X-type Soliton, Resonance Y-type Soliton and Hybrid Solutions of a (2+1)-dimensional Hirota-Satsuma-Ito System for the Shallow Water Waves


 Water waves are observed in the rivers, lakes, oceans, etc. Under investigation in this paper is a (2+1)-dimensional Hirota-Satsuma-Ito system arising in the shallow water waves. Via the Hirota method and symbolic computation, we derive some X-type and resonance Y-type soliton solutions. We also work out some hybrid solutions consisting of the resonance Y-type solitons, solitons, breathers and lumps. Graphics we present reveal that the hybrid solutions consisting of the resonance Y-type solitons and solitons/breathers/lumps describe the interactions between the resonance Y-type solitons and solitons/breathers/lumps, respectively. The obtained results rely on the water-wave coefficient in that system.


Introduction
Waves, one of the most common natural phenomena, have attracted people's attention [1][2][3][4][5][6]. It has been known that mechanical waves consist of the water waves, seismic waves, sound waves in air, electromagnetic waves, quantum waves, etc [7][8][9][10][11][12]. Among them, the subject of water waves has been the most fascinating and varied of all areas in the study of wave motions [13]. The core of fluid dynamics in general, especially ocean dynamics, has been the study of the water waves and their various ramifications [13]. It has been a fact that most of the fundamental ideas and results for nonlinear dispersive waves and solitons originated in the investigation of the water waves [14,15].
Solitons, one type of the nonlinear waves, have been found in optics, fluid mechanics, plasma physics and so on [31][32][33]. Soliton resonance phenomenon has been actively studied, and initially defined as "at the critical angle of intersection, two obliquely directed solitons interact strongly to make a branch soliton from a point at which the wave fronts of the two solitons meet together" [34]. For certain NLEEs, it has been believed that the resonances of solitons might give rise to various types of new excitations such as the breathers, lumps, web solitons, solitonic fissions and fusions, rogue and rational-exponential waves [35].
The Hirota-Satsuma shallow water wave system [36,37], has been introduced to describe the unidirectional shallow water waves, where ϕ(x, t) and ψ(x, t) are the functions of the variables x and t, while the subscripts denote the partial derivatives. For the shallow water waves, Refs. [37][38][39][40][41][42][43][44] have considered the following generalization of Eqn. (1) called the Hirota-Satsuma-Ito (HSI) system: with u(x, y, t), v(x, y, t) and w(x, y, t) being the functions of the scaled spatial coordinates x, y and temporal coordinate t, α being a real non-zero constant, u(x, y, t) meaning the physical field, while v(x, y, t) and w(x, y, t) representing the potentials of physical field derivatives.
Via the dependent variable transformations as follows: Ref. [43] has derived a bilinear form for System (2) as where f is a real differentiable function of x, y and t, while D x , D y and D t are the bilinear operators defined by [36] with G(x, y, t) being a differentiable function with respect to x, y and t, H(x ′ , y ′ , t ′ ) being a differentiable function of the independent variables x ′ , y ′ and t ′ , while a 1 , a 2 and a 3 being the non-negative integers. N -soliton solutions for System (2) have been presented as [38] where with N being a positive integer, ı, , ℓ = 1, 2, · · · , N , k ı 's, p ı 's and ζ ı 's being the complex constants, N ı< implying the summation over all possible combinations of N elements under the condition ı < , and N ı< meaning the product over all possible combinations of N elements under the condition ı < . There have been some works closely related to System (2): complexiton solutions [37], localized wave interaction solutions [38], rational localized waves and their Absorb-Emit interactions [39], multi-wave, breather wave and hybrid solutions [40], resonant multi-soliton solutions [41], higher-order breathers, lumps and semi-rational solutions [42], lump and lump-soliton solutions [43], Alice-Bob HSI system and its Bäcklund transformation, bilinear form, lump and breather solutions [44].
However, to our knowledge, X-type soliton solutions, resonance Y-type soliton solutions different from those in Ref. [41], and hybrid solutions composed of the resonance Y-type solitons and solitons/breathers/lumps for System (2) have not been reported. In Sec. 2, two X-type soliton solutions for System (2) will be obtained based on the Hirota method and symbolic computation. In Sec. 3, resonance Y-type soliton solutions which differ from those in Ref. [41] will be worked out through the Hirota method and symbolic computation. In Sec. 4, hybrid solutions composed of the 2-resonance solitons and two solitons/the first-order breathers/the first-order lumps for System (2) will be presented via the Hirota method and symbolic computation. Section 5 will be our conclusions.

X-type soliton solutions for System (2)
To search for some X-type soliton solutions for System (2), we consider the two-soliton solutions for System (2) obtained from N = 2 in N -Soliton Solutions (5), i.e., where while k i 's, p i 's and ζ i 's (i = 1, 2) are the real constants. Setting certain conditions on A 12 in Two-Soliton Solutions (6), we are able to derive two X-type soliton solutions for System (2) as follows: where the other parameters are determined via Two-Soliton Solutions (6). For the shallow water waves, X-Type Soliton Solutions (7) and (8)

4.3
Hybrid solutions consisting of the M -resonance Y-type solitons and higher-order lumps for System (2) Setting the following conditions on N -Soliton Solutions (5): we are able to obtain the hybrid solutions consisting of the M -resonance Y-type solitons and Qth-order lumps for System (2). When we take N = 4, i.e., M = 2, Q = q = 1, hybrid solutions composed of the 2-resonance Y-type solitons and first-order lumps for System (2) are derived as K 1 = K * 2 = K 11 + iK 12 , P 1 = P * 2 = P 11 + iP 12 , ζ 1 = ζ 2 = πi, For the shallow water waves, hybrid solutions composed of the 2-resonance Y-type solitons and first-order lumps, i.e., (16), are dependent on the coefficient α in System (2). Figure 5 shows the interaction between the 2-resonance Y-type soliton and first-order lump via Solutions (16).

Conclusions
Since waves play a pervasive role in nature, people have paid attention to the study of waves, and to the water waves in particular. Shallow-water models have been applied in fluid dynamics, ocean dynamics, etc. For the shallow water waves, making use of the Hirota method and symbolic computation, we investigated a (2+1)-dimensional HSI system, i.e., System (2). Our results depend on α, the coefficient in System (2). Main results obtained before can be summarized here: • Two X-type soliton solutions, i.e., Solutions (7) and (8) have been obtained. Propagations of the X-type solitons have been displayed in Figs. 1.
• Resonance Y-Type Soliton Solutions (10) describing the interaction between the two 2-resonance solitons have been derived. Figure 2 has shown the interaction between the two 2-resonance Y-type solitons.
• Hybrid solutions composed of the 2-resonance Y-type solitons and two solitons, the 2-resonance Y-type solitons and first-order breathers, and the 2-resonance Y-type solitons and first-order lumps, i.e., Solutions (12), (14) and (16), have been worked out.
Interactions between the 2-resonance Y-type soliton and two solitons, the 2-resonance Y-type soliton and first-order breather, and the 2-resonance Y-type soliton and firstorder lump have been exhibited in Figs. 3, 4 and 5, respectively.