Abstract
Most nonlinear partial differential equations have many applications in the physical world. Finding solutions to nonlinear partial differential equations is not easily solvable and hence different modified techniques are applied to get solutions to such nonlinear partial differential equations. Among them, we considered the modified Korteweg–de Vries third order using the balance method and constructing its models using certain parameters. The method is successfully implemented in solving the stated equations. We obtained kind one and two soliton solutions and their graphical models are shown using mathematical software-12. The obtained results lead to shallow wave models. A few illustrative examples were presented to demonstrate the applicability of the models. Furthermore, physical and geometrical interpretations are considered for different parameters to investigate the nature of soliton solutions to their models. Finally, the proposed method is a standard, effective, and easily computable method for solving the modified Korteweg–de Vries equations and determining its perspective models.
1. Introduction
It is significantly important in nonlinear phenomena to search for exact solutions to nonlinear partial differential equations (NLPDEs). Exact solutions play a vital role in understanding various qualitative and quantitative features of nonlinear phenomena. There are diverse classes of interesting exact solutions, traveling wave solutions, and soliton solutions, but it often needs specific mathematical techniques to construct exact solutions due to the nonlinearity present in their dynamical nature [1–3]. The NLPDEs are widely used as models to depict many important complex physical phenomena in a variety of fields of science and engineering. Some nonlinear partial differential equations can be written as follows:
Burger’s equation is as follows:
The modified Korteweg–de Vries (KdV) equation is as follows:
The Kadomtsev–Petriashvili (KP) equation is as follows:
The NLPDEs are fundamentally important because lots of mathematical physics models are often described by such wave phenomena and the investigation of traveling wave solutions is becoming more and more attractive in nonlinear sciences nowadays. However, NLPDEs are very difficult to solve explicitly, specifically, or in detail. As a result, many powerful methods have been proposed and developed for finding analytical solutions to nonlinear problems.
Some of the researchers used to solve NLPDEs are the simple equation method [4], the tanh-function and balance methods [5], the inverse scattering transform method [6], Backland and transformation [7], and other mathematical procedures such as combinations of the equation, transformation procedure, bilinear method, integration, and so on. Of these methods, the balance method was based on the higher order of partial derivative terms, the highest nonlinear terms, and parameters of the intended technique to solve the proposed problem of the modified KdV equation.
The Kadomtsev–Petviashvili (KP) equation is a partial differential equation that describes nonlinear wave motion [8], which is usually written as
that can be applied to mathematical physics as a way to model water waves of long wavelengths. It is a two-dimensional generalization of the one-dimensional KdV (2). Like the KdV equation, the KP equation is completely integrable.
KdV were among the scholars who derived the KdV equation and one of the most famous nonlinear PDEs that arise in a great number of physical situations. It was derived from fluid mechanics to describe shallow water waves in a rectangular channel.
The positive parameter refers to a dispersive effect.
The generalized modified KdV equation is given by the following equation:
, where is a positive parameter. Formulated in the moving frame , the generalized modified KdV equation reads as follows:where denotes the wave speed.
The inverse scattering transform method is a method introduced that yields a solution to the initial value problem for an NLPDE with the help of the solutions to the direct and inverse scattering problems for an associated linear ordinary differential equation. The balance method (BM) is a powerful method for finding exact or approximate solutions to given NLPDEs which was presented by Wang and other scholars in recent years and was improved by Fan with others to make it more straight forward and simple [9]. In addition to this, as El-Wakil and others [10], used the balance method and auto-Backlund transformation. We used balance method for solving nonlinear partial differential equations, the modified Korteweg–de Vries equation. It is well known the KdV equations describe the unidirectional propagation of shallow water waves and a number of generalizations.
Let us consider a general nonlinear PDE, say, in two variables,where is a polynomial function of its arguments and the subscripts denote the partial derivatives. The balance method consists of the following steps:
Step 1. Suppose that the solution of (8) is in the form of the following equation:where , , , and balance coefficients are constants to be determined. By balancing the highest nonlinear terms and the higher order partial derivative terms in this expression; , and can be determined.
Step 2. Substituting (9) into (8) and arranging it at each order of yield an equation as follows:where are differential-algebraic expressions of and . Setting and using a compatible condition yield a set of differential-algebraic equations (DAEs).
Step 3. Solving the set of DAEs, and can be determined. By substituting , and into (9), the exact solutions of (8) can be obtained.
Nonlinear type partial differential equations can be solved by many different methods such as the Hirota method [11, 12]. Inverse scattering transform method and similarity reductions method [13]. Among those methods, the modified Korteweg–de Vries equation was solved by a few of them with their limitations as it is. The limitation of complexities to find the solutions are one core of the problem. This study aimed to find an alternative solution to the modified KdV equation which simplifies the complexity of solving NLPDEs showing the 1 and soliton solutions of the modified KdV equation, by applying the balance method.
A long wave characterizes geophysical fluid dynamics in shallow waters and deep oceans [14, 15]. From another perspective, the KdV equation appeared for the first time in 1895 as a one-dimensional evolution equation describing the waves of an along surface gravity propagation in a water shallow canal [16]. It also appeared in a numeral of diverse physical phenomena such as hydromagnetic collision-free waves, ion acoustic waves, stratified waves interior, lattice dynamics, and physics of plasma [17]. By applying the differential transform technique, the approximate result of coupled KdV has been studied in [5]. One of the most attractive and surprising wave phenomena is the creation of solitary waves or solitons. An adequate theory for solitary waves was developed, in the form of a modified wave equation known as KdV [18–21]. The modified KdV equation has defined a wide variety of physical phenomena used to model the interaction and evolution of nonlinear waves [22–24]. Korteweg and de Vries pursued the work done by Rayleigh and including the effect of surface tension leading to the now famous KdV equation given as follows: , where is the surface elevation, of the wave, above the equilibrium level , is a constant related to the uniform motion of the liquid, with the unit of length, is the gravitational acceleration and , with is surface tension.
The permanent profile of a soliton solution of the KdV equation results from the equilibrium between two effects; the nonlinearity which is proportional to or and Dispersion which is proportional to or . The nonlinearity tends to comprise, to constitute the wave and dispersions spread it out. The dispersive term is proportional to , it decreases whereas, the nonlinearity term, is proportional to , increases leading to a large wave. The balance method has been developed for the analytical solutions of nonlinear partial differential equations. Compared to other methods, the balance method gives high accuracy and has a wide range of applications in many mathematical problems and many physical phenomena. In this paper, we used the balance method in combination with the Hirota bilinear equation method to solve the modified form of the KdV equation considered using the balance method and its respective models.
2. Mathematical Formulation and Main Results
2.1. Preliminaries
2.1.1. Bilinear Operators
Bilinear differential operator mapping a pair of functions . Unlike usual linear differential operators like , which maps a single function into a single function . That is
Remark 1. For all integers ,where is a function of and , whereas is a function of and ; for which .
2.1.2. Fractional Derivative
A fractional derivative is a derivative of any order, real or complex in applied mathematics or mathematical analysis.
2.1.3. The ELzaki Transform
The Elzaki transform is a function of the form with respect to where is the complex number, .
2.2. Main Results
2.2.1. Method of Solution
Considering the general nonlinear PDEs given in (8), let us consider the modified KdV equation that was chosen as an example to illustrate the balance method. Suppose that the solution of (5) is (9). Now considering , balancing it in the (5), it is required that (by letting , , , and , where 1, 2, and 3 represent the order of derivatives for this case or condition). Again balancing and in (9), it gave that , which implies . Choosing , and then solving for the rest of the variables, we obtained that and . That means from the equation , by simplifying it we found . Since we chosen , then we have . Again from the second condition , we have .
Hence (9) , …
Also, this equation can also be expressed as follows:
From equation (15), we get
From (18), we get
Now substituting equations (14), (15), (16), and (20) into (5), we have;
Now equating the coefficients of and to zero on the left-hand side, we have and . This is to mean that; equating , , , , and . Now considering the two equations and , it is possible to solve for . Since and and , we solved for the reset as
Adding these two equations together and solving for , as . From this, we can get
Again considering the equations;
And adding them together using a system of equations as we have . Simplifying it we get, . Substituting into this equation we get, . Which give us , . And hence it implies that
Substituting and into the following equation: , we get , where is any arbitrary constant.
Putting (22) into (5), we have the following results, where, and .where
Simplifying (25) and integrating once with respect to x, we get
Equations (20) and (26) are identical withwhere is an arbitrary function of , and is an arbitrary constant. Especially, taking as zero in equation (27), we get the bilinear equation of (5) as follows:
Equation (28) can be written concisely in terms of -operator aswhere .
Applying Hirota’s method, the bilinear equation of (5) can be written as
Equation (30) is obtained by setting in equation (29). Obviously, equation (30) is a special case of equation (29). Therefore, a more general bilinear equation of the modified KdV equation is obtained by using Hirota’s method.
Let us consider the modified KdV equation by using Hirota’s method.
Substitutingwhich gives the following new equation: . Now integrating both sides with respect to x, we have;where is constant of integration. Likely we can get, without loss of generality, the integration “constant” with respect to , can be observed by a redefinition of that does not change the KdV field, , is the time derivative.
Using the new , we have
Now considering the 1-soliton solution of the modified KdV equation, letwithwhere is the velocity of the wave propagating.
It can be written as , with
In fact, we can integrate the right-hand side of equation (36) once again, using tanh log cosh y. Therefore, equation (34) can be written as follows:
Let
Then, , where
That is, which is the 1-soliton solution of the modified KdV equation, that we use in the following:
Considering equations (31) and (33), the KdV equation in a bilinear form, and rewriting it in a quadratic form, we have the following equation. Inspired by equation (34), let us substitutewhere . Substituting in equation (33), we get
Equation (43) is quadratic in .
Thus, equation (33) for becomes
Multiplying by throughout, we have
This is the bilinear form of the modified KdV equation. Despite the fact that some nontrivial cancellations took place it looks more complicated than the original problem, however, its special form makes it possible to solve using bilinear operators.
Claim 1.
To express equation (45), we use the following expressions:Like the first two terms in (45). all functions of .
Again this looks a bit like the differentiation of a product. But, we have even though . So importantly the operators are not associative since .
In fact, the right-hand side is meaningless, because is a single function, but the outer needs to act on a pair of functions.Again it looks just like a differentiated product of and but with significant changes on the odd terms. .Here, we have to notice that (49) is like , but we have an alternative signs;So, the KdV equation is in a quadratic form in equation (45) and can be recast as is a bilinear form of the KdV equation.(1) Solutions of the Bilinear KdV Equation. For example (1) and considering , we have two cases.
Case 1. For , then , which is a vacuum.
For , then solution. Since equation (51) is a bilinear equation, then this suggests that multisoliton solutions might be the sums of exponents of linear functions of and .
(2). The 1-Soliton Solution. In Hirota form the multisoliton solutions are sums of exponential of linear expressions in and .
For the one-soliton solution trywhere , , and
Theorem 1. For any and , we have = , where .
Proof. By induction.
Let if . Then,Then,and similarly for operator.
Finally, the equation is obviously true for and so by induction must be true for all n and m. In particular, we findUnless Then, the bilinear form of the KdV equation for isSince , we have two opportunities. These two cases or opportunities are as follows:(1). If , then is independent of . So , (which is a boundary condition).(2). If , then and hence , , which is This is the one-soliton solution with .
Figure 1 shows the disturbance of the water surface having kink-shaped traveling waves with amplitude and antibell soliton. The follow of this wave is to the front.
Figure 2 shows the propagation of waves having kink shaped and like some periodic soliton. And follow the wave from back to front.
Figure 3 shows surface waves of long wave length having kink-shaped and bright periodic soliton. And the wave is from left to right.
Figure 4 surface waves of long wave length having kink shaped and bright periodic soliton. And the wave is from left to right. This is the other form of Figure 3, but it is different by the value of , that is why it is broken from the above.
(3) Multisoliton Solutions.
Case 2. For the power series in an auxiliary with parameter With in which the series will terminate at some power of to give exact results rather than an infinite series in . So, we can take finite = 1.
To begin with, we put and collect powers of . We start with 1 as the term as it appeared in the one-soliton solution. DefiningWe want to find solutions such thatSubstituting (59) into (60), we have , with and gathering terms of the same degree and also expanding as for the one soliton case we haveso we have to solve an infinite set of equations order by order in for all n = 1, 2, ….
Equation (63) can be written as(Expression involving in ).
Now we can solve (63) which is equivalent to (64) recursively to determine the coefficients . That is we begin with and solve iteratively to get hoping that it terminates at some point. To do this we need the following:
Theorem 2. For any given , we have; , where
Proof. By considering to interchange primed and unprimed values as a simple label; i.e.,where is any function of and
Thus, (70) becameFor , it reduced toBeginning with we may iterate(repeat) to find all the and note that at any order we are now solving linear partial differential equations. A simple particular solution to the last equation would beUsing (63) for , we have , where this result follows (55). Here is the fact, with , the expansion of (59), terminated at order or (the all vanish trivially). The series has terminated withAll higher-order equations with are solved by for . We can always absorb into the constant and we recognize the single soliton solution. Our hope is to find other solutions where the series terminates in this way, so that the solution is exact. Consider beginning with. We have already seen the single soliton solution. Let’s verify that we recover the two soliton solutions. Take for ,where are as above chosen to satisfy the equation, and . The equation becameSince the equation is linear in it has an obvious solution of the form . So, we just have to determine . Substituting into the equation; that is, letThe solution is then, , which is a two-soliton solution of the KdV equation. Setting , we haveFigure 5 shows follow of the presence of a solitary wave to the front, having two kinks shaped with long amplitude, dark of high speed.
Figure 6 shows the solitary wave to the front, having two kinks shaped, a valley of the wave, and dark high speed.
Figure 7 shows the solitary wave from left to right, having two kink-shaped solitons, a valley-shaped wave, long amplitudes, and dark high speed.
Figure 8 shows follow of the solitary wave to the front, having two kink--shaped soliton profiles, periodic, and dark and bell slow speed.
Finally, we just need to differentiate to get the original or using . It is easier in this format to study the asymptotic of the two soliton solutions. As in the BT discussion, first, we choose a coordinate system with the origin at the center of the around soliton; taking with , where and let , we have Near the soliton, we have (by definition) , so that is small but as follows: (, if , is fixed and , and if , ), (, if , , and if , ). When the clearly reduces to the single soliton case, . When , we have to do a little more work. To lead order we havewhere the ellipsis indicates terms of order . But then Since gives equivalent solutions we might ignore the constant piece. The remaining solution is equivalent to an of the forThis is the original single -soliton solution but with a phase shift ofPrecisely what we found for the 2 KdV soliton solution using the much more complicated BT method. Finally, it was just presented the general solution which might be proved by induction. Define an matrix with elements. Considering the 2-soliton solution for equation (68), and . Then, the -soliton solution , where .
2.3. Illustrative Examples
Example 1. Consider the fractional-order nonlinear KdV system is given as follows:with the initial condition,For , the exact results of the KdV scheme (82) are given byUsing the ELzaki transform to (83), we obtain,Applying the inverse ELzaki transform of (85), we haveNow, by using the suggested analytical method, we getGenerally, the solution result was obtained as follows:The series form result is as follows:For , the exact results of the KdV scheme (81) are given by
Example 2. Consider the fractional-order nonlinear dispersive long wave scheme.and the initial condition;While using ELzaki transform of (91), we obtainedApplying the inverse ELzaki transform of equation (93), we getNow, by using the current analytical method, we getThus, the series form of the result is as follows:For , the exact results of the KdV scheme (91)is as follows:
Example 3. Considering the nonlinear fractional-order new coupled modified KdV system,we getand the initial conditions areUsing the ELzaki transform to (98), we getApplying the inverse ELzaki transform of (101)Now, by using the suggested analytical method, we getThe series form result is given as follows:The exact result of (98) is as follows:
3. Conclusion
Nonlinear partial differential equations can be solved using different methods based on their complexities to determine the exact solutions. In this work, the balance method is applied to solve the modified KdV of the NLPDEs of third order kind analytically. The modified Korteweg–de Vries equation is solved using the balanced method considered with the bilinear and Hirota methods. The obtained soliton solutions of the first and second kinds with certain parameters illustrated their physical models. The obtained results were illustrated graphically to show the geometrical interpretation of models and a few illustrative examples were presented to check the applicability of the method. The obtained solution will serve as a very useful event in the study of nonlinear partial differential equations. This work reveals that the balance method is sufficient, effective, and convenient to simplify the complexities in solving NLPDEs and can be applied appropriately for solving other modified KdV equations of solitary wave type.
Data Availability
The required data are included in the manuscript and cited as references when required.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
The authors are thankful to the referee for their invaluable suggestions and comments which put the article in its present shape.