M lump and interaction between M lump and N stripe for the third-order evolution equation arising in the shallow water

In this paper, we use the Hirota bilinear method for investigating the third-order evolution equation to determining the soliton-type solutions. The M lump solutions along with different types of graphs including contour, density, and three- and two-dimensional plots have been made. Moreover, the interaction between 1-lump and two stripe solutions and the interaction between 2-lump and one stripe solutions with finding more general rational exact soliton wave solutions of the third-order evaluation equation are obtained. We give the theorem along with the proof for the considered problem. The existence criteria of these solitons in the unidirectional propagation of long waves over shallow water are also demonstrated. Various arbitrary constants obtained in the solutions help us to discuss the graphical behavior of solutions and also grants flexibility in formulating solutions that can be linked with a large variety of physical phenomena. We further show that the assigned method is general, efficient, straightforward, and powerful and can be exerted to establish exact solutions of diverse kinds of fractional equations originated in mathematical physics and engineering. We have depicted the figures of the evaluated solutions to interpret the physical phenomena.

The Hirota bilinear method for finding the lump soliton solutions and interaction of a lump solution with some of one-line, two-line, and three-line and even kink-breathersoliton solutions of evolution equations was initially introduced by Ma [45] by assuming the solution to be a series of functions including lump (combination of two positive functions as polynomial), lump-kink (combination of two positive functions as polynomial and exponential functions), called the interaction between a lump and one-line soliton, lump-soliton (combination of two positive functions as polynomial and hyperbolic cos functions), called the interaction between lump and two-line solitons, kinky breathersoliton (combination of two exponential functions and trigonometric cos function), and finally the stripe soliton function only with exponential solution function. The method received considerable attention and underwent through many improvements. It is important to note that the later improvements were given different names by different authors. For getting the lump solutions and their interactions, the authors have conjugated sufficient time to search the exact rational soliton solutions, for example, the Kadomtsev-Petviashvili (KP) equation [23], the B-Kadomtsev-Petviashvili equation [42], the reduced p-gKP and p-gbKP equations [25], the (2 + 1)-dimensional KdV equation [26], the (2 + 1)dimensional generalized fifth-order KdV equation [33], the (2 + 1)-dimensional Burger equation [34], the nonlinear evolution equations [22], the generalized (3 + 1)-dimensional Shallow water-like equation [35], the (2 + 1)-dimensional Sawada-Kotera equation [30], and the (2 + 1)-dimensional bSK equation [31,32]. Various types of work for finding the periodic solitary wave solutions of the (2 + 1)-dimensional extended Jimbo-Miwa equations [37], interaction between lump and other kinds of solitary, periodic and kink solitons for the (2 + 1)-dimensional breaking soliton equation [27], lump and interaction between different types of those on the variable-coefficient Kadomtsev-Petviashvili equation [28], and periodic type and periodic cross-kink wave solutions [29] are achieved through the Hirota bilinear operator.
Take the third-order evolution equation of the form where α, ε are small parameters, τ is the Bond number, and o Ψ is the surface elevation in the x-direction, which is a model for the unidirectional propagation of long waves over shallow water, obtained via asymptotic expansion around simple wave motion of the Euler equations up to first-order in the small-wave amplitude [61,62]. Assume the Hirota derivatives based on the functions ρ(x) and (x) given as where the vectors λ = (λ 1 , λ 2 , λ 3 ), μ = (μ 1 , μ 2 , μ 3 ), and o 1 , o 2 , o 3 are arbitrary nonnegative integers. It is known that this third-order evolution equation possesses a Hirota bilinear form We utilize the following relation between the functions φ(x, y, t) and Ψ (x, y, t): . Based on the Bell polynomial theories of soliton equations, we get to the relation (1.5) Then, by considering ρ = exp(∂ -1 x Γ ) > 0, the derivatives ρ x , ρ y , ρ t , ρ xx , ρ yy , ρ xy , ρ xt , ρ xxy , ρ xyy , ρ xxx , ρ xxyy , and ρ xxxx can be written as which can be rewritten as is the third-order evolution-type equation, and the theorem has been proved.
We clearly confirm that other published papers do not cover ours, and made work is really new. Here our purpose is discovering the exact solutions of the third-order evaluation equation under consideration by the Hirota bilinear method for gaining the M lump, the interaction between 1-lump and two-stripe solutions, and the interaction between 2lump and one-stripe solutions, which arise in more classes. We give a discussion about the third-order evaluation equation and the Hirota bilinear method. We also offer graphical illustrations of some solutions of the considered model along with the obtained solutions.
After that, we deal with the probe of solutions and finish by conclusion.

New M-lump solutions of the third-order evolution equation
According to analysis in [39], based on the Hirota operator, the solution of nonlinear differential equation (1.3) can be written as 3) The notation σ = 0, 1 shows summation over all conceivable compositions of σ 1 = 0, 1, σ 2 = 0, 1, . . . , σ N = 0, 1; the summation N i<j is over all conceivable compositions of N values of i < j. For example, the first three expressions of (2.1) are as follows:

Interaction between lumps and stripe solitons of equation (1.1)
In the following subsections, we further treat diverse solitons.

Interaction between 1-lump and 2-stripe soliton of Equation (1.1)
To treat 1-lump and 1-stripe solitons of equation (1.1), we catch f as a blend of the following functions: y, t, 1), (x 5 , x 6 , x 7 , x 8 ) = (x, y, t, 1), where (x 1 , x 2 , x 3 ) = (x, y, t), and Ω i , i = 1, . . . , 8, r 1 , r 2 , r 3 are free parameters to be found later. Plugging (3.1) into Eq. (1.3), collecting the coefficients at the diverse polynomial functions including the functions e Ω i x i and their products, and solving the obtained algebraic system containing 26 equations, we obtain the following solutions: Set I: which should satisfy the condition Plugging (3.3) into (3.2), we achieve te following interactive wave solution of Eq. (1.1): Moreover, by selecting suitable values of the parameters the graphic representation of periodic wave solution is presented in Fig. 11, including the 3D plot, density plot, and 2D plot when three spaces arise at spaces x = -1, x = 0, and x = 1. Set II: which should satisfy the condition Plugging (3.6) into (3.2), we achieve the following interactive wave solution of Eq. (1.1): Likewise, by selecting suitable values of the parameters the graphic representation of periodic wave solution is presented in Fig. 12, containing the 3D plot, density plot, and 2D plot when three spaces arise at spaces x = -1, x = 0, and x = 1.
Set III: ,  Plugging (3.9) into (3.2), we achieve the following interactive wave solution of Eq. (1.1): 1-lump and 2-stripe solitons, the energy of the 1-lump is more robust than that of the 2stripe soliton; as t → 0, the 1-lump commences to be swallowed by the 2-stripe soliton gradually, its energy commences to move from one place to another into the 2-stripe soliton progressively, until it is swallowed by the stripe soliton completely. These two types of solutions move into one soliton and continue to spread.

Interaction between 2-lump and 1-stripe solitons of equation (1.1)
To search treatment between 2-lump and 1-stripe solitons of equation (1.1), we catch f as a blend of the following functions: y, t, 1), where Ω i , i = 1, . . . , 4, r, k are free elements to be defined later. Plugging (3.12) into Eq. (1.3), collecting the coefficients at the diverse polynomial functions including e 8 i=5 Ω i x i and their products, and solving the obtained algebraic system of 11 equations, we get the following solutions: Set I: , (3.14) To ensure the positivity of ρ, we need the following determinant condition: (12τ + 6)(3τ -1) < 0, τ = -1 2 .

Conclusions
We employed the Hirota bilinear method, along with some Hirota derivatives and the Bell polynomial theories of soliton equations, to find abundantly many exact lumps and interaction lumps with two types of typical local excitations, which occurred between a lump and a stripe soliton of soliton solutions to third-order evaluation equation. We investigated M lump solutions and made different types of graphs, including the contour, density, and three-and two dimensional plots. We also obtained an interaction between 1-lump and two-stripe solutions and an interaction between 2-lump and one-stripe solutions and found more general rational exact soliton wave solutions of the third-order evaluation equation. This approach has been successfully applied to obtain some real rational soliton wave solutions to third-order evaluation equation with constant coefficients. We proved a theorem for the considered problem. We also obtained existence criteria of these solitons in the unidirectional propagation of long waves over shallow water. The attained solutions are in broad-ranging form, and the definite values of the included parameters of the attained solutions yield the soliton solutions and are helpful in analyzing the water waves mechanics, the quantum mechanics, the water waves in gravitational force, the signal processing waves, the optical fibers, and so on. This paper showed that the Hirota bilinear method, combined with Hirota derivatives, gives a unified approach to constructing the exact rational lump soliton wave solutions to many nonlinear partial differential equations. Our results allowed us to understand the dynamics of nonlinear propagation in fluid mechanics, plasma, and so on. Moreover, the established results have shown that the Hirota bilinear method is general, straightforward, and powerful and helped us to examine traveling wave solutions of NLPDEs.