A New Nonlinear Equation with Lump-Soliton , Lump-Periodic , and Lump-Periodic-Soliton Solutions

An extended (2+1)-dimensional Calogero-Bogoyavlenskii-Schiff-like equation is proposed by using the generalized bilinear operators based on a prime number p = 3. By combining multiexponential functions with a quadratic function, the interaction between lumps andmultikink soliton is generated. In themeanwhile, the interaction of lumpwith periodicwaves and the interaction among lumps, periodic waves, and multikink soliton can be obtained by introducing the ansätz forms. The dynamics of these interaction solutions are analyzed graphically by selecting appropriate parameters.

Besides, new NPDEs can be constructed by extending Hirota's bilinear form, which involve different prime numbers and possess potential applications [40].It is interesting to study lump solutions of these kinds of the NPDEs associated with the generalized bilinear operators [41][42][43].In this paper, the main purpose is to construct an NPDE by using generalized bilinear operators which possesses lump solutions.In the meanwhile, interaction between a lump and multikink soliton, and interaction of lump with periodic waves, and interaction among a lump, periodic waves, and multikink soliton are obtained by introducing the ansätz forms.The interaction among a lump, periodic waves, and multikink soliton to the NPDEs has not been reported in other studies.
Based on the generalized bilinear operators [40], a generalized bilinear form on a prime number  = 3 has  3,  3, +  3  3, Employing a dependent variable transformation (8) the generalized bilinear form of (5) can be cast as an eCBS like (eCBSL) equation in the following form: The purpose of the present paper is to study some novel interaction solutions of the above eCBSL equation ( 9).This paper is organized as follows.In Section 2, the interaction between lump waves and multikink soliton is obtained by adding additional exponential functions to a quadratic function.In Section 3, the interacted lumps with periodic waves and interaction among lumps, periodic waves, and multikink soliton are derived by using the ansätz forms.Some concluding remarks will be devoted to the last section.

Interaction between Lumps and Multisoliton
. .Interaction between a Lump and One-Kink Soliton.In order to obtain interaction between lump waves and one soliton of (9), we assume that an interaction solution is determined by a sum of a quadratic function and an exponential function: with   ( = 1, 2, ⋅ ⋅ ⋅ , 9),  1 ,  1 ,  1 , and  1 being thirteen undetermined real parameters.By substituting (10) into (5) and vanishing the different powers of , , , and exp, we present two families of lump solutions for the parameters.
Case I.
The solution of  and  will be localized in all directions of spaces, while the parameters satisfy the conditions By substituting ( 10) into ( 8) and combining the parameters relations (11), the first class of the interacted lump with onekink soliton of ( 9) can be obtained: where Case II. with the solution of  and  will be localized in all directions of spaces.By substituting (10) into (8) and combining the relations (15), we get the second class of interacted lump with one-kink soliton of (9): where Remark.The interaction between lump waves and one-kink soliton will become lump waves with  1 = 0.
. .Interaction between a Lump and Two-Kink Soliton.For interaction between lump waves and two-kink soliton, we use the form of a quadratic function with two exponential functions: By inserting ( 19) into ( 5) and gathering the coefficients of , , , and exp, the parameters are yielded by solving the algebraic equations The parameters need to satisfy the conditions then, the solution of  and  is localized in all directions of spaces.By substituting (19) into (8) and combining the parameters relations (20), we get the following interaction solution of ( 9): where The ansätz form of ( 19) is a combination of a quadratic function and two exponential functions.This kind of interaction solution is thus called a special rogue wave [46].Figure 1 shows this kind of a special rogue wave by selecting the parameters as The spatial structure of a lump wave caught in two-kink soliton is described in Figure 1(a) at  = 0.

Interaction among Lumps, Periodic Waves, and Multisoliton
. .Interaction between a Lump and Periodic Waves.Similar to the lump-soliton solutions, the interaction between lump waves and triangular periodic waves for (5) can be assumed as follows: By substituting (24) into (5) and collecting the coefficients of , , , sin, and cos, the parameters are determined: The parameters satisfy then, the solution of  and  is localized in all directions of spaces.By substituting ( 24) into (8) and combining the parameters relations (25), we get the following interaction solution of ( 9): where The 3D plot, density plot, and curve plot for the interaction between a lump and triangular periodic waves of ( 27) are depicted in Figure 2. The specific parameters are chosen as The interaction between a lump and periodic waves of  is presented in Figure 2(a) at  = 0. . .Interaction among a Lump, Periodic Waves, and One-Kink Soliton.The interaction among lump waves, triangular periodic waves, and one-kink soliton for (5) can be assumed as follows: Substituting ( 29) into (5) leads to The parameters satisfy then, the solution of  and  is localized in all directions of spaces.By substituting (29) into (8) and combining the parameters relations (30), we get the following interaction solution of (9): where The parameters are selected as  . .Interaction among a Lump, Periodic Waves, and Two-Kink Soliton.The interaction among lump waves, triangular periodic waves, and two-kink soliton for (5) can be assumed as follows: By substituting (34) into ( 5) and collecting the coefficients of , , , sin, cos, and exp, the parameters are stated by solving the algebraic equations The parameters satisfy then, the solution of  and  is localized in all directions of spaces.Substituting (34) into (8) and combining the parameters relations (35), we get the following interaction solution of ( 9): where The parameters are selected as This kind of interaction solution has not been given in other studies.

Conclusion
In summary, some novel interaction solutions of a (2+1)dimensional eCBSL equation ( 9) were considered by using the ansätz forms.By combining a quadratic function with multiexponential functions, interaction between lumps and multikink soliton were found.The interacted lumps with periodic waves and interaction among lumps, periodic waves, and multikink soliton were derived by introducing additional trigonometric and exponential functions.The phenomena of interaction between a lump and two-kink soliton, interaction of lump with periodic waves, and interaction among a lump, periodic waves, and multikink soliton for the eCBSL equation were generated as illustrative examples.In particular, the behavior of interaction among a lump, periodic waves, and two-kink soliton has not been found in other studies.The results presented in this paper can be applied to other high-dimensional NPDEs.It may be helpful for the theories of the associated Hirota bilinear equation.In the meanwhile, the multisoliton, one-breather, and rogue wave can be obtained by the Darboux transformation [47].These kinds of solutions for (9) are worthy of further study.

Figure 1 :
Figure 1: Profile of the interaction between a lump wave and two-kink soliton in (22).(a) 3-dimensional plot with  = 0, (b) the corresponding density plot, and (c) the curve plot with various values of time.

Figure 2 :
Figure 2: Profile of the interaction between a lump wave and triangular periodic waves in (27).(a) 3-dimensional plot with  = 0, (b) the corresponding density plot, and (c) the curve plot with various values of time.

Figure 2 (
b) displays the corresponding density plot of the lump-periodic wave.Figure 2(c) represents the curve plot with  = 0 and various values of time  = −3,  = 0,  = 3.

Figure 3 :
Figure 3: Profile of the interaction among a lump wave, triangular periodic waves, and one-kink soliton in (32).(a) 3-dimensional plot with  = 0, (b) the corresponding density plot, and (c) the curve plot at various values of time.