Interactions of rogue and solitary wave solutions to the (2 + 1)-D generalized Camassa–Holm–KP equation

This research explores a (2 + 1)-D generalized Camassa–Holm–Kadomtsev–Petviashvili model. We use a probable transformation to build bilinear formulation to the model by Hirota bilinear technique. We derive a single lump waves, multi-soliton solutions to the model from this bilinear form. We present various dynamical properties of the model such as one, two, three and four solitons. The double periodic breather waves, periodic line rogue wave, interaction between bell soliton and double periodic rogue waves, rogue and bell soliton, rogue and two bell solitons, two rogues, rogue and periodic wave, double periodic waves, two pair of rogue waves as well as interaction of double periodic rogue waves in a line are established. Among the results, most of the properties are unexplored in the prior research. Furthermore, with the assistance of Maple software, we put out the trajectory of the obtained solutions for visualizing the achieved dynamical properties.


Introduction
Developing few precise results as well as new features to nonlinear partial evolution models (NLPEMs) is an important topic throughout soliton theories. In latest decades, researchers have been spending efforts to become skilled at solitonic solutions and interactions to NLPEMs and established several analytical approaches to extract such solutions, for instance the Darboux revolution [1], ðG 0 =GÞ-expansion [2], expfunction [3,4], sub-equation [5,6], Hirota bilinear [7][8][9], bilinear network [10,11] and Lie symmetry reductions [12] techniques. The Hirota bilinear technique is one of the major undeviating as well as expedient technique to achieve the precise solitonic elucidation to NLPEMs. When NLPEM be able to reach in bilinear formulation, then the rogue, multisoliton and more such type of breather wave solutions to the model are possible.
Last decades, scientists are greatly astonished to various dynamical signal solutions to their riveting category of rogue-lump solutions that be capable to originate in stream bearing wave of deep ocean [13,14], plasma and nonlinear optics [15,16] as well as Bose-Einstein solidifies [17]. Lou et al. [18] investigated lump wave solutions by means of variable separation technique in 2002. Few years ago, Ma and his co-investigators [19] recognized a common category of lump wave solution to the KPI equation. State shift lump propagations of (2 ? 1)-D gKdV model were investigated by Wang et al. [20]. Additional integral forms that preserved lump wave solutions are Boussinesq [21], BKP [22], Davey-Stewartson II [23], Ishimori-I [24] model equations. Due to coherent rationale results, lump propagation is analytical rational function solutions confined to a small area in every route in space whereas lump waves are limited to a small area in nearly every but not every route in space [25][26][27][28]. Rogue waves are bizarrely massive, erratic even rapidly occurring surface waves which can be immensely hazardous to hits the ships in deep ocean, even invert it due to massive ones. Rogue waves are confined in mutual space as well as time, arise from nowhere even vanish lacking a trace [29][30][31][32], have engaged with the liability of abundant oceanic disasters. More complicated phenomena and effects of nonlinearities can be described from multisoliton solutions of NLPDEs [33,34] using complex values to the free parameters involving in the multiwave soliton solutions of the models [35,36]. Peng [37] has investigated fractional-order chaotic system, while controlling of tentative nonlinear systems was studied by Gao et al. [38] and diagnosis of nonlinear systems was investigated by Rigatos et al. [39] with neural networks. Recently, enormous efforts have been salaried on diverge nonlinear models to derive collision among solitons, lump, rogue and hybrid solutions by dynamical researchers [40][41][42]. Motivated by these works in [40][41][42], we aim to derive such massive-type wave solution even more complicated nonlinear dynamical structures-like interactions of bell soliton with double periodic rogue waves, rogue with one and two bell soliton, double periodic waves as well as two pair of rogue waves in a line of the gCHKP model [43]. Those solutions are still unexplored for any nonlinear models in the previous studies.
In this manuscript, we sport light on the (2 ? 1)-D gCHKP model [43] to detect distinguish behaviors of where a; b and c are free invariables. Equation (1) is utilized to illustrate the task of spreading during creation of patterns inside fluid drops. In Ref. [43], authors have derived diverse wave structures as well as analysis stability of the solutions to the model. The various processes have been used to investigate the gCHKP model [Eq.
(1)] such as sine-cosine as well as tanh method [44] and derived compact and noncompact wave solutions; homoclinic breather limit method [45] and derived breather, rogue and solitary waves solutions; Lie group [46], exp-function [47], ansatz [48] methods and presented single solitary as well as periodic wave solutions. Here, we are highly interested to establish collisions of bell soliton with double periodic rogue waves, rogue with bell soliton, rogue with two bell solitons, two rogues, double periodic waves as well as two pair of rogue waves in a line for the model. The residual parts of manuscript are planned as: In Sect. 2, the bilinear structure to the gCHKP model is obtained by utilizing transformation u ¼ 12c b ½ln f xx and then a theorem is presented. We discuss lump surge propagations using positive quadratic function in next part 3. In part 4, the resulting N solitons are specified as well as various deformed waves with interactions are achieved. Conclusions are given with future task in the last part 5.
iff 6c=b is a constant.

Rogue-type solution of gCHKP equation
To explore quadratic polynomial solutions of the (2 ? 1)-D bilinear gCHKP model in (10), we initiate with in which where a i ; 1 i 9 ð Þ are free invariables to be calculated. Inserting Eq. (12) to bilinear structure equation (10) gives several polynomials in terms of x; y; t. Collecting coefficients of x; y; t as well as like invariable to be zero leads to a system of constraints in terms of a i ; 1 i 9 ð Þ , and solving the constraints by Maple software yields: where 4 5 . Therefore, substituting Eq. (14) into Eq. (12) with Eq. (13) and transformation equation (11) yields most wanted rogue wave solution to Eq. (1). We promised that any specific time t; the lump wave result tends to zero, iff g 2 þ h 2 ! 1; i.e., the same as to x 2 þ y 2 ! 1: At most or at least point in the lump wave, i.e., central point taking t ¼ t 0 is x ¼ a 2 a 7 t 0 À a 3 a 6 t 0 þ a 2 a 8 À a 4 a 6 a 1 a 6 À a 2 a 5 ; where a 1 a 6 À a 2 a 5 6 ¼ 0: Inserting Eq. (15) into Eq. (11), the maximum height of rogue reached at as of which anyone can detect that height of the rogue wave is dependent on a 1 ; a 5 ; a 9 ; b and c: Figure 1 shows the sketch of the 3-D view of rogue wave result to Eq. (1) in distinct planes, due to some values Fig. 1 looks like eye-shaped rogue profile that consisted of two valleys as well as one confined hump. Besides this, it is noticed that rogue has upmost crest comparing with surrounding as well as it can be appearances within a short time.

Multi-soliton solutions and various deformed wave structures
This section is going to present the solitary wave solutions analytically from the bilinear form. We utilized the Hirota scheme to derive the N-soliton solution of the model from its bilinear representation [Eq. (10)]. Moreover, we ascertain rational solutions of model through a long wave limit technique. Consider f has multiple solitons with the structure

Two solitons and their deformed structures into rogue and periodic waves
Owing to N ¼ 2; f 2 takes the structure where Replace Eq. (17) into Eq. (11) in addition to a few over-simplifications, then linked with every part of the coefficients of exp to zero, one can gain solution as: Including Eqs. (17)- (19) into Eq. (10) yields twosoliton solution to Eq. (1). Putting a ¼ À0: one be capable to achieve a 2-bell solitons solution that is depicted in Fig. 2a. Stand on over technique [Eq. (17)] presents breather as a result of picking proper invariables. Breather waves to Eq. (1) are possible in x; y ð Þ; x; t ð Þ as well as y; t ð Þ plane from Eq. (19), under the parametric conditions For instance, setting parameters as follows.
we obtain breather waves and its outlines in distinct planes (Fig. 2b-d). This expression [Eq. (17)] can also be written as.
Owing to uncover coherent elucidation to Eq. (10), a long limit of f in Eq. (17) have to exercise. For the purpose, set the parameters as to Eq. (17) with captivating limit as e ! 0, then f be able to write: where Plugging Eqs. (24), (25) interested in Eq. (10), resulting u yields We cover the rogue wave shapes for constants p 1 ¼ 2; p 2 ¼ À2 (Fig. 3). Evidently, Fig. 3 with r i ¼ k i x þ p i y þ q i t ð Þþr 0 i ; i ¼ 1; 2; 3 and q i ; B ij in which i\j; i; j ¼ 1; 2; 3 comes from Eq. (16).
The values of a ¼ À0: Fig. 4a. Figure 4a shows that it is

yield interaction of three bell solitons solution which is depicted in
and a 8 ¼ À2; a outlook of the wave for t ¼ 0; b outlook of the wave for x ¼ 0; c outlook of the wave for y ¼ 0; as well as analogous contour profiles (below) correspondingly an interaction of three bell solitons and interact at the origin.
Besides, the values a ¼ À0:005; b ¼ 3; c ¼ 0:5; where Fig. 4b, c. It is actually complex valued solution, and both real and imaginary parts present interaction of a bell soliton with double periodic rogue waves, in which bell soliton arises along diagonal. We provide also an extra density plot separately in Fig. 4c with different color combinations for better understanding.
For the complex conjugate values of the parameters ffiffiffiffiffiffi ffi À1 p . and r 0 1 ¼ r 0 2 ¼ r 0 3 ¼ r 0 4 ¼ 0 yields interaction of two periodic rogue waves in Fig. 5c Fig. 3, which comes from two-soliton solution. The interaction of double periodic line rogue with a bell soliton via Eq. (28) in Fig. 4b, a periodic line rogue with a bell soliton via Eq. (29) in Fig. 4b, a periodic line rogue with a bell soliton via Eq. (30) in Fig. 4f are derived from three-soliton solution. From the four soliton solution, we constructed interaction of two bell solutions with a periodic rogue wave depicted in Fig. 5b, two periodic line rogue wave via Eq. (33) visualized in Fig. 5c, periodic line rogue with a periodic wave via Eq. (34) displayed in Fig. 5d, two periodic waves illustrated in Fig. 5e. We also established colliding of two pair of periodic curvy rogue waves via Eq. (36) shown in Fig. 6a, b, two periodic rogue waves in a line solution via Eq. (37) depicted in Fig. 6c, d. Most of these dynamics are found for the first time in the literature.

Conclusions
We have successfully look over the (2 ? 1)-D gCHKP model, that has claimed processing of collisions of exponentially confined edifices exact solutions. By exploitation the Hirota bilinear structure, we have acquired the lump-, breather-, solitary-as well as vast colliding-wave solutions among the solutions. Such colliding wave's solutions arises as interaction between bell soliton and double periodic rogue waves, rogue and bell soliton, rogue and two bell solitons, two rogues, rogue and periodic wave, double periodic waves, two pair of rogue waves as well as interaction of double periodic rogue waves. In addition, some coherent solutions are stated charming by a long limit. Discussion of the outcomes reveals to comprehend multifaceted corporal occurrences of the gCHKP equation with sufficient graphical diagrams. The achieved results disclosed that the imposed investigation process advocates a straight as well as simple methodical tackle to handle any nonlinear harm over and done with the bilinear method occurring in engineering and applied problems. From experiences of these studies, we also aim to analyze the chaotic behavior of the gCHKP model following the detection using chaos analysis in [51][52][53] in near future. Our future task will also be to draw dynamical behavior of variable coefficient gCHKP model in this similar procedure.
Data availability The authors sketched the dynamical interactions of solitons and rogue with Maple. So, supported data are included inside this article and not taken from outside sources.

Declarations
Conflict of interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.