A Novel Dynamic of Localized Solitary Waves for the Hirota-Maccari System

This paper investigates a particular family of semi-rational solutions in determinant form by using the KP hierarchy reduction method, which describe resonant collisions among lumps or resemble line rogue waves and dark solitons in the Hirota-Maccari system. Due to the resonant collisions, the line resemble rogue waves are generated and attenuated in the background of dark solitons with line proﬁles of ﬁnite length, it takes a short time for the lumps to appear from and disappear into the dark solitons background. These novel dynamic of localized solitary waves may be help to understand some physical phenomena of nonlinear localized waves propagation in many physical settings.


Introduction
It is a hot topic to investigate the dynamics of nonlinear-wave interactions, which lead to rich dynamics in many fields of nonlinear physical systems such as fluids, plasmas, nonlinear optics, field optics and so on. Seeking the coherent solutions of nonlinear-wave interactions are fast becoming a key in properly understanding their theoretical description and experimental observations in a variety of physical settings.
In the past years, many nonlinear evolution equations (NEEs) can be used to describe the rather complicated interactions of the nonlinear waves, which have been analyzed and numerically studied. For the analytic solutions for the NEEs, certain methods such as the inverse scattering transformation [1], the bilinear method [2], Darboux transformation [3], Bäcklund transformation [4] are proposed. Some hybrid solutions which exhibit the interactions of the soliton, breather, rogue wave and lump are studied for nonlinear Schrödinger equation [5,6], Davey-Stewartson equations [7,8], Mel'nikov equation [9,10], Yajima-Oikawa systems [11,12], KP equation [13,14], etc. It is interesting that the lump solutions can be transformed into rogue waves under certain condition [15][16][17]. Rogue waves are waves of enormous energy that unpredictably arise transient and its appearing time are very short-lived. Actually, for most of the two-dimensional equations, the rogue waves are mostly manifested as line waves with profiles of infinite length generated by the wave background, and then rapidly attenuate to the same background. In this case, the rogue waves are divided as the line rogue waves that are not localized in the two-dimensional space but in time.
In this paper, we investigate a novel dynamic of localized solitary waves for the Hirota-Maccari (HM) system [18]: By using various methods and techniques soliton solutions, periodic waves solutions and exact travelling wave solutions of the Eq. (1) have been obtained [19][20][21]. Besides, we are devoted to the study of nonlinearwave interactions of the Eq. (1) in [22], which capture a resonant interaction of lumps with dark solitons.
Recently, Rao has derived the so-called doubly localized two-dimensional rogue waves, which described by semi-rational type solutions are doubly localized in two-dimensional space and time [23][24][25]. To our knowledge, these waveform structures have not been investigated in the HM system. Thus one motive of our work is to get doubly localized two-dimensional rogue waves to the HM system. We consider a new type of semi-rational solutions of the HM system which are expressed in determinant form through KP reduction technique, which describes resonant collisions among lumps or resemble line rogue waves and dark solitons.
Compared with the previous results that the interaction between lumps and dark solitons in [22], the lumps studied in the present paper lead to a intriguing soliton behaviours: the lumps are generated from the dark soliton background, and then decay to the dark soliton background equally quickly. Namely that the lumps have the characteristics of two-dimensional spatial localization and one-dimensional temporal localization.
The organization of this paper is as follows. In Section 2, we obtain a new family of semi-rational solutions to the HM system by means of KP hierarchy reduction method. In Section 3, we investigate a novel dynamic of localized solitary waves for the HM system (1), which displays the interaction between three types of localized solitary waves and dark line solitons. Our conclusions will be given in Section 4.

Semi-rational solutions to the HM system
It is cleared that the HM system (1) can be rewritten in the following form by using the coordinate transformation x → ix, y → −y, t → −it: In order to construct a family of semi-rational solutions of the HM system (2) by utilizing the KP hierarchy reduction method. We employ the variable transformation then (2) can be transformed into the bilinear form where D is Hirota operator and * represents the complex conjugation. Consider the conversion criteria (∂ x − ∂ s )f = cf, h = g * and c is a arbitrary constant. Then (4) can be generalized as a (3 + 1)-dimensional system where s is an auxiliary variable. Now, we begin with characterizing the following result on the KP hierarchy [26][27][28]: where the elements of matrix m The following bilinear equations in the KP hierarchy are satisfied by these tau functions: Assuming x −1 , x 1 , x 3 are real and x 2 is pure imaginary. Considering parametric constraints q = p * , d jl = c * jl , (ξ j ) * = η j , then we can obtain Change the variable to x and treat x −1 just as a parameter that can be substituted any value, in particular, the value here is zero. Then Eq. (8) can be reduced to the bilinear equations (4) of the HM system for τ 0 = f, τ 1 = g, τ −1 = g * , and the tau functions (6) would be turned into the solution of Eq. (4). Hence, we obtain a new family of semi-rational solutions to the HM system (1) by the following theorem. Theorem 1. The HM system (1) has semi-rational solution (3) with f, g, g * given by where the matrix elements are presented by and p, c rk , ξ r are complex parameters, N is an arbitrary positive integer and δ rj is the Kronecker delta.

A novel dynamic of localized solitary waves and dark line solitons
The semi-rational solutions in Theorem 1 illustrate the interaction between N th-order lumps or resemble line rogue waves and (N + 1) dark line solitons, which lead to the true localization of N th-order lumps or resemble line rogue waves in two-dimensional space and time. In this section, we study in detail the novel dynamics of these doubly localized solitary waves and dark line solitons for the HM system.

The interaction between fundamental lump or resemble line rogue wave and two dark line solitons
Taking N = 1 in Theorem 1, we have the first-order semi-rational solution for the HM system as where m rj are given by (10). After simple algebra calculations, the final form of the first-order semi-rational solution is given as where It should be emphasized that u = (ln(F )) x is the first-order rational solution for the HM system, which is As studied in Ref. [22], the resemble line rogue wave in the HM system with features is similar to line rogue waves, which appears and disappears with profiles of infinite length in (x, t) planes. That means it is not

The interaction between non-fundamental lump or resemble line rogue wave and three dark line solitons
By taking N = 2, Theorem 1 generates the second-order semi-rational solution where m rj are given by (10). This semi-rational solution also exhibits two different dynamical scenarios depended on p I = 0 and p I = 0, respectively.
When p I = 0, the second-order semi-rational solution (15) yields to more rich wave patterns. According to the three choices of relations between phase parameters ξ j (j = 1, 2, 3) choices, we get two types dynamics expressed by the resonant collision among localized solitary waves and three dark line solitons. , where it appears determine whether the resemble lump-type rogue wave splits from the leftmost or rightmost dark line soliton in the above evolution process. We will discuss it in two cases. Case 1, when ξ 1 >> 0, ξ 1 >> ξ 3 >> ξ 2 , ξ 2 << 0, ξ 3 = 0, the resemble lump-type rogue wave first separates from a leftmost dark line soliton shown in Fig. 4(a). Case 2 is a fully symmetric structure with case 1. When ξ 2 >> 0, ξ 2 >> ξ 1 >> ξ 3 , ξ 3 << 0, ξ 1 = 0, the resemble lump-type rogue wave first separates from a rightmost dark line soliton shown in Fig. 4(b).
When p I = 0, we obtain two lump-type rogue waves on a background of three dark line solitons. With the increase of N (N > 2), the higher-order semi-rational solution is the superposition of several first-order semi-rational solutions with more complex dynamics.

Conclusion
In this paper, we have investigated a novel dynamic of localized solitary waves for the HM system. By Type III: the three-dimensional plots of the resonant collision among two lump-type rogue waves and three dark line solitons ξ 1 = 0, ξ 2 = 4π, ξ 3 = −4π.
structures. It is interesting that we also unearth a new type of localized resonant wave that is localized in temporal-spatial, which termed as resemble lump-type rogue waves. Comparing with lump-type rogue waves, the wave pattern of resemble lump-type rogue waves is very similar, but appear in different planes.
We have shown the dynamics of the two resemble lump-type rogue waves and three dark line solitons in

Data Availability
The data that support the findings of this article are available from the corresponding author, upon reasonable request.