A deterministic approach to investigate nonlinear evolution equations for large balance numbers

When the balance number is greater than one, the modified simple equation (MSE) method typically fails to yield analytical wave solutions for nonlinear evolution equations (NLEEs) that appear in engineering and mathematical physics. We have addressed this shortcoming in this article and established a technique to implement the MSE approach to investigate NLEEs for balancing number two. Two NLEEs, namely, the regularized long wave and the Jimbo-Miwa equations, have been investigated in order to affirm the approach. Through this method, we found further generic wave solutions related to physical parameters, and when the parameters receive particular values, solitons emerge from the exact solutions. Graphs are used to investigate the solitary wave features of the attained solution functions, which illustrate the usefulness, validity, and compatibility of the scheme.


Introduction
The physical characteristics of reality can be expressed mathematically, which is determined by studying ordinary and partial differential equations.Nonlinear evolution equations (NLEEs) are the most competent and acceptable structure to describe nature through partial differential equations.Analogous models are established in various arenas of research, from population ecology to economics, ranging from the physical and natural sciences, infectious disease epidemiology, biology, neural networks, mechanics, etc.The theory of evolution equations is a branch of functional analysis that studies nonlinear evolution equations (NLEEs).Many mathematicians have come up with various methods to investigate the models established on different issues.Consequently, different groups of mathematicians, physicists, and engineers have been working ceaselessly to develop exact solutions to NLEEs.Recently, they have developed several techniques  to explore different types of NLEEs to find the exact solutions that can protect nature from various kinds of destruction.Among the analytical methods, the modified simple equation approach [15][16][17][18][19][20][21] is a lately established expanding process.Its computation is simple and organized, and symbolic computation software does not require in modifying the obtained algebraic equations.But, there are some limitations: the approach is unable to provide a solution for the balance number higher than one.To our foremost knowledge, only a small amount of research has been conducted in the literature addressing higher balance numbers.In [22], Salam examined the modified Liouville equation through the MSE method for the balance number two and found few solutions.However, the attained solutions do not satisfy the desired equation.Furthermore, Zayed and Arnous have applied the same method to unravel the KP-BBM equation [23].In the above-discussed articles, there are no directives on how to examine other NLEEs for the balance numbers greater than one.When the balance number is higher than one, it is often hard to unravel the NLEEs using the MSE approach, and one cannot use this method directly.Therefore, in this article, we have devised a technique to exceed this issue and considered two NLEEs, namely the regularized long wave and Jimbo-Miwa equations are successfully examined to find some broad-ranging soliton solutions to validate this newly developed strategy.Furthermore, we have figured out the wave profile of the attained solutions by choosing different values of the associated free parameters with the help of MATLAB software.

Algorithm
Let us suppose the succeeding nonlinear evolution equation: where H is a polynomial of an unknown function u = u(t, x, y, z) and its subscripts refer to partial derivatives, including the maximum order derivatives and nonlinear terms In order to investigate equation (1) using the MSE approach [17][18][19][20][21][22][23][24], we first accomplish the following steps: Step 1: Applying the traveling wave variable, into equation (1) gives the following nonlinear differential equation: where G is a polynomial of u(ξ) and its derivatives, wherein ¢ = u du dxi .
Step 2: Consider the solution to equation (3) is the subsequent form where a i , (i = 0, 1, 2, 3,K, N) are unfamiliar constants such that a N ≠ 0, to be evaluated, and s(ξ) is an unrevealed function to be determined.Here is to be noted that there is no need a pre-defined differential equation to determine the value of s(ξ).
Step 3: we can evaluate the positive integer N performing in equation ((3) using the balancing principle in equation (3) between the highest number of derivatives and the order of nonlinear terms For the degu(ξ) = N, the degree of the other expressions is as follows: Step 4: On account of the function s(ξ) we attain the polynomial of and its derivatives by substituting the solution (4) into equation (3).Equalizing the like power of the coefficients of s(ξ) −i , (i = 0, 1, 2, 3,K, N) to zero in the obtained polynomial.We gained a system of algebraic and differential equations that can be unraveled for receiving a i , (i = 0, 1, 2, 3,K, N), s(ξ) and the associate parameters are determined.

Solutions analysis
The aim of this section is to execute the MSE method to the regularized long wave equation and the Jimbo-Miwa equation to establish some fresh and broad-ranging solitary traveling wave solutions.

The regularized long wave equation
The regularized long wave equation was established to describe the behavior of the undular bore.This is an important physical phenomenon because it describes the dispersive wave with weak nonlinearity as well as nonlinear transverse waves in magneto-hydrodynamic waves in plasma, ion-acoustic, shallow water waves, etc The purpose of this sub-section is to obtain the travelling wave solutions through the stated MSE scheme to the general regularized long wave (GRLW) equation where β and γ are real constraints and q is a positive integer.For q = 1, the equation ( 6) is termed regularized long wave (RLW) and have solitary wave solutions, named solitons whose shape is not affected by a collision.
To construct solitons by using the MSE method to the equation (6), we consider the wave transformation Thus, equation (6) converts into the nonlinear equation as follows: Inserting the values of U, ¢ U and U″ into (8) and equating the cohorts of s 0 , s −1 , s −2 , s −3 , s −4 to zero, we attain respectively From equations (12) and ( 16), we obtain a 0 = 0, - , since a 2 ≠ 0. Therefore, based on the values of a 0 , there arise the following two cases: Case 1: When a 0 ≠ 0, then from equations (13)-( 15), we get where c 1 and c 2 are integration constants.Now, replacing the values of a 0 , a 1 , a 2 and s(ξ) into solution (9) gives After rearranging the solution (17), we develop the subsequent close-form solution of the regularized long wave equation ( 6 ) If we pick c 1 = -1 + ω and c 2 = -k 2 βω, then we obtain from solution (18) Again, considering c 1 = -1 + ω and c 2 = -k 2 βω for solution (18) then we obtain the bell-shaped soliton solution like Setting the values of c 1 = -1 + ω and c 2 = ± ik 2 βω , the solution (18) turns into the subsequent solitary wave solution as follows: x t 3 1 Also setting c 1 = -1 + ω and c 2 = ± ik 2 βω then solution (18) yields the following solitary wave solution: then from equations (13)- (15), we obtain , and where c 1 and c 2 are integral constants.
To obtain the solitary soliton we insert the values of a 0 , a 1 , a 2 and s(ξ) into solution (9) provides the soliton function as follows: ) Thus, we attain the general exact solution function (24) to the regularized long wave equation (6).Consider the values of arbitrary constant as c 1 = ω − 1 and c 2 = k 2 βω then the solitary wave solution (24) Besides, if we set c 1 = ω − 1 and c 2 = m i k 2 βω, then from (24), we get the solution function in the manner: Again, if c 1 = ω − 1 and c 2 = m i k 2 βω, then solution (24) turns into the form: Similar to this, different analytical soliton estimations can be generated by selecting different values of the integral constants and other relevant parameters.But, to avoid terseness, the solutions have not been documented.

Physical Interpretation of the Solutions
In this part, we discuss the wave profile and the significance of the obtained solutions to the regularized long wave equation through the 3D plot with the help of Matlab.The solution functions (19)  It is noted that the other obtained solutions to represent the same type of waves that are not asserted here for sagacity.

The Jimbo-Miwa equation
The Jimbo-Miwa equation is used to illustrate some interesting wave profiles in physics but does not pass any common test of integrability.In this sub-section, the MSE method is exploted to obtain standard explicit and solitary wave solutions to the Jimbo-Miwa equation in the form

xxxy x xy y xx yt xz
To investigate the solitons through the MSE method to equation (29) we apply the wave transformation By using equation (30) in equation (29) we attain the nonlinear equation as follows where ¢ represents the derivatives with regard to ξ. Substituting in equation (31) and integrating then we get the following differential equation: Applying the balancing principle between the highest degree of the nonlinear term U 2 and the derivative term U″ and gives the value of N = 2. Using the value of N we attain the shape of the solution function of equation (32) as in solution (10).Consequently, we obtain the subsequent system of equation by replacing the values of U and U″ into equation (32) and equating the same power of coefficients of s as follows From equations (33) and (37), we obtain a 0 = 0, , a 2 = − 2k, since a 2 ≠ 0. Hereafter we achieve the following cases related to the values of a 0 .10 Therefore, the solution of the Jimbo-Miwa equation in terms of the the exponential function is, Simplifying solution (39), we attain the solution to the Jimbo Miwa equation (29) as  x

y z t x y z t x y z t x y z t x y z t x y z t x y z t x y z t
By considering the several values of the arbitrary constant, we achieve the more general exact solution from the solution (40).
Also, if we set , then from equations (34)-(36), we obtain , and 2 , where c 1 and c 2 are integral constants.
Now using the values of a 0 , a 1 , a 2 and s(ξ) in equation (9), we have Therefore, we obtain the solution function of the Jimbo-Miwa equation as follows: The solutions (48)-(51) are drawn in figures 8 to 9 respectively for y = 0, z = 0 and ω ≠ 2. The most important convenience of the MSE method is that the calculations are straightforward and very spontaneous.It does not need any symbolic computation software to facilitate the calculations, as it needs for the sine-cosine method, the exp-function method, the tanh-function method, the ¢ ( ) G G -expansion, the homotopy analysis method, etc.It is imperative to note that the solutions obtained by the MSE method are equivalent to those solutions attained by the above-mentioned methods.Whereas c 1 and c 2 are arbitrary constants, we might attain many fresh and broad-ranging exact solutions to the stated Jimbo-Miwa equation by this scheme avoiding any help of symbolic computation software.

Conclusions
In this study, we have been able to overcome the limitations of the MSE method and make it suitable for the NLEEs whose balance number is greater than one.Furthermore, we have successfully applied the newly developed method to the regularized long wave equation and the Jimbo-Miwa equation; the balance number for both is two.Generally, the implemented method does not work to deliver any travelling solution for those NLEEs whose balance number is greater than one.For this case, we have successfully settled the technique for balance number two of any NLEEs to investigate the MSE method to appear the solutions.There is no solitary wave solution when the solution of s(ξ) is the polynomial of the wave variable ξ, meanwhile it does not satisfy the condition |u| → 0 as ξ → ± ∞ for solitary wave solution.As a consequence, each coefficient of the polynomial must be zero and this constraint is essential to solving NLEEs for greater balance numbers.We have implemented the achieved procedure to stated NLEEs and attained some fresh traveling wave solutions.When the parameters receive special values, solitary wave solutions are derived from the exact solutions.We analyzed  the solitary wave profile via 3D graphs of the obtained solutions for the generic values of the associated parameters.By successfully applying this method to two stated equations, it can be said that this method can be used very easily in the future to many other NLEEs whose balance number is greater than one which will make the research more advanced and fruitful.

Figure 2 .
Figure 2. Plot of periodic solutions u 3 in (21) and u 4 in (22) to the regularized long wave equation. w

+ c 3 2 3
we select c 1 = 3 + 2ω, c 2 = − k 2 and w =then the solution function (40) represents We attain the succeeding trigonometric function solution by simplifying solution (46) to the Jimbo-Miwa equation as Which is the more general travelling solution of the Jimbo Miwa equation ((29).Whereas c 1 , c 2 and c 3 are constant of integrations, if we pick the values of c 1 = (3 + 2ω), c 2 = k 2 and w (47) gives the subsequent solitary wave solutions:

3. 3 . 1 .
Physical interpretation of the attained solitons The purpose of the sub-module is to deliberate the physical interpretation of the obtained results to the Jimbo-Miwa equation.Solutions (41) demonstrate the kink-shaped solitons that are shown in figure 6.The solution function (42) represent the same wave as the solution (41).When a travelling wave ascends from one asymptotic state to another is called kink and this approach to continuous at infinity.Moreover, figure 7(a) shows the spike type wave for the solution (44) within −4 x, t 4 for ω = -0.6,β = 1, γ = 1, y = 0, z = 0 and 7(b) depicts the 2D profile of it for t = -1, t = 0, t = 1.On the other side, the solutions (48) and (50) represent the wave profile shown in figures 8 and 9 for −4 x, t 4 for ω = 2, β = -1, γ = 1, y = 0, z = 0 and −4 x, t 4 for ω = − 1, β = − 1, γ = 1, y = 0, z = 0 respectively.Besides, the other obtained solutions represent the same type of waves for the different values of the free parameters but we have not documented them here to avoid the repetition.