## Abstract

The study of nonlinear waves is a significant topic in the field of fluid Physics. In this paper, we study the nonlinear transformed waves, interactions and transformed molecular waves of the (3+1)-dimensional Konopelchenko-Dubrovsky-Kaup-Kupershmidt equation in plasma or ocean dynamics. Based on the condition of state transition, the transformed wave solutions derived from breath waves are obtained. The dynamics of the transformed wave solutions change with time, including peaks and shapes. In addition, we report two kinds of interactions, namely long- and short-lived collisions. The long-lived collision mode reveals the two constituent elements are always intertwined with each other. However, there is only disposable collision in short-lived collision mode, which is also considered as a catch-up case. As special interactions, the transformed molecular waves considered as a collision free mode whose atoms are transformed waves are investigated. The breather-breather molecule, breather-M-shaped molecule and double M-shaped molecule via velocity resonance condition are given. In summary, graphical investigations indicate that the novel transformed wave structures are shown for application in fluid mechanics.

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## 1 Introduction

The nonlinear waves are significant research hotspots for describing the dynamics of various physical phenomena in optics, Bose-Einstein condensates, fluid mechanics, plasma and other fields [1,2,3,4,5,6,7]. The related research of nonlinear evolution equations can be beneficial to describe the dynamics of nonlinear waves [8,9,10,11,12]. Solitons are balanced by nonlinear effects and dispersion, and their shapes remain unchanged during propagation [13]. However, unlike the solitons, the breathers undergoes periodic changes with evolution, which diverge and converge periodically. Generally, two types of breathers named the Kuznetsov-Ma breather [14, 15] and Akhmediev one [16] have been widely studied in both experimental and theoretical studies. The characteristics of them are manifested as periodic in time and localized in space (Kuznetsov-Ma breather) [14, 15] and localized in time and periodic in space (Akhmediev breather) [16], respectively. If the period of time or space is taken to be infinite, then the rogue wave can be derived from two types of breathers. Such waves are famous for the characteristics of “appear from nowhere and disappear without a trace” [17]. In addition to the low-dimensional dynamic models, various novel nonlinear waves appear in high-dimensional ones. In recent years, There are many results about state transition derived from breathers, superregular breathers or rogue waves. Specifically, the transformed waves have attracted lots of attention in high-dimensional nonlinear evolution equations [18,19,20,21,22,23,24,25]. Unlike other nonlinear waves, the transformed waves in high-dimensional systems exhibit time-varying characteristics, where the peaks and shapes of such waves change with evolution. By virtue of certain parameters regulations, the transformed waves derived from lump waves have been obtained in the (2+1)-dimensional generalized KdV equation [18]. Based on phase shift analysis and characteristic lines method, Wang *et* *al*. have studied the the locality, oscillation, time-varying characteristics in the (2+1)-dimensional Ito equation [19], the (3+1)-dimensional B-type Kadomtsev-Petviashvili equation [20] and so on [21,22,23]. Furthermore, some wave interactions modes have been reported, which cotain inelastic, semi-elastic, completely elastic collisions, and collision-free case [19, 21, 24, 25].

Recently, the nonlinear molecular waves have attracted extensive attention, including soliton molecule, breather molecule, transformed wave molecule and mixed molecule. Soliton molecules (also named soliton complexes) have been studied in experimental and theoretical research [26,27,28,29,30,31]. Stephanie Willms *et* *al*. have reported the rearrangement and final distribution ofenergy among the heteronuclear soliton molecules [32]. The heteronuclear dissipative Kerr soliton molecules have been discovered by generating superpositions of distinct soliton states [33]. In addition, the breather molecule has been revealed in the form of abound state of breathers, which is the generalization of soliton molecule [34, 35]. Kibler *et* *al*. have reported the propagation of breather molecules and the theoretical description of the synthesis in the focusing nonlinear Schrödinger equation [34]. The dichromatic breather molecules have been created for the the population inversion in a synchronized mode-locked fiber laser [36]. Moreover, some breather molecules have been given in integrable systems, including low-dimensional system [37] and high-dimensional ones [22, 23, 38]. As the extension of the concept of molecular waves, the constituent atoms of transformed molecular waves are transformed waves, which exists as a special form of interaction [22, 23]. The dynamics of transformed molecular waves also can reflect the time-varying characteristics of transformed waves.

It is well known that the high-dimensional nonlinear evolution equations have played an important role on revealing the significant mechanisms [39,40,41,42,43,44]. The (3+1)-dimensional generalized Konopelchenko-Dubrovsky-Kaup-Kupershmidt equation, which is applied in plasma or ocean dynamics, can be written as [45,46,47,48,49,50]

where \(u = u(x, y,z, t)\) is a real differentiable function of the variables *x*, *y*, *z* and *t*, and \(\alpha _j\) (\(j = 1, 2,..., 9\)) are the real constant coefficients. When \(\alpha _j\) takes different parameters combinations, Eq. (1) can be reduced to different equations, including a (2 + 1)-dimensional B-type Kadomtsev-Petviashvili equation (\(\alpha _1=\alpha _2=\alpha _9=0,\alpha _3=1\), \(\alpha _4=\alpha _5=-5,\alpha _6=-15,\alpha _7=15\) and \(\alpha _8=45\)) [51], a (2+1)-dimensional Bogoyavlensky-Konopelchenko equation (\(6\alpha _1=\alpha _2, 4\alpha _5=\alpha _6\) and \(\alpha _3=\alpha _4=\alpha _7=\alpha _8=\alpha _9=0\)) [52], the KdV equation (\(\alpha _1=1, \alpha _2=-6\) and \(\alpha _3=\alpha _4=\alpha _5=\alpha _6=\alpha _7=\alpha _8=\alpha _9=0\)) [53] in fluid mechanics and plasma physics. Some important research results, including solitons [45, 49, 50], breathers [45,46,47, 49, 50], lump solutions [45,46,47,48, 50], periodic line waves [45], half periodic kink solution [48] and some interaction solutions [45,46,47, 49], have been obtained based on analytical methods. Gao *et* *al*. have reported that the amplitudes of solitons are related to the ratio of \(\alpha _1\) to \(\alpha _2\) and the specific controlled parameters of breathers, periodic line waves and lumps are introduced in detail [45]. The multiple-lump, hybrid solutions and pfaffian solutions have been reported by Tian *et* *al*. [46, 49]. Furthermore, soliton molecules and the interactions between such waves and T-breathers (or M-lumps) have been obtained via velocity resonant principle [50]. For Eq. (1), however, there has been no research on transformed waves and their corresponding molecular waves in previous studies as yet. The discussions of such novel waves will undoubtedly provide theoretical support for the relevant applications of this model in the field of fluid mechanics. In addition, the exploration of collisions and molecular waves based on transformed waves can enrich the interaction patterns of nonlinear waves, which is beneficial for explaining diverse bound nonlinear waves in natural world.

Our work is devoted to explore novel transfomed waves, interactions and molecular wave modes related to state transition. The paper is organized as follows. In Sect. 2, the Painlevé test of Eq. (1) has passed with certain coefficient constraints. In Sect. 3, the N-soliton solution obtained through the Hirota bilinear method and phase transformation mechanism are given. In Sect. 4, we introduce the long- (permanent interaction) and short-lived collisions (catch-up case). In Sect. 5, three molecular waves regarded as special interactions where the brathers and M-shaped waves are constituent atoms are given. Finally, some results will be explicated in Sect. 6.

## 2 The Painlevé analysis

The discussion on the integrability of Eq. (1) is a meaningful point to predict some properties. As is well known to us, Painlevé analysis is a very effective and widely applied tool to test the integrability of nonlinear evolution equations [43, 54,55,56]. Now, the Eq. (1) can be equivalent to writing in another form, as follows

where \(v=v(x,y,z,t)\) is a real differentiable function and \(v_x = u_y\) represents the flow is not rotational in the \(x-y\) plane. Based on the Weiss-Tabor-Carnevale-Kruskal method [43], we set Laurent expansion about

where \(\varepsilon _1\), \(\varepsilon _2\) are negative integers and \(u(x,y,z,t)_r\), \(v(x,y,z,t)_r\), \(\vartheta _(x,y,z,t)\) are arbitrary analytical functions. To determine the first coefficient, we set

and

where *k* is an arbitrary positive integer. Through substituting Eq. (4) into Eq. (2) and analyzing for balancing the main terms, we get \(\varepsilon _1 = -2\), \(\varepsilon _2 = -2\) and two branches

Next, to solve the set of resonance points, we truncate the following expansion shown as

Substitute Eq. (7) into Eq. (2), and then make the coefficients of \((\vartheta ^{-7}, \vartheta ^{-3})\) equal to 0, i.e.,

with

with

By calculating Eq. (8) and Eq. (10), the resonance points are obtained, i.e., (−1, 2, 3, 6, 10) and (−2, −1, 2, 5, 6, 12), respectively.

Finally, we verify whether the compatibility conditions are satisfied [43]. For the **Case1**, we find the explicit expressions for \(u_1\) and \(u_3\) and the arbitrary expressions for \(u_2\), \(u_4\), \(u_5\), \(u_6\), \(u_7\), \(u_8\), \(u_9\) and \(u_{10}\). By analyzing explicit expressions, the coefficient constraints

are obtained to pass the Painlevé integrability test, which is also applicable to the **Case2**. To sum up, the (3+1)-dimensional Konopelchenko-Dubrovsky-Kaup-Kupershmidt equation has passed the Painlevé integrability test with the integrable coefficient constraints Eq. (12).

## 3 The nonlinear transformed waves analysis

In this section, we study the nonlinear transformed waves derived from breath waves via the Hirota bilinear method [57]. Under the Eq. (12) and the transformation

the Hirota bilinear form of Eq. (1) is written as

where *f*(*x*, *y*, *z*, *t*) is a real function and \(D_x\), \(D_y\), \(D_z\) and \(D_t\) are defined in Ref [57]. The N-soliton solution is given by [45]

with

Now, we consider two-soliton solution with the following expression

with

We pluralize the following parameters shown as

where complex conjugate is represented by \(*\), \(a_{1}\), \(b_{1}\), \(c_{1}\), \(d_{1}\), \(o_{1}\) and \(p_{1}\) can be selected as arbitrary real constants. Substitute Eq. (19) into Eq. (17) and the breath wave solution is obtained by

with

Next, we plug Eq. (20) into Eq. (13) and the breath wave solution in another form can be written as

with

From Eq. (22), it is obvious that the expression of breath wave is composed of nonlinear superposition between trigonometric functions (\(\sin \theta _{1I}\) and \(\cos \theta _{1I}\)) and hyperbolic functions (\(\sinh (\theta _{1R}+\frac{1}{2} \ln (\delta _2))\) and \(\cosh (\theta _{1R}+\frac{1}{2} \ln (\delta _2))\)). From physical perspective, the locality and oscillation of breath waves are determined by hyperbolic and trigonometric functions, respectively. In three planes (\(x-y\), \(x-z\), \(y-z\)), the dynamics of nonlinear waves have similar properties, so we choose the case in \(x-y\) plane for analysis. Moreover, the characteristic lines of solitary wave component (\(L_1\)) and periodic wave component (\(L_2\)) are given by

where the angle between two characteristic lines determines the information of breath waves. We select appropriate parameters to draw the dynamics of breath wave shown as Fig. 1. The three-dimensional diagram and density diagram at \(t=0\) are given to show that the breath wave are also periodic lump chains (see Figs. 1a-b). In addition, Fig. 1c displays that the characteristic lines of solitary wave component and periodic wave component maintain a invariant angle with evolution (\(\gamma _1=\gamma _2=\gamma _3\)).

Next, we consider a special case considered as state transition where two characteristic lines remain parallel throughout the evolution process, i.e.,

The formations of certain transformed wave solutions depend on different wavenumber ratio (\(\frac{a_1}{b_1}\)). As the wavenumber ratio increases, M-shaped wave, oscillatory M-shaped wave, multi-peaks wave, and quasi-sine wave gradually appear. As shown in Fig. 2, the transformed wave solutions exhibit three different dynamic characteristics at \(t=-5\), \(t=0\), and \(t=4\), respectively. This phenomenon confirms the unique time-varying characteristics in high-dimensional nonlinear evolution equations. In other words, the peaks and shapes of such waves always change during evolution process. Figure 2d is cross sectional view corresponding to the above three-dimensional diagram for the aim of better displaying time-varying feature. As shown in Fig. 2e, the same color characteristic lines of two wave components are always parallel to each other at a specific moment. Compared to Fig. 2, as the wavenumber ratio decreases, the oscillations of transformed waves become more pronounced. Figures 3a–c show that the shapes of the oscillating M-shape waves are still changing during evolution, as evidenced by Fig. 3d. Similarly, the characteristic lines of oscillating M-shape waves remain parallel during the evolution process. As the wavenumber ratio increases, the oscillations continue to increase [see Fig. 4]. Furthermore, the quasi-sine waves undergoes minimal shape changes while moving forward and their peaks and sub peak values have almost no ups and downs. In addition, quasi-sine waves exhibit rich periodic oscillations shown in Fig. 5. The cross section view displays a quasi periodic image with evolution.

## 4 Interactions

In this section, we discuss two types of interactions whose constituent components are solitons, breath waves or transformed waves. First, the two-order breath wave is obtained by four-soliton solution. The parameters are represented in pluralized form as follows

where complex conjugate is represented by \(*\), \(a_{2}\), \(b_{2}\), \(c_{2}\), \(d_{2}\), \(o_{2}\) and \(p_{2}\) can be selected as arbitrary real constants. Then, substituting Eq. (25) into Eq. (15), the expansion of \(f_{4}\) is given as follows

with

where \(\Theta _{13}\), \(\Theta _{14}\), \(\Theta _{12}\) and \(\Theta _{34}\) are obtained by substituting Eq. (19) and Eq. (25) into Eq. (16). The four-soliton solution (Eq. (26)) is composed of trigonometric and hyperbolic functions, which represents the oscillation and isolation of each wave components, respectively. Moreover, the characteristic lines of solitary wave component (\(L_{11}\) and \(L_{21}\)) and periodic wave component (\(L_{12}\) and \(L_{22}\)) are given by

where \(\lambda _{1R}\), \(\lambda _{1I}\), \(\lambda _{2R}\) and \(\lambda _{2I}\) are given in Eq. (27).

Figure 6 shows the evolution process of collisions between a soliton and a breath wave in different moments. From the evolutionary density figures, it can be seen that the shapes of the soliton and breath wave do not change. Moreover, the two waves contact with each other at any time, which represents a long-lived collision mode. Similarly, there are two breath waves colliding with each other shown in Fig. 7, where the shapes of each constituent elements are unchanged. In some long-lived collision modes, due to the time-varying property of the transformed waves, the shapes of them change with the evolution process which is shown in Fig. 8. Compared to solitons, the peak and frequency of the transformed waves change with evolution, which is shown in Figs. 8g–i. In addition, two transformed waves (oscillatory M-shaped waves) collided with each other are shown in Fig. 9. The density and cross section view are shown in Figs. 9b and c.

Different from the long-lived collision mode, the short-lived one means that the collision of two kinds of waves is concentrated in a relatively short period of time with evolution. The short-lived collisions between a breath and a soliton are shown in Fig. 10. When \(t=-1.5\), the soliton lag behind breath wave, meaning soliton is on the left side of breath wave [see Fig. 10a]. As evolution progresses, the soliton and breath wave collide and merge into integrated form, manifested as splitting one breath wave into two ones shown in Fig. 10b. Afterwards, the two waves reflect their original characteristics separately shown in Fig. 10c. Figure 11 reflects the same collision mode as Fig. 10, but the difference is that the breath wave is split into two different ones by the transformed wave during the collision [see Fig. 11b]. Moreover, the parallel interaction of a soliton and a oscillatory M-shaped wave are shown in Fig. 12, which reveals the process of the soliton catching up with the oscillatory M-shaped wave and surpassing it.

## 5 Transformed molecular waves

As is well known, molecular waves exhibit as stable bound state, which has been extensively studied based on theoretical research and experimental exploration in various fields such as optics, fluid mechanics, plasma physics and so on [26, 27, 36]. Under the velocity resonance condition, the atoms that are components of the molecular wave maintain a fixed distance and propagate with an equal speed in the same direction. In this section, we report a kind of special interaction that presents in the form of molecular waves, which is different from collision interaction modes. The molecular waves are composed of nonlinear wave atoms including soliton, breather or tranformed waves, which satisfies velocity resonance condition. In this case, the molecular distance between the two characteristic lines remains unchanged. The propagation velocities of each atoms are equal, which means that the characteristic line velocities of solitary wave components are equal, i.e.,

with

At the same time, the two characteristic lines remain parallel.

Next, three different molecular waves are introduced shown in Figs. 13, 14 and 15, which include breather-breather molecule, breather-M-shaped wave molecule and double M-shaped wave molecule. Figure 13 shows the evolution of molecular wave which cotains two breather atoms. It can be clearly seen that the molecular distance of two atoms is unchanged, which is assisted by the invariance of distance between characteristic lines of solitary wave components [see Fig. 13d], i.e.,

Similarly, a breather and a transformed wave are selected as the constituent atoms of molecular waves in Fig. 14, which can also reflect the time-varying characteristics of transformed waves. The shapes and peaks of transformed wave atoms are changed with evolution. In addition, the molecular distance of molecular wave remains unchanged, which means that the characteristic lines of solitary wave components for breather and M-shaped wave remain parallel and a constant distance [see Fig. 14d], i.e.,

In addition, two M-shaped waves are selected as the constituent atoms of the molecular wave shown in Fig. 15. Unlike Fig. 15, both atoms undergo deformation during the propagating process shown in Fig. 15d. As explained by Fig. 15e, the molecular distance between the two M-shaped waves remains unchanged, i.e.,

To sum up, the three types of molecular waves mentioned above have different initial phases, which reflects as non-zero molecular distance. The molecular waves including breather-breather molecule, breather-M-shaped wave molecule and double M-shaped wave molecule, have been discussed by visually demonstrating their evolutionary process. The molecular waves of such novel components enrich the molecular wave family and provide important theoretical support for experimental applications in plasma or ocean dynamics.

## 6 Conclusion

It was reported in previous research work that the solitons, breathers, lumps and soliton molecules have been investigated [45,46,47,48,49,50]. In this article, the phase transformation mechanism of the nonlinear waves in the (3+1)-dimensional Konopelchenko-Dubrovsky-Kaup-Kupershmidt equation has been reported with Hirota bilinear method. Based on the characteristic lines method, the breath wave and transformed waves are introduced by distinguishing the intersecting or parallel of the characteristic lines between the solitary wave component (\(L_1\)) and periodic wave component (\(L_2\)) [19]. Specifically, breathers can be modulated with two intersecting characteristic lines [see Fig. 1], while the generations of transformed waves need to meet the constraint of that such two lines are parallel. In addition, we show various transformed wave with different wavenumber ratio (\(\frac{a_1}{b_1}\)), including the M-shaped wave [see Fig. 2], oscillatory M-shaped wave [see Fig. 3], multi-peaks wave [see Fig. 4], and quasi-sine wave [see Fig. 5], where the time-varying characteristics are vividly displayed. Moreover, two types of collision modes are shown, which means intersecting (long-lived) and parallel (short-lived) collisions. The long-lived collision modes show that the two waves contact with each other at any time [see Figs. 6, 7, 8 and 9]. Contrary to the above, short-lived collision modes only focus on a small period of time that entangle with each other, separate gradually and then move away from each other [see Figs. 10, 11 and 12]. As a special type of interaction, molecular waves are given due to their constant molecular distance and stable propagation characteristics, including breather-breather molecule [see Fig. 13], breather-M-shaped wave molecule [see Fig. 14] and double M-shaped wave molecule [see Fig. 15]. The transformted wave molecules also reveal the time-varying characteristics of the transformed waves while ensure that the atomic spacing remains constant. In a word, these results enrich nonlinear wave solutions for providing practical guidance in plasma or ocean dynamics. In the future research, we will further focus on the solutions of other high-dimensional equations, including nonlinear transformted waves, soliton molecules, breather molecules, transformed molecules, lump molecules or other novel solutions.

## Data availability

Data sharing is not applicable to this article as no datasets were generated or analyzed during the current study.

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## Acknowledgements

We express our sincere thanks to the editors, reviewers, each member of our discussion group for their suggestions. This work has been supported by the Hebei Province Higher Education Reform and Practice Project under Grant No. 2023GJJG427, by the Natural Science Foundation of Hebei Province under Grant No. A2021502004, and by the Fundamental Research Funds for the Central Universities under Grant No. 2021MS071.

## Funding

This work has been supported by the Hebei Province Higher Education Reform and Practice Project under Grant No. 2023GJJG427, by the Natural Science Foundation of Hebei Province under Grant No. A2021502004, and by the Fundamental Research Funds for the Central Universities under Grant No. 2021MS071.

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Xuemin Yao: Writing – original draft, Methodology, Software. Jinying Ma: Writing – review & editing, Funding acquisition, Supervision. Gaoqing Meng: Writing – review & Funding acquisition. All authors reviewed the manuscript.

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Yao, X., Ma, J. & Meng, G. The phase transition of control parameters for the (3+1)-dimensional Konopelchenko–Dubrovsky–Kaup–Kupershmidt equation in plasma or ocean dynamics.
*Nonlinear Dyn* **112**, 18435–18451 (2024). https://doi.org/10.1007/s11071-024-09971-4

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DOI: https://doi.org/10.1007/s11071-024-09971-4