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Novel Localized Waves and their Interaction Solutions for a Dimensionally Reduced (2+1)-dimensional Boussinesq Equation from N-soliton Solutions

10 The Boussinesq equation (BqE) has been of considerable interest in coastal and ocean 11 engineering models for simulating surface water waves in shallow seas and harbors, tsunami 12 wave propagation, wave over-topping, inundation, and near-shore wave process in which 13 nonlinearity and dispersion effects are taken into consideration. The study deals with the 14 dynamics of localized waves and their interaction solutions to a dimensionally reduced (2+1)- 15 dimensional BqE from N -soliton solutions with the use of Hirota’s bilinear method (HBM). 16 Taking the long-wave limit approach in coordination with some constraint parameters in the 17 N -soliton solutions, the localized waves (i.e., soliton, breather, lump, and rogue waves) and 18 their interaction solutions are constructed. The interaction solutions can be obtained among 19 localized waves, such as (i) one breather or one lump from the two solitons, (ii) one stripe and 20 one breather, and one stripe and one lump from the three solitons, and (iii) two stripes and one 21 breather, one lump and one periodic breather, two stripes and one lump, two breathers, and two 22 lumps from the four solitons. It is to be found that all interactions among the solitons are elastic. 23 The energy, phase shift, shape, and propagation direction of these localized waves and their 24 interaction solutions can be influenced and controlled by the involved constraint parameters. 25 The dynamical characteristics of these localized waves and their interaction solutions are 26 demonstrated through some 3D and density graphs. The outcomes achieved in this study can 27 be used to illustrate the wave interaction phenomena in shallow


Introduction
The research of localized waves is one of the foremost topics in the areas of nonlinear 33 science and mathematical physics. In nonlinear science, solitons, lumps, breathers, and rogue 34 waves are localized waves, which are important objects in nonlinear physical systems [1]. Due 35 to its importance in nonlinear science, researchers have paid deep attention to the exploration 36 of these localized waves to a family of nonlinear evolution equations (NLEEs). Generally, 37 solitary waves are localized waves that propagate at a constant speed without changing their 38 shape [2]. Such a wave shape preserves its identity after pair-wise collisions, caused by a 39 cancellation of nonlinear and dispersive effects in any nonlinear models [2]. With numerical 40 simulations, Hirota [3] validated the preservation of soliton identities after pair-wise collisions. 41 But the detailed analysis of the numerical results exposed the existence of some ripples after a 42 collision meaning that the original identity is not completely recovered [4]. Therefore, it is 43 directive to explore exact solutions of NLEEs admitting soliton solutions for proper scrutiny 44 of collisions. 45 It is worth mentioning that John Scott Russell, in 1834, first observed a solitary wave 46 travelling along a Scottish canal, and thereafter in 1965, Zabusky and Kruskal introduced the 47 term "soliton" [5,6]. However, on the other hand, lumps are rationally decaying solutions that 48 are localized in all spatial directions, while breathers are partially localized breathing waves 49 with a periodic structure in one certain direction [7]. A rogue wave is a special kind of rational 50 solution that is localized both in spatial and temporal directions [8,9]. The mechanism of 51 rogue/freak waves can be regarded as the high amplitude waves generated by the interaction of 52 solitons and breathers [1,10]. The concept of rogue/ freak waves in the ocean was first 53 proposed by Draper in 1965 [1]. The nonlinear localized waves mentioned above (solitons, 54 lumps, breathers, and rogue waves) appear in several fields, such as oceanography, fluids, 55 plasma physics, mathematical physics, optical fibers, nonlinear optics, cold atoms, Bose- 56 analyzing the wave propagation dynamics of the localized waves and their interaction 82 structures in shallow water through some relevant models. Although, in the past, Wazwaz [49] 83 mentioned that the study of wave propagation of long waves in shallow water has become a 84 dynamic research field in nonlinear science. As in ref. Jawad et al. [50], the dynamics of wave 85 propagation in shallow water are seen in lakes, rivers, ocean beaches and sea, and described by 86 the BqE, which leads to (1) 88 The BqE was first introduced by Joseph Boussinesq in the 1870s for modeling the 89 propagation of shallow water waves in multiple directions [51]. In this regard, the BqE 90 contributes to a dominant role in explaining different physical aspects in the field of ocean 91 engineering, fluid dynamics, and plasma physics dealing with a variety of wave phenomena.

92
In ocean and coastal engineering, oceanographers and coastal engineers use the BqEs for 93 simulating surface water waves in shallow seas and harbors, dune, ocean basin-scale tsunami 94 propagation, wave over-topping, inundation, and near shore wave process modeling in which 95 nonlinearities and dispersion are seen to be taken into consideration [52][53][54][55][56][57]. It is of interest to 96 note here that the physical behaviors of any nonlinear models can be explored by analyzing integrable (3+1)-dimensional BqE [52], which leads to 103 where , , , are arbitrary constants, and is an unknown function that depends on the 104 spatial coordinates ( , , ), and temporal coordinate . Eq. (2) explains the propagation of 105 gravity waves on the water surface [52]. Wazwaz and Kaur [52] proposed the extended form  We use the following dependent variable transformation [52]: 158 where , , , and are non-negative integers.

159
By virtue of the definition of HBDO and its properties, Eq. (5) is turned into the following 160 form: 163 Based on the simplified HBM and the transformation given by Eq. (4), -soliton solutions are 164 obtained from

166
where and stand for denoting the dispersion variables and generalized phase shifts, 167 respectively, and are given by where , , , and are arbitrary constants associated with the amplitude and phase of the 171 th-soliton. It is pertinent to mention that Wazwaz and Kaur [52] have recently constructed the 172 multiple solitons for Eq. (1) by using the above expression. So, we will not repeat its physical 173 importance.

174
Without loss of generality, = can be set in Eq. (5). Then, Eq. (5) is converted to the 175 following HBE: 177 Also, the functions for obtaining -soliton solutions of Eq. (10), the dispersion variables ( ), 178 and the generalized phase shifts are changed, respectively, into 186 where ( , , ) is a function involving spatial coordinates , , and temporal coordinate .  In this section, we will discuss localized waves and their interaction solutions in the case of 202 two-stripe, three-stripe, and four-stripe solitons.  204 Here, we discuss two cross-stripe solitons, parallel-stripe solitons, -periodic breather, -205 periodic breather, lump, and rogue wave solutions of Eq. (3). By taking = 2 in Eq. (11), one 206 can derive a function for two-soliton solutions given by

Interaction solutions between two-stripe solitons
208 where and are given by Eqs. (12) and (13), respectively. Substitution of Eq. (15) into Eq.  given by Eq. (16). To ensure such behavior, one -periodic breather is illustrated in Figs.

Case IV: Lump soliton and rogue wave solutions
Taking the long-wave limit as → 0 in Eq. (24) along with Eq. (16), the lump soliton of Eq.

310
On restricting certain conditions on the specified parameters and a long-wave limit approach, 311 one can construct several types of localized interaction solutions. Exploration of these solutions 312 is described in detail as follows. 314 Here, we would like to interact the two parallel stripe solitons with other one stripe soliton  separated (see Fig. 7(a1)). Then, at time = 0, they are merged together (see Fig. 7(a2)).     Table 2. The first column in Table 2 shows 405 the order of -soliton, while the second column shows the types of localized wave solutions.

406
The last column provides the corresponding restrictions of parameters for the interaction of 407 localized wave solutions.

419
where and are specified by Eqs. (12) and (13), respectively, and the phase shifts follow 420 = and = . plots through Fig. 9(b) and Fig. 9(c), respectively. When and are purely imaginary 448 numbers, and the others are real numbers, the breather wave is periodic only along the -axis.   one lump and one breather are displayed in Fig. 11. Figs. 11(a1)-(a5) illustrate the 3D plots of  propagate from the negative side of the -axis to its positive side, which is clearly realized from Figs. 539

13(b)-(f). It can be visualized from Figs. 13(a)-(f) that the two lumps propagate with the same velocity 540
along the -axis. During the propagation, the two lumps get closer, then collide with each other, and 541 finally move apart. It is also perceived from the figures that during the propagation, two lumps preserve 542 their amplitudes, but they altered their positions. The overview of the constraint parameters for the four-543 soliton solutions of Eq. (3) is presented in Table 3. The first column in Table 3 shows the order of -544 soliton, the second column shows the localized wave structures, and the last column provides the 545 corresponding parameters of the localized wave interaction solutions in the second column. 546  Conflict of interests. 602 The authors declare that they have no known competing financial interests or personal 603 relationships that could have appeared to influence the work reported in this paper.