The mixed solutions of the (2+1)-dimensional Hirota–Satsuma–Ito equation and the analysis of nonlinear transformed waves

In this paper, we obtain the N-soliton solution for the (2+1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(2+1)$$\end{document}-dimensional Hirota–Satsuma–Ito equation by the Hirota bilinear method. On this basis, the breathers and lumps can be obtained using the complex conjugate parameter as well as the long wave limit method, and the mixed solutions containing them are investigated. Then, different nonlinear transformed waves are obtained from breathers and lumps under specific conditions, which include quasi-anti-dark soliton, M-shaped soliton, oscillation M-shaped soliton, multi-peak soliton, quasi-periodic soliton and W-shaped soliton. Finally, on the basis of the two-breather solutions, we discuss in detail the mixed solutions consisting of one breather and one nonlinear transformed wave, and the mixed solutions formed by two nonlinear transformed waves.


Introduction
The intersection and integration of different disciplines have become more pronounced as a product of the Y.-N. An · R. Guo (B) School of Mathematics, Taiyuan University of Technology, Taiyuan 030024, China e-mail: gr81@sina.com inevitable trend in the development of science and technology, and the emergence and development of nonlinear science typically reflect this characteristic [1][2][3][4][5][6][7][8][9]. As an important branch of nonlinear science, the research of soliton theory has not only promoted the development of nonlinear science but also provided new ideas and methods for research fields such as plasma [10,11], hydrodynamics [12,13], nonlinear optics [14], atmospheric science [15] and marine science [16]. Taking nonlinear transformed waves as an example, quasianti-dark solitons can be manipulated and controlled as information carriers and have potential applications in optical communications and signal processing [17]. Oscillation M-shaped solitons have little distortion or signal loss when transmitting data over long distances and can be used in the design of high-speed electronic circuits [18]. In addition, lumps have been extensively studied in the field of nonlinear dynamics and have potential applications in areas such as data transmission, energy storage and quantum computing [19][20][21]. Therefore, the study of exact solutions of nonlinear systems is a very important task. So far, some effective methods have been proposed for solving nonlinear systems, such as the inverse scattering transform [22,23], the Darboux transform [24][25][26], the Bäcklund transform [27,28], the Riemann-Hilbert method [29][30][31], the Hirota bilinear method [32][33][34][35][36][37][38][39] and so on.
The Hirota bilinear method is a mathematical approach for solving nonlinear partial differential equations, which uses a bilinear operator that transforms the partial differential equation into a bilinear equation and solves it through a series of recurrence relations to obtain a functional representation of the soliton. The soliton is a nonlinear wave that maintains its propagation velocity as well as its state unchanged during propagation [40][41][42]. Further, using the complex conjugate parameters, we can obtain breathers, which are periodic in their propagation along a particular direction [43][44][45][46][47]. Lumps are localized and can be obtained from soliton by the long wave limit method [48,49]. Finally, by analyzing the characteristic lines, we can obtain the nonlinear transformed waves under specific conditions [50][51][52][53][54]. Ref. [55] has investigated the nonlinear transformed waves by the quintic equation of the nonlinear Schrodinger hierarchy and Ref. [56] has shown that such waves can occur in the Hirota equation.
Hirota and Satsuma introduced equations describing the unidirectional propagation of shallow water waves in one spatial dimension and in time, namely the (1 + 1)-dimensional Hirota-Satsuma shallow water waves equation [57,58] (1.1) In this paper, we will focus on the (2 + 1)-dimensional Hirota-Satsuma-Ito (HSI) equation in the form as follows: where ς is a real nonzero constant, x and y denote spatial variables, t denotes the time variable, the physical field u = u(x, y, t) and the potentials of physical field derivatives v = v(x, y, t), w = w(x, y, t) are real functions [59,60]. The (2 + 1)-dimensional HSI equation, which is of great importance in the study of shallow water waves, was originally proposed by Hirota and Satsuma from a Bäcklund transformation of the Boussinesq equation and is known to describe prop, agitation of unidirectional shallow water waves just like the Korteweg-de Vries equation [61,62]. Therefore, studying the exact solution of this system will help us to have a better understanding for shallow water waves [43]. It is well known that some exact solutions of Eq. (1.2) have been studied. For example, Ref. [63] has obtained N -solitons solutions for the equation using the Hirota bilinear method, and Ref. [64] has got a degeneracy from Nsoliton solutions to Y -resonant solitons by introducing new conditions. Ref. [65] has acquired breather solutions by taking conjugate complexes for the soliton parameters and Ref. [66] has explored lump solutions. However, the mixed solutions formed by solitons, breathers and lumps, as well as nonlinear transformed waves and the connections between them, have not been investigated.
The paper is organized as follows. In Sect. 2, we will obtain the bilinear form and the N -soliton solutions of Eq. (1.2) using the Hirota bilinear method. In Sect. 3, we will obtain breathers and lumps from the N -soliton solutions by taking the complex conjugate parameter and the long wave limit method, and discussing separately the mixed solution containing breather, lump, and both at the same time. Further, we will obtain the nonlinear transformed waves generated by the breathers and lumps under specific conditions. Thus, we will show six different kinds of nonlinear transformed waves and their existence conditions in Sect. 4, including quasi-anti-dark soliton, Mshaped soliton, oscillation M-shaped soliton, multipeak soliton, quasi-periodic soliton and W -shaped soliton. Moreover, we will discuss the mixed solutions containing one nonlinear transformed wave and one breather, and the mixed solutions formed by two nonlinear transformed waves. Finally, the conclusions will be summarized in Sect. 5.

N-soliton solutions of the (+ 1)-dimensional Hirota-Satsuma-Ito equation
In this section, we will use the Hirota bilinear method to find the exact solutions of Eq. (1.2). Firstly, we introduce the logarithmic transformation According to Eq. (1.2), the other potential functions have the following forms: Thus, Eq. (1.2) can be transformed into the bilinear form [35][36][37] D 3 where D x , D y and D t are bilinear differential operators and have been defined as follows , t x =x,y =y,t =t.

(2.3)
Then we can obtain the N -soliton solutions of Eq. (2.2) as follows: where In the above expressions, k i , l i and ψ i are arbitrary constants, μ=0,1 is the summation of all possible combinations of μ i , μ j = 0, 1. For the sake of illustration, we take ς = 1 and focus on solutions for different forms of the function u. The solutions for the potential functions v and w can be given similarly and will not be analyzed in detail.

The mixed solutions of the (2 + 1)-dimensional Hirota-Satsuma-Ito equation
In the previous section, we obtained the N -soliton solutions of Eq. (1.2). By taking different values for the parameters, the soliton solutions can be converted into breather solutions and lump solutions. In this section, we will focus on the mixed solutions that contain breathers, lumps, and both, respectively. Finally, the conclusions are extended to the higher-order case by analyzing the simple case where N takes small values.

The mixed solutions contain breathers
In order to find the two-soliton solution of Eq. (1.2), it follows from Eq. (2.4) that we can take Substituting it into Eq. (2.1) and taking values for k 1 , k 2 , l 1 and l 2 in different domains, we are able to obtain a two-soliton solution and a one-breather solution as shown in Fig. 1a and b, respectively.

(3.2)
Similarly, taking values for k i , l i , i = 1, 2, 3, the mixed solution consisting of one soliton and one breather can be obtained as illustrated in Fig. 1c.
3 , l 3 = 1 and l 4 = 2, and take different values for the parameter t, the shapes of the mixed solutions obtained above will be significantly different as presented in Fig. 2. It is clear that as the parameter t increases, the overall direction of the mixed solution moves in the positive direction of the x-axis and the negative direction of the y-axis.
In addition to this, when taking different values for k i and l i , we can obtain different forms of the twobreather solutions-intersecting, parallel and coincident. To illustrate this with the two-breather solutions shown in Fig. 3, the two breathers are intersecting when l 1 = l 3 , and parallel when l 1 = l 3 . Further, if k 1 = k 3 is Fig. 1 The parameters are selected as ψ 1 = 0, ψ 2 = 0, t = 0. a is a two-soliton solution with k 1 = 1 2 , l 1 = − 4 5 , k 2 = 3 5 , l 2 = 1. b is a one-breather solution with k 1 = 2 5 + 3 10 i, l 1 = 1 5 + 9 10 i, i. c is the mixed solution composed of one soliton and one breather with The mixed solutions consist of two solitons and one breather with satisfied, the two breathers are coincident. The effect of the parameters t, k i and l i on the shape of the solution have similar properties in other cases, so we will not discuss them further.
(3.4) By analyzing the above solutions, we summarize the restricted conditions of the mixed solutions containing breathers in Table 1, where * represents the complex conjugation. Fig. 3 The parameters are selected as The parameters are selected as a is the mixed solution composed of three solitons and one breather with k 5 = 1 and l 5 = 1. b is the mixed solution composed of one soliton and two breathers with Three-soliton One breather and one soliton Four-soliton One breather and two solitons Five-soliton One breather and three solitons Two breathers and one soliton Six-soliton Three breathers

The mixed solutions contain lumps
The long wave limit method is a mathematical approach in which the limit is taken as the wavelength approaches infinity and the amplitude of the waves can remain constant. It is used to approximate systems when the wavelength of the waves involved is much longer than the size of the system. It is particularly useful for studying the behavior of waves in fluids, such as water waves, whose wavelength may be much greater than the depth of the water [67][68][69][70][71]. When we take ψ 1 = ψ 2 = iπ in Eq. (3.1) and apply the long wave limit method, the f 2 can be reduced to the following form Substituting Eqs. (3.5) into (2.1) and taking l 1 = 4 5 + 9 10 i, l 2 = 4 5 − 9 10 i, the one-lump solution can be obtained as shown in Fig. 5a.
If we want to get the mixed solutions consisting of one soliton and one lump, without loss of generality, we take ψ 1 = ψ 2 = iπ in Eq. (3.2) and apply the long wave limit method. Then f 3 can be simplified as follows We can acquire the mixed solution composed of one soliton and one lump as depicted in Fig. 5c, after substituting Eqs. (3.6) into (2.1) and taking values for the parameters.
Similarly, when we take ψ 1 = ψ 2 = iπ in Eq. (3.3) and use the long wave limit method, the f 4 can be rewritten in the following form with and l 4 = 2, the mixed solution containing two solitons and one lump can be obtained as shown in Fig. 6a.
3), the f 4 can be simplified as Eq. (3.8). Then substituting it into Eq. (2.1), when l 1 = l * 2 , l 3 = l * 4 are satisfied, the two-lump solution can be found as visualized in Fig. 6b. where Similarly to the case of N = 4, we can obtain the mixed solution containing lumps when N = 5. Simplifying f 5 by the long wave limit method gives Eqs. (3.9) and (3.10), and substituting them into Eq. (2.1), we obtain the mixed solution with three solitons and one lump, and the mixed solution with one soliton and two lumps, respectively, as depicted in Fig. 7a, b.
For the case of N = 6, when ψ j = iπ, j = 1, 2, . . . , 6 and apply the long wave limit method, f 6 can be written in the following form, substituting it into Eq. (2.1) and assigning values to l i , where i = 1, 2, . . . , 6, the three-lump solution can be obtained as presented in Fig. 7c. with Finally, when ψ j = iπ, j = 1, 2, . . . , N , where N = 2M, we can deduce the M-lump solution as follows. Moreover, the restricted conditions of the mixed solutions containing lumps are given in Table 2.

The mixed solutions contain breathers and lumps
Now, we will focus on the mixed solutions of Eq. (1.2) containing both breathers and lumps, where the order of the soliton solution satisfies N ≥ 4. In Sect. 3.1, we have obtained the forms of f 4 and f 5 , when ψ 1 = ψ 2 = iπ and under the action of the long wave limit method. Then, by taking values for the parameters, we can obtain the mixed solution containing one breather and one lump for the case of N = 4 as shown in Fig. 8a, and the mixed solution consisting of one soliton, one breather, and one lump for the case of N = 5 as revealed in Fig. 8c.
With the above results, we obtain the mixed solutions that contain both breathers and lumps. In particular, we obtain the form of the coexistence of soliton, Three-soliton One lump and one soliton Four-soliton One lump and two solitons Five-soliton One lump and three solitons k 1 , k 2 → 0, l 1 = l * 2 Two lumps and one soliton breather, and lump in Fig. 8c. Thus, for higher-order soliton solutions, we can obtain more mixed solutions with complex styles. Finally, we summarize this section in Table 3.

The nonlinear transformed waves of the (2 + 1)-dimensional Hirota-Satsuma-Ito equation
In this section, we focus on the nonlinear transformed waves of Eq. (1.2) when N takes values from 2 to 4. Through the following discussion, we can obtain different forms of nonlinear transformed waves and the mixed solutions composed of them.

Combination of nonlinear transformed waves
with soliton solutions in the case of N = 3 By means of Eq. (4.1), f 3 can be written in the following form: with  Quasi-anti-dark soliton we can get the mixed solution containing an oscillation M-shaped soliton and one soliton as displayed in Fig. 13a. When α 1 = 0.12, β 1 = 1.2, we can derive the mixed solution including one multipeak soliton and one soliton as presented in Fig. 13b. When α 1 = 0.001, β 1 = 2, we can derive the mixed solution made up of one quasi-periodic soliton and one soliton as visualized in Fig. 13c. Next, when κ 1 = π and using the long wave limit method to Eq. (4.12), we can obtain When N = 4, we introduce the new variable where α 2 , β 2 , τ 2 , δ 2 = 0, λ 3 > 0, ρ 2 and κ 2 are arbitrary real constants, k * 4 , l * 4 and ψ * 4 are complex conjugates of k 4 , l 4 and ψ 4 , respectively. Then, substituting where 1 , 1 , λ 2 are determined by Eq. (4.3).

The mixed solutions with one nonlinear transformed wave and one breather
When δ 1 = 0 and δ 2 = 0, the two-breather solution can be converted to a mixed solution consisting of one nonlinear transformed wave and one breather. At this moment, the f 4 can be expressed as follows with

Case A. The mixed solutions contain quasi-periodic soliton
Quasi-periodic solitons are localized wave packets that not only propagate without changing shape but also have a quasi-periodic oscillatory behaviour. This means that the soliton exhibits a complex, non-repetitive oscillatory pattern in its waveform, due to the interaction between nonlinear effects and dispersion in the medium. Let α 1 = 0.001, β 1 = 1 and τ 1 = 1.1, assigning values to α 2 , β 2 and τ 2 according to Table 4, we can acquire the mixed solution formed by the quasiperiodic soliton with quasi-anti-dark soliton, M-shaped soliton, oscillation M-shaped soliton and multi-peak soliton, respectively. Since the amplitude of the quasiperiodic soliton is much smaller compared to the amplitude of the other nonlinear transition waves that form the mixed solution, we enlarge its small amplitude region and place it in the cross-sectional view of the corresponding mixed solution, as shown in Fig. 17. In particular, when α 1 = 0.001, β 1 = 1, τ 1 = 3, α 2 = 0.002, β 2 = 2 and τ 2 = −1, we can obtain the solution formed by two quasi-periodic solitons as presented in Fig. 18.
is the cross-sectional view of (a) at y = 10 Case B. The mixed solutions contain multi-peak soliton A multi-peak soliton is a type of soliton with multiple peaks or humps. Unlike single-peaked solitons, which have a single maximum amplitude, a multi-peak soliton has several peaks of different amplitudes. Similar to Fig. 17, the mixed solutions containing one multipeaked soliton are depicted in Fig. 19.

Case C. The mixed solutions contain oscillation Mshaped soliton
An oscillating M-shaped soliton is similar in shape to an M-shaped soliton but has a small oscillating tail. We take α 1 = 0.5, β 1 = 1.5 and τ 1 = 1 as fixed values to find the mixed solutions containing oscillation M-shaped soliton. In Fig. 20, when we take values for α 2 , β 2 and τ 2 , we obtain mixed solutions consisting of oscillation M-solitons with quasi-anti-dark solitons, M-shaped solitons, and oscillation M-solitons, respectively.

Case E. The mixed solutions contain quasi-antidark soliton
An anti-dark soliton is a type of soliton that has a localized wave packet with a depression or gap in its centre, rather than a peak. Taking suitable values for α i , β i and τ i , where i = 1, 2, we can obtain the solution formed by two quasi-anti-dark solitons as shown in Fig. 22a.

Conclusions
In this paper, solutions of different forms for the (2 + 1)-dimensional Hirota-Satsuma-Ito equation have been studied and their properties have been analyzed by means of images.
We have obtained the N -soliton solutions for Eq. (1.2) using the Hirota bilinear method, and on this basis, the breathers and lumps have been obtained by using complex conjugate parameters and the long wave limit method. Further, we discuss the mixed solutions containing breathers, lumps, or both of the two, respectively. At N = 2 and taking δ 1 = 0, six different kinds of nonlinear waves have been obtained. For the case of N = 4, when we take δ 1 = 0 and δ 2 = 0, the two-breather solutions are converted into the mixed solutions of one nonlinear transformed wave and one breather; when δ 1 = 0 and δ 2 = 0, the two-breather solutions are turned into solutions consisting of two nonlinear transformed waves.
For the (2 + 1)-dimensional HIS equation, we will investigate the higher-order nonlinear transformed waves, as well as the collision properties of the components in the mixed solutions in our future works. Publisher's Note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.