

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Figure 8.Plots of the quasi periodic Bell-shaped and quasi periodic W-shaped waves with (α, β, γ, a₁, b₁, c₁, d₁, g₁, h₁, λ₁, β₁, η₁, z)=(−0.1, −4, 0.5, 0.01, 2, 0.01, 2, 4, 1, 2, 0, 0, 0). (a) t=−0.4. (b) t=0. (c) t=0.63. (d) Cut plots at different times. (e) The characteristic lines at different moments. (f) The schematic diagram of analysis of wave components.


Next, we consider another extreme case where a₁=c₁=0 in equation (16). Under this circumstance, the SWC vanishes completely, so the characteristic line of the SWC also vanishes, and only the PWC exists. Consequently, the breath wave turns into the pure periodic wave, as depicted in figures 9(a)–(c). In figure 9(d), one can observe that the periodic wave exhibits the short-lived localized property similar to the quasi periodic wave (see figures 8(a)–(c)). It should be noted that the periodic wave solution is a pure solution similar to the Bell-shaped wave one, while the quasi periodic wave solution is a mixed one. The variation of the characteristic lines in figure 9(e) is similar to that of the Bell-shaped wave (see figure 7(e)). The schematic diagram of the superposition principle of the pure periodic wave has been exhibited in figure 9(f). In contrast to figure 8(f), the SWC disappears completely. Besides, since the function cosh(M) appears in the expression equation (18), the shape of the periodic wave changes over time.

Figure 9.

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Figure 9.Plots of the periodic Bell-shaped and periodic W-shaped waves with (α, β, γ, a₁, b₁, c₁, d₁, g₁, h₁, λ₁, β₁, η₁, z)=(−0.1, −4, 0.5, 0, 2, 0, 2, 4, 1, 2, 0, 0, 0). (a) t=−0.4. (b) t=0. (c) t=0.63. (d) Cut plots at different times. (e) The characteristic lines at different moments. (f) The schematic diagram of analysis of wave components.


G. Rational W-shaped waves

Now let us consider the case a₁→0, b₁→0, and the following parameter conditions:(33)$\begin{eqnarray}\begin{array}{rcl}{\theta }_{i} & = & {k}_{i}x+{l}_{i}y+{m}_{i}z+{\omega }_{i}t+{\delta }_{i},\\ {k}_{i} & = & {s}_{i}\epsilon ,\,\,\,\,\,\,{l}_{i}={r}_{i}{s}_{i}\epsilon ,\,\,\,\,\,\,{m}_{i}={n}_{i}{s}_{i}\epsilon ,\\ {\omega }_{i} & = & -\displaystyle \frac{\alpha {k}_{i}^{4}+\gamma {l}_{i}^{2}+\gamma {m}_{i}^{2}}{{k}_{i}},\\ {\delta }_{1} & = & {\rm{i}}\pi ={\delta }_{2}{}^{* },\,\,\,\,\,\,(i=1,2).\end{array}\end{eqnarray}$Then taking the long-wave limit of the second-order soliton solution (ε→ 0), the lump solution can be given by [45](34)$\begin{eqnarray}{u}_{{\rm{L}}}=\displaystyle \frac{12\alpha \left(2{D}_{12}-\left({\psi }_{1}+{\psi }_{2}\right){}^{2}+2{\psi }_{1}{\psi }_{2}\right)}{\beta \left({D}_{12}+{\psi }_{1}{\psi }_{2}\right){}^{2}}\end{eqnarray}$with(35)$\begin{eqnarray}\begin{array}{l}{D}_{12}=\displaystyle \frac{12\alpha }{\gamma \left(\left({r}_{1}-{r}_{2}\right){}^{2}+\left({n}_{1}-{n}_{2}\right){}^{2}\right)},\\ {\psi }_{i}=x+{r}_{i}y+{n}_{i}z-\gamma \left({r}_{i}^{2}+{n}_{i}^{2}\right)t,\,\,\,\,\,\,(i=1,2).\end{array}\end{eqnarray}$Similar to the case of the first-order breath wave, we can present the transition condition of the lump solution (34) as follows:(36)$\begin{eqnarray}{r}_{1}={r}_{2}.\end{eqnarray}$Under the above condition, we have the expression of the rational W-shaped wave,(37)$\begin{eqnarray}{u}_{\mathrm{RW}}=\displaystyle \frac{12\alpha \left(2{D}_{12}^{{\prime} }-\left({\psi }_{1}+{\psi }_{2}\right){}^{2}+2{\psi }_{1}{\psi }_{2}\right)}{\beta \left({D}_{12}^{{\prime} }+{\psi }_{1}{\psi }_{2}\right){}^{2}}\end{eqnarray}$with(38)$\begin{eqnarray}{D}_{12}^{{\prime} }=\displaystyle \frac{12\alpha }{\gamma \left({n}_{1}-{n}_{2}\right){}^{2}}.\end{eqnarray}$As shown in figures 10(a)–(c), the waves also exhibit the short-lived localized property similar to the (quasi-) periodic waves (see figures 8 and 9). When t=−2, we observe a rational W-shaped wave with a lower amplitude [see figure 10(a)]. As time changes, the amplitude of the wave gradually increases, and reaches the maximum value at t=0, as displayed in figure 10(b). Then when t=2, the rational W-shaped wave retreats back to the original shape and there is little change in the location (see figure 10(c)). Moreover, the rational W-shaped wave almost does not propagate in the (x, y)-plane over time. Through the analyses, we can conclude that the rational W-shaped wave is completely different from the non-rational waves that propagate in the (x, y)-plane. Some similar results (the short-lived structures) have also been reported in other nonlinear evolution equations [67–71].

Figure 10.

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Figure 10.Plots of the rational W-shaped waves with (α, β, γ, r₁, r₂, n₁, n₂, z)=(−1, −3, 2, 0.5, 0.5, 1+i, 1 − i, 0). (a) t=−2. (b) t=0. (c) t=2.

3. Classification of converted waves

Through the above analyses, we find that the wave numbers a₁ and b₁ have the important impacts on the types of converted waves. We introduce the physical quantity $\tau ={{\rm{e}}}^{-\delta }$ ($\tau \in [0,1]$, $\delta =| \tfrac{{a}_{1}}{{b}_{1}}| $) to roughly classify the converted waves, as shown in figure 11. It should be pointed out that there are no strict boundaries for this classification. Let us further discuss the gradient relationships of these converted waves. The value of the parameter τ describes the weights of the soliton states and periodic states of the converted waves. As the value of a₁ increases and that of b₁ is fixed, the value of τ will decrease and the soliton state of the converted wave becomes stronger. Conversely when the value of b₁ increases and that of a₁ remains unchanged, the value of τ will increase and the converted wave gradually shows periodic state. From left to right on the coordinate axis, the soliton state of the converted wave gradually weakens and starts to show periodic state. The point (τ=0) stands for the pure Bell- and W-shaped waves. And the converted waves eventually become pure periodic waves at the point (τ=1). Except for the two endpoints, other converted waves contain both soliton state and periodic state. When τ≪e⁻¹ ($| \tfrac{{a}_{1}}{{b}_{1}}| \gg 1$), the converted waves are quasi Bell-shaped and quasi W-shaped waves that are displayed in the red line area. When τ∼e⁻¹ ($| \tfrac{{a}_{1}}{{b}_{1}}| \sim 1$), the dark blue segment accounts for the W- and M-shaped waves. Further, when ${{\rm{e}}}^{-a}\lt \tau \lt {{\rm{e}}}^{-b}$ [$b\lt | \tfrac{{a}_{1}}{{b}_{1}}| \lt a$, (b<a<1)], the W- and M-shaped waves are transferred to the oscillating W- and oscillating M-shaped waves. They are shown in the purple line segment of the figure. When τ>e^−b [$| \tfrac{{a}_{1}}{{b}_{1}}| \lt b$, (b<a<1)], the value of τ is going to be around the dark green segment and the multi-peak solitary waves appear. Further over, when τ≫e⁻¹ ($| \tfrac{{a}_{1}}{{b}_{1}}| \ll 1$), the converted waves show the quasi periodic state. Consequently, we can realize the continuous transitions among different converted waves by controlling the value of τ.

Figure 11.

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Figure 11.The classification of the converted waves (schematic diagram). ‘B’=Bell-shaped wave, ‘W’=W-shaped wave, ‘QB’=Quasi Bell-shaped wave, ‘QW’=Quasi W-shaped wave, ‘M’=M-shaped wave, ‘OW’=Oscillating W-shaped wave, ‘OM’=Oscillating M-shaped wave, ‘MPS’=Multi-peak solitary wave, ‘QPW’=Quasi periodic wave, ‘PW’=Periodic wave. We introduce a function $\tau ={{\rm{e}}}^{-\delta }$ ($\tau \in [0,1]$, $\delta =| \tfrac{{a}_{1}}{{b}_{1}}| $). The closer to the left end point (point 0), the stronger the soliton state and the weaker the periodic state. On the contrary, the closer to the right end point (point 1), the weaker the soliton state and the stronger the periodic state. Except for the two endpoints, all other converted waves contain both soliton and periodic states.

4. Interactions between the converted waves

In this section, we will study the nonlinear interactions between the converted waves. To achieve this, we pluralize the parameters in the fourth-order soliton solution to derive the second-order breath-wave one. Equation (11) gives the definitions of ${k}_{i},{l}_{i},{m}_{i},{\delta }_{i}\,(i=1,2)$, and(39)$\begin{eqnarray}\begin{array}{l}{k}_{3}={a}_{2}+{\rm{i}}{b}_{2}={k}_{4}{}^{* },\,\,{l}_{3}={c}_{2}+{\rm{i}}{d}_{2}={l}_{4}{}^{* },\\ {m}_{3}={g}_{2}+{\rm{i}}{h}_{2}={m}_{4}{}^{* },\\ {\delta }_{3}=\mathrm{ln}\displaystyle \frac{{\lambda }_{3}}{2}+{\beta }_{2}+{\rm{i}}{\eta }_{2}={\delta }_{4}{}^{* },\end{array}\end{eqnarray}$where a₂, b₂, c₂, d₂, g₂, h₂, λ₃ (>0), β₂,η₂ are some real constants. And then we can obtain the expression of f₄, i.e.(40)$\begin{eqnarray}\begin{array}{rcl}{f}_{4} & = & 1+{\lambda }_{1}{{\rm{e}}}^{{\xi }_{1}}{\rm{\cos }}{{\rm{\Lambda }}}_{1}+\displaystyle \frac{{\lambda }_{1}^{2}{d}_{12}}{4}{{\rm{e}}}^{2{\xi }_{1}}\ +{\lambda }_{3}{{\rm{e}}}^{{\xi }_{2}}{\rm{\cos }}{{\rm{\Lambda }}}_{2}\\ & & +\displaystyle \frac{{\lambda }_{3}^{2}{d}_{34}}{4}{{\rm{e}}}^{2{\xi }_{2}}\\ & & +\displaystyle \frac{{\lambda }_{1}^{2}{\lambda }_{3}^{2}{d}_{12}{d}_{34}}{16}{{\rm{e}}}^{2{\xi }_{1}+2{\xi }_{2}}{d}_{13}{d}_{24}{d}_{14}{d}_{23}\\ & & +\displaystyle \frac{{\lambda }_{1}{\lambda }_{3}}{4}{{\rm{e}}}^{{\xi }_{1}+{\xi }_{2}}\left({d}_{13}{{\rm{e}}}^{i\left({{\rm{\Lambda }}}_{1}+{{\rm{\Lambda }}}_{2}\right)}+{d}_{14}{{\rm{e}}}^{i\left({{\rm{\Lambda }}}_{1}-{{\rm{\Lambda }}}_{2}\right)}\right.\\ & & \left.+{d}_{23}{{\rm{e}}}^{-i\left({{\rm{\Lambda }}}_{1}-{{\rm{\Lambda }}}_{2}\right)}+{d}_{24}{{\rm{e}}}^{-i\left({{\rm{\Lambda }}}_{1}+{{\rm{\Lambda }}}_{2}\right)}\right)\\ & & +\displaystyle \frac{{\lambda }_{1}^{2}{\lambda }_{3}{d}_{12}}{8}{{\rm{e}}}^{2{\xi }_{1}+{\xi }_{2}}\left({d}_{13}{d}_{23}{{\rm{e}}}^{i{{\rm{\Lambda }}}_{2}}+{d}_{14}{d}_{24}{{\rm{e}}}^{-i{{\rm{\Lambda }}}_{2}}\right)\\ & & +\displaystyle \frac{{\lambda }_{1}{\lambda }_{3}^{2}{d}_{34}}{8}{{\rm{e}}}^{{\xi }_{1}+2{\xi }_{2}}\left({d}_{13}{d}_{14}{{\rm{e}}}^{i{{\rm{\Lambda }}}_{1}}+{d}_{23}{d}_{24}{{\rm{e}}}^{-i{{\rm{\Lambda }}}_{1}}\right),\end{array}\end{eqnarray}$with(41)$\begin{eqnarray}\begin{array}{rcl}{\xi }_{i} & = & {a}_{i}x+{c}_{i}y+{g}_{i}z+{\omega }_{{iR}}t+{\beta }_{i},\\ {{\rm{\Lambda }}}_{i} & = & {b}_{i}x+{d}_{i}y+{h}_{i}z+{\omega }_{{iI}}t+{\eta }_{i},\end{array}\end{eqnarray}$(42)$\begin{eqnarray}{\omega }_{{iR}}=\displaystyle \frac{-\alpha {a}_{i}^{5}+2\alpha {a}_{i}^{3}{b}_{i}^{2}+{a}_{i}\left(3\alpha {b}_{i}^{4}+\gamma \left(-{c}_{i}^{2}+{d}_{i}^{2}-{g}_{i}^{2}+{h}_{i}^{2}\right)\right)+{b}_{i}\left(-2\gamma \left({c}_{i}{d}_{i}+{g}_{i}{h}_{i}\right)\right)}{{a}_{i}^{2}+{b}_{i}^{2}},\end{eqnarray}$(43)$\begin{eqnarray}{\omega }_{{iI}}=\displaystyle \frac{-3\alpha {a}_{i}^{4}{b}_{i}-2\alpha {a}_{i}^{2}{b}_{i}^{3}-2{a}_{i}\gamma \left({c}_{i}{d}_{i}+{g}_{i}{h}_{i}\right)+{b}_{i}\left(\alpha {b}_{i}^{4}+\gamma \left({c}_{i}^{2}-{d}_{i}^{2}+{g}_{i}^{2}-{h}_{i}^{2}\right)\right)}{{a}_{i}^{2}+{b}_{i}^{2}},\,\,\,\,\,(i=1,2),\end{eqnarray}$and(44)$\begin{eqnarray}{d}_{{ij}}=-\displaystyle \frac{\alpha \left({k}_{i}-{k}_{j}\right){}^{4}+\left({k}_{i}-{k}_{j}\right)\left({\omega }_{i}-{\omega }_{j}\right)+\gamma \left({l}_{i}-{l}_{j}\right){}^{2}+\gamma \left({m}_{i}-{m}_{j}\right){}^{2}}{\alpha \left({k}_{i}+{k}_{j}\right){}^{4}+\left({k}_{i}+{k}_{j}\right)\left({\omega }_{i}+{\omega }_{j}\right)+\gamma \left({l}_{i}+{l}_{j}\right){}^{2}+\gamma \left({m}_{i}+{m}_{j}\right){}^{2}},\,\,\,\,\,(1\leqslant i\lt j\leqslant 4),\end{eqnarray}$and ${d}_{12},{d}_{34}\gt 1$.

To obtain the the second-order converted waves, we consider the following conditions(45)$\begin{eqnarray}{a}_{i}{d}_{i}-{b}_{i}{c}_{i}=0\,\,\,\,\,(i=1,2).\end{eqnarray}$In this case, the second-order breath wave is converted into a variety of converted ones, and we will investigate their interactions in detail below. Compared with the first-order waves, the second-order ones are composed of two series of SWCs and PWCs [S_i, P_i (i=1, 2), S_i stands for the SWC and P_i stands for the PWC]. Because of the complexity of the higher-order solutions, it is difficult to extract the expression of each wave component. However, we can easily calculate the velocities of the four wave components, which are given by(46)$\begin{eqnarray}{V}_{{s}_{i}-x}=\displaystyle \frac{{\omega }_{{iR}}}{{a}_{i}},\,\,\,\,{V}_{{s}_{i}-y}=\displaystyle \frac{{\omega }_{{iR}}}{{c}_{i}},\,\,\,\,\,\,\,\,(i=1,2),\end{eqnarray}$(47)$\begin{eqnarray}{V}_{{p}_{i}-x}=\displaystyle \frac{{\omega }_{{iI}}}{{b}_{i}},\,\,\,\,\,{V}_{{p}_{i}-y}=\displaystyle \frac{{\omega }_{{iI}}}{{d}_{i}},\,\,\,\,\,\,\,\,(i=1,2).\end{eqnarray}$Moreover, the characteristic line equation of each wave component can be readily expressed as(48)$\begin{eqnarray}\begin{array}{l}{L}_{{s}_{1}}:\,\,{a}_{1}x+{c}_{1}y+{g}_{1}z+{\omega }_{1R}t+{\beta }_{1}+\displaystyle \frac{1}{2}\mathrm{ln}\displaystyle \frac{{\lambda }_{1}^{2}{d}_{12}}{4}=0,\\ {L}_{{p}_{1}}:\,\,{b}_{1}x+{d}_{1}y+{h}_{1}z+{\omega }_{1I}t+{\eta }_{1}=0,\end{array}\end{eqnarray}$(49)$\begin{eqnarray}\begin{array}{l}{L}_{{s}_{2}}:\,\,{a}_{2}x+{c}_{2}y+{g}_{2}z+{\omega }_{2R}t+{\beta }_{2}+\displaystyle \frac{1}{2}\mathrm{ln}\displaystyle \frac{{\lambda }_{3}^{2}{d}_{34}}{4}=0,\\ {L}_{{p}_{2}}:\,\,{b}_{2}x+{d}_{2}y+{h}_{2}z+{\omega }_{2I}t+{\eta }_{2}=0.\end{array}\end{eqnarray}$The interactions involve many kinds of waves, and we choose some representative ones for further study. Types of interaction include the semi-elastic collision, perfectly elastic collision, inelastic collision and one-off collision. On the surface, it seems that two converted waves interact with each other directly. In fact, the collision between them can be understood from the nonlinear superposition principle. In other words, it can be regarded as a collision of two SWCs and a collision of two PWCs. The collisions produce two new converted waves with different phase shifts (or different shapes).

4.1. Perfectly elastic collision

In this part, we will study the perfectly elastic collision between two converted waves. In fact, this type of interaction is only suitable for the waves with the single characteristic line and single wave component such as the Bell-shaped wave, pure periodic wave and rational W-shaped wave. We hereby use the rational W-shaped wave as an example to illustrate this phenomenon. We set(50)$\begin{eqnarray}\begin{array}{rcl}{r}_{1} & = & {l}_{1R}+{\rm{i}}{l}_{1I}={r}_{2}{}^{* },\,\,\,{r}_{3}={l}_{2R}+{\rm{i}}{l}_{2I}={r}_{4}{}^{* },\\ {n}_{1} & = & {n}_{1R}+{\rm{i}}{n}_{1I}={n}_{2}{}^{* },\,\,\,{n}_{3}={n}_{2R}+{\rm{i}}{n}_{2I}={n}_{4}{}^{* }.\end{array}\end{eqnarray}$Then we consider the parameter conditions as follows:(51)$\begin{eqnarray}{a}_{i}\to 0,\,\,{b}_{i}\to 0,\,\,\,(i=1,2),\end{eqnarray}$and(52)$\begin{eqnarray}{r}_{1}={r}_{2},\,\,{r}_{3}={r}_{4}.\end{eqnarray}$In this case, two lump waves can be converted into two rational W-shaped ones. As shown in figure 12(a), we observe that the shapes of these two rational W-shaped waves do not change except for the phase shifts after the collision, like the general soliton interaction. Figure 12(b) shows the sectional views of (a). Additionally, the two rational W-shaped waves show the similar time-varying feature (short-lived) like the first-order one in figure 10. We omit the relevant discussions.

Figure 12.

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Figure 12.(a) The perfectly elastic collision between the two rational W-shaped waves with (α, β, γ, l_1R, l_1I, l_2R, l_2I, n_1R, n_1I, n_2R, n_2I, z, t)= (−3, −6, 3, 0.5, 0, −1, 0, 1, 1, 0.5, 1, 0, 0). (b) The sectional views of (a). ‘RW₁’ and ‘RW₂’ represent the two rational W-shaped waves, respectively. And ‘${\mathrm{RW}}_{1}^{\prime} $’ and ‘${\mathrm{RW}}_{2}^{{\prime} }$’ represent the waves after the collision. For the two rational W-shaped waves, they both have only one SWC and one characteristic line, so there is no deformation except for the phase shifts after the collision.

4.2. Semi-elastic collision

To implement such mode, we utilize two sets of parameter conditions, one satisfies the condition of the Bell-shaped waves and the other leads to the formation of the M-shaped waves, i.e.(53)$\begin{eqnarray}| \displaystyle \frac{{a}_{1}}{{b}_{1}}| \sim \infty ,\qquad \qquad | \displaystyle \frac{{a}_{2}}{{b}_{2}}| \sim 1.\end{eqnarray}$Figure 13(a) shows the 3D plot of the collision between the M- and Bell-shaped waves. From figure 13(b), we can clearly see that there is no change in the shape of the Bell-shaped wave after the collision, while that of the M-shaped wave changes significantly. However, since the frequency of the PWC and the localization of the SWC in the M-shaped wave keep constant, the localization of the M-shaped wave is almost invariable except for the phase shift after the collision. We further use the analyses of the characteristic lines and phase shifts to explain such phenomenon. As depicted in figure 13(c), one can observe that the Bell-shaped wave contains only one SWC (S₁) and one characteristic line (${L}_{{S}_{1}}$). So the shape of the Bell-shaped wave does not change except for the phase shift after the collision. Instead, the M-shaped wave consists of two wave components (S₂, P₂) and two characteristic lines (${L}_{{S}_{2}}$, ${L}_{{P}_{2}}$). The SWC S₂ becomes ${S}_{2}^{{\prime} }$ after the collision while the PWC P₂ develops into ${P}_{2}^{{\prime} }$ after the collision. Due to the different phase shifts of the two components, the distance (D) between these two characteristic lines of M-shaped wave will change. Further, the superposition region will also change, resulting in a change in the shape of the M-shaped wave.

Figure 13.

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Figure 13.(a) The semi-elastic collision between the Bell-shaped wave and M-shaped wave with $(\alpha ,\beta ,\gamma ,{a}_{1},{b}_{1},{c}_{1},{d}_{1},{g}_{1},{h}_{1}$, ${\lambda }_{1},{\beta }_{1},{\eta }_{1},{a}_{2},{b}_{2},{c}_{2},{d}_{2},{g}_{2},{h}_{2},{\lambda }_{3},{\beta }_{2},{\eta }_{2},z,t)$ = $\left(-1,-4,10,1,0,-1,0,4,1,2,0,0,1,2,1,2,4,1,2,0,0,0,0\right)$. (b) The sectional views of (a). ‘B’ and ‘M’ represent the Bell-shaped wave and M-shaped wave, respectively. ‘B’ and ‘M’ represent the two waves after the collision respectively. (c) The schematic diagram of analysis of wave components. The superposition region of these two wave components for the M-shaped wave changes after the collision, which leads to the change in shape.

4.3. Inelastic collision

We now consider the case in which the shapes of both converted waves change after the collision, namely the inelastic collision. Here we take the following parameters, that is,(54)$\begin{eqnarray}| \displaystyle \frac{{a}_{i}}{{b}_{i}}| \sim 1,\qquad (i=1,2).\end{eqnarray}$Under the above conditions, the W- and M-shaped waves can be generated. In figure 14(a), we can view it as either an interaction between a W-shaped wave and an M-shaped one, or an interaction between two M-shaped waves. The sectional views are displayed in figure 14(b). In the x direction, we can clearly see that the M-shaped wave turns into a W-shaped one, and the W-shaped wave changes into an M-shaped one after the collision. The superposition analysis is shown in figure 14(c). Both the W-shaped wave (S₁, P₁, ${L}_{{S}_{1}}$, ${L}_{{P}_{1}}$) and M-shaped wave (S₂, P₂, ${L}_{{S}_{2}}$, ${L}_{{P}_{2}}$) contain two components and two characteristic lines. In fact, the phase shifts occur between the two SWCs (S₁, S₂) after the collision, and the collision also causes the phase shifts to change between the two PWCs (P₁, P₂). At the same time, the distances (D₁, D₂) between the corresponding characteristic lines also change. Therefore, the superposition regions of these two groups of superposition components (S₁, P₁ and S₂, P₂) change after the collision, which leads to the deformation of the converted waves. However, the frequency of this PWC and the localization of this SWC remain the same after the collision. Thus the localization of the two converted waves remains unchanged.

Figure 14.

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Figure 14.(a) The inelastic collision between the W-shaped wave and M-shaped wave with $(\alpha ,\beta ,\gamma ,{a}_{1},{b}_{1},{c}_{1},{d}_{1},{g}_{1},{h}_{1}$, ${\lambda }_{1},{\beta }_{1},{\eta }_{1},{a}_{2},{b}_{2},{c}_{2},{d}_{2},{g}_{2},{h}_{2},{\lambda }_{3},{\beta }_{2},{\eta }_{2},z,t)$ = $\left(-1,-4,10,1,2,-1,-2,4,1,2,0,0,1,2,1,2,4,1,2,0,0,0,0\right)$. (b) The sectional views of (a). ‘W’ and ‘M’ represent the W-shaped wave and M-shaped wave, respectively. ‘W′’ and ‘M′’ represent the two waves after the collision respectively. (c) The schematic diagram of analysis of wave components. For the wave components constituting the M-shaped wave and W-shaped wave, due to the different phases, the superposition regions of the corresponding two wave components are different, which leads to obvious deformation of the two converted waves.

4.4. One-off collision

In the above three cases, we observe that the two converted waves have different characteristic directions and collide at the origin of coordinate axis. Next, we will study a new mode in which two converted waves have the same characteristic direction. In other words, the two converted waves propagate with the same direction, also known as the one-off collision. We give the corresponding parameter conditions as follows:(55)$\begin{eqnarray}\begin{array}{l}b\lt | \displaystyle \frac{{a}_{1}}{{b}_{1}}| \lt a\,\,(b\lt a\lt 1),\,\,\,\,\,\,\,\,| \displaystyle \frac{{a}_{2}}{{b}_{2}}| \sim \infty ,\\ {a}_{1}{c}_{2}-{a}_{2}{c}_{1}=0.\end{array}\end{eqnarray}$The first parameter condition in (55) produces an oscillating W-shaped wave, the second one leads to a Bell-shaped wave. And the third equation results in the same characteristic direction of the two waves. In figure 15(a), the oscillating W-shaped wave precedes the Bell-shaped wave. Over time, the Bell-shaped wave catches up with the oscillating W-shaped one, and the two waves merge, as displayed in figure 15(b). We could call it a nonlinear wave complex structure [72], which is the nonlinear superposition of two kinds of converted waves. By calculating the velocities of the Bell-shaped wave and oscillating W-shaped wave via equations (46) and (47), we find that the velocity of the Bell-shaped wave is much greater than that of the oscillating W-shaped one. Thus as time goes by, one can discover that the Bell-shaped wave exceeds the oscillating W-shaped one, as demonstrated in figure 15(c). Similar to the time-varying feature of the Bell-shaped wave, the shapes of the two converted waves change during the propagations. We omit the detailed analyses here.

Figure 15.

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Figure 15.The one-off collision between the oscillating W-shaped wave and Bell-shaped wave with (α, β, γ, a₁, b₁, c₁, d₁, g₁, h₁, λ₁, β₁, η₁, a₂, b₂, c₂, d₂, g₂, h₂, λ₃, β₂, η₂, z)=(−1, −4, 10, 1, 4, 1, 4, 4, 1, 2, 5, 0, 1, 0, 1, 0, 4, 1, 2, −5, 0, 0). (a) t=−0.3. (b) t=−0.1. (c) t=0.032. (d) Cut plots of (a)–(c). The two converted waves show the time-varying feature.

5. Conclusion

In this article, we have studied the transition dynamics of equation (1). We have shown that equation (1) admits various converted waves including (oscillating-) W- and M-shaped waves (see figures 3 and 4), multi-peak solitary waves (see figure 5), (quasi-) Bell-shaped and W-shaped waves (see figures 6 and 7), (quasi-) periodic waves (see figures 8 and 9) and rational W-shaped waves (see figure 10). We have decomposed the breath wave into an SWC and a PWC, and found that the breath wave and converted waves have different superposition modes. The characteristic lines of the former are intersecting (see equation (29) and figure 2(a)), while those of the latter are parallel to each other (see equation (30) and figure 2(b)). We have also discovered that the shapes of the converted waves of equation (1) vary with time, which is regarded as the time-varying feature. The feature can be explained by the phase shift and nonlinear superposition analyses. Based on the relation of wave numbers of wave components, we have given the corresponding classification for different kinds of converted waves. Finally, we have studied the interactions of several categories of converted waves, for instance, the perfectly elastic collision (see figure 12), semi-elastic collision (see figure 13), inelastic collision (see figure 14) and one-off collision (see figure 15). The essence of the inelastic collision can be illustrated by using the combination of the phase shifts, characteristic lines and nonlinear superposition analyses.

Acknowledgments

This work is supported by the National Natural Science Foundation of China (Grant Nos. 11 875 126, 61 705 006, and 11 947 230), and the China Postdoctoral Science Foundation (No. 2019M660430).

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