Analytical solutions of the extended Kadomtsev–Petviashvili equation in nonlinear media

Saad Althobaiti; Ali Althobaiti

doi:10.1515/phys-2023-0106

Article Open Access

Analytical solutions of the extended Kadomtsev–Petviashvili equation in nonlinear media

Saad Althobaiti and Ali Althobaiti

Published/Copyright: August 31, 2023

Published by

Become an author with De Gruyter Brill

Submit Manuscript Author Information Explore this Subject

From the journal Open Physics Volume 21 Issue 1

Abstract

This manuscript attempts to construct diverse exact traveling wave solutions for an important model called the (3+1)-dimensional Kadomtsev–Petviashvili equation. In order to achieve that, the Jacobi elliptic function technique and the Kudryashov technique are chosen in favor of their noticeable efficacy in dealing with nonlinear dynamical models. As expected, the used approaches lead to a variety of traveling wave solutions of different types. Finally, we have graphically illustrated some of the obtained wave solutions to further make sense of their representation. Also, we provide an overview of the main results at the end.

Keywords: traveling waves; exact solutions; Jacobi functions; exponential solution; KP equation

1 Introduction

Real and complex-valued evolutionary equations are a vital type of nonlinear partial differential equations (NPDEs) that appear in nonlinear sciences. The field of fluid dynamics is an example, where several nonlinear evolution models emerge as a result of modeling diverse physical processes of fluid propagation/transmission, traveling/flow, and other situations in various media. At this junction, it will be appropriate to mention some of the well-known nonlinear dynamical equations that play an imperative role in the evolution of fluid flow, including the Korteweg–de-Vries equations [1–4], the Ablowitz–Kaup–Newell–Segur equation [5], the Davey–Stewartson equation [6], and the Kadomtsev–Petviashvili (KP) equation [7–11] to mention a few. In particular, the equation of great concern in this study is the KP equation which was initiated in 1970 and serves as an important model in nonlinear wave theory [12]. Moreover, the model is also applicable for modeling water waves with insignificant surface tension as well as in modeling waves in thin films with significant surface tension [13]. Also, some of the applications of the model are in plasma physics (see [13]).

The literature on nonlinear evolution equations contains numerous mathematical approaches for the complete treatment of their resulting exact solutions, which are commonly referred to as solitary wave solutions [14,15]. Here, let us recall some of these famous analytical approaches such as the extended auxiliary equation method [16,17], the extended simplest equation method [18], the Jacobi elliptic function method [19], the Kudryashov method [20], and others (see [21–31]). On the other hand, some of the notable numerical approaches are used and developed in the treatment of nonlinear evolution equations and nonlinear fractional differential equations, including the homotopy analysis method [32], the Laplace-homotopy perturbation method [33], and the Adomian decomposition method [34] (see [35,36]).

Exact solutions of NPDEs may shed light on our understanding of many nonlinear phenomena. Moreover, they can be used to verify the accuracy and quantify the errors of different numerical, asymptotic, and approximate analytical methods. This article aims at constructing diverse exact traveling wave solutions of the KP equation [9–11], via the application of two promising analytical approaches. The approaches are the Jacobi elliptic function method [19] (also called the modified auxiliary equation method [16,17]) and the Kudryashov technique [20]. The choice of these approaches is motivated by their noticeable efficiency in tackling different classes of NPDEs. Moreover, it is very pertinent to note here that the Jacobi elliptic function technique gives generalized solutions that can be recast to several periodic and hyperbolic function solutions upon playing with the Jacobi elliptic parameter involved. Thus, the novelty of this work is to go beyond what is known in the literature by finding exact solutions in terms of Jacobi functions. In addition, the exact traveling wave solutions to be acquired will be systematically examined with regard to their constrain conditions (when existed). We also plot the three-dimensional plots of certain solutions, besides the description of the shapes of the constructed exact profiles. This study takes the following organization. Section 2 contains some details about the adopted methodologies. Section 3 contains the application of the adopted methodologies on the KP equation; however, Sections 4 and 5 give the discussion of results and the conclusion, respectively.

2 Analytical techniques

This section gives the steps for finding a number of exact solutions via the application of the Jacobi elliptic function technique [19], also called the modified auxiliary equation technique [16,17] and the Kudryashov technique [20].

Therefore, we first consider the following NPDE:

(1) P 1 ( u , u x , u y , u z , u t , u x t , u y t , u z t , u x x , … ) = 0 ,

the steps of the techniques are presented in what follows:

Step 1. We start by assuming that:

(2) u ( x , y , z , t ) = U ( ζ ) , ζ = x + a y + b z + c t ,

where a , b , and c are the constants. Therefore, on using the transformation given in Eq. (3) into Eq. (1), the following nonlinear ordinary differential equation is obtained:

(3) P 2 ( U , U ′ , U ″ , U ‴ … ) = 0 .

Step 2. Jacobi elliptic function technique

The Jacobi elliptic function technique assumes that the solution of Eq. (3) has the form:

(4) U ( ζ ) = ∑ i = − n n μ i ψ i ( ζ ) ,

where μ i ’s are the arbitrary undetermined constants, and n is a positive integer. Also, ψ ( ζ ) is said to be satisfied by the following equation:

(5) ψ ′ 2 ( ζ ) = α 0 + α 1 ψ 2 ( ζ ) + α 2 ψ 4 ( ζ ) ,

where α 0 , α 1 , and α 2 are the constants. Also, Eq. (5) satisfies the following solution cases:

Case 1. When α 0 = 1 , α 1 = − ( 1 + k 2 ) , α 2 = k 2 , then Eq. (5) admits the following solution:

(6) ψ ( ζ ) = sn ( ζ , k ) ,

where sn ( ζ , k ) is the Jacobi function and k represents the elliptic modulus such that 0 < k < 1 .

Case 2. When α 0 = 1 − k 2 , α 1 = 2 k 2 − 1 , and α 2 = − k 2 , then Eq. (5) admits the following solution:

(7) ψ ( ζ ) = cn ( ζ , k ) ,

and cn ( ζ , k ) is the Jacobi function, and k is as stated earlier.

Case 3. When α 0 = k 2 − 1 , α 1 = 2 − k 2 , and α 2 = − 1 , then Eq. (5) admits the following solution:

(8) ψ ( ζ ) = dn ( ζ , k ) ,

where dn ( ζ , k ) is the Jacobi function.

Case 4. When α 0 = k 2 , α 1 = − ( 1 + k 2 ) , and α 2 = 1 , then Eq. (5) admits the following solution:

(9) ψ ( ζ ) = ns ( ζ , k ) ,

where ns ( ζ , k ) is also the Jacobi function.

Case 5. When α 0 = 1 − k 2 , α 1 = 2 − k 2 , and α 2 = 1 , then Eq. (5) admits the solution of the form:

(10) ψ ( ζ ) = cs ( ζ , k ) ,

where cs ( ζ , k ) is the Jacobi function c s .

Case 6. When α 0 = 1 , α 1 = 2 k 2 − 1 , and α 2 = k 2 ( k 2 − 1 ) , then Eq. (5) admits the solution of the form:

(11) ψ ( ζ ) = sd ( ζ , k ) ,

where sd ( ξ , k ) is the Jacobi function s d .

Step 3: Kudryashov technique

The Kudryashov technique expresses the solution of Eq. (3) in the form:

(12) U ( ζ ) = ∑ i = 0 n μ i Φ i ( ζ ) ,

where μ i ’s are non-zero constants, and n is a positive integer. Moreover, Φ ( ζ ) is said to be satisfied by the following equation:

(13) Φ ′ ( ζ ) = Φ 2 ( ζ ) − Φ ( ζ ) ,

In addition, Eq. (13) admits the following solution:

(14) Φ ( ζ ) = 1 h e ζ + 1 ,

where h is an arbitrary constant.

Step 4. The value of n in both Eqs (4) and (12) is carried out by the use of homogeneous balancing principle.

Step 5. Finally, upon substituting Eqs (4) or (12), as the case may be using the Jacobi elliptic function technique or the Kudryashov technique, together with Eq. (5) or Eq. (13) into Eq. (3) and equating all the coefficients of different powers of ψ ( ζ ) or Φ ( ζ ) to zero, we find a system of algebraic equations for μ i . Accordingly, a solution of Eq. (1) is therefore determined.

3 Exact solutions for the extended KP equation

Consider the extended (3+1)-dimensional KP given by [9–11]:

(15) ( u t + 6 u u x + u x x x ) x + β ( u y y + u z z ) = 0 ,

where β is a real constant.

Therefore, upon using the transformation given in Eq. (3) that reads

u ( x , y , z , t ) = U ( ζ ) , ζ = x + a y + b z + c t ,

where a , b , and c are the constants to be evaluated; thereafter, Eq. (15) thus transforms into the following:

(16) c U ″ + 6 ( U U ″ + U ′ 2 ) + U ″ ″ + β ( a 2 U ″ + b 2 U ″ ) = 0 .

Next, upon making use of the homogeneous balancing principle on Eq. (16), we find n = 2 .

3.1 Exact solutions via Jacobi elliptic function technique

Therefore, the solution of Eq. (16) is expressed via the application of the Jacobi elliptic function technique as follows:

(17) U ( ζ ) = μ 0 + μ 1 ψ ( ζ ) + μ 2 ψ 2 ( ζ ) + μ − 1 ψ ( ζ ) + μ − 2 ψ 2 ( ζ ) ,

where ψ ( ζ ) is said to satisfy Eq. (5). Then, on putting Eq. (17) with the use of Eq. (5) into Eq. (16) and thereafter putting the coefficients of ψ ( ζ ) to zero, we obtain a system of algebraic equations. Solving this system for μ 0 , μ 1 , μ 2 , μ − 1 , and μ − 2 , the following solution sets are acquired:

Set 1.

(18) μ 0 = 1 6 ( a 2 ( − β ) − 4 α 1 − b 2 β − c ) , μ 1 = 0 , μ 2 = 0 , μ − 1 = 0 , μ − 2 = − 2 α 0 .

Using the result in Eq. (18), we obtain the following.

Case 1. If α 0 = 1 , α 1 = − ( 1 + k 2 ) , and α 2 = k 2 , then the KP equation in Eq. (15) has a solution in the form:

(19) u ( ζ ) = 1 6 − β ( a 2 + b 2 ) − 12 sn ( ζ , k ) 2 − c + 4 k 2 + 4 .

Therefore, the aforementioned solution becomes the following:

(20) u ( x , y , z , t ) = 1 6 ( − β ( a 2 + b 2 ) − 12 coth 2 ( a y + b z + c t + x ) − c + 8 )

and

(21) u ( x , y , z , t ) = 1 6 ( − β ( a 2 + b 2 ) − 12 csc 2 ( a y + b z + c t + x ) − c + 4 ) ,

when k → 1 and k → 0 , respectively. In addition, the three-dimensional (3D) plots are depicted in Figure 1. The figure describes the solutions determined in Eqs (20) and (21), for graphical visualization.

$Figure 1 3D plots (a) and (b) for solutions reported in Eqs (20) and (21), respectively, when t = z = 0 t=z=0 and a = b = c = β = 1 . a=b=c=\beta =1.$

Figure 1

3D plots (a) and (b) for solutions reported in Eqs (20) and (21), respectively, when t = z = 0 and a = b = c = β = 1 .

Case 2. If α 0 = 1 − k 2 , α 1 = 2 k 2 − 1 , and α 2 = − k 2 , then Eq. (15) admits the following solution:

(22) u ( ζ ) = 1 6 − β ( a 2 + b 2 ) + 12 ( k 2 − 1 ) cn ( ζ , k ) 2 − c − 8 k 2 + 4 .

Moreover, the aforementioned solution transforms into the following:

(23) u ( x , y , z , t ) = 1 6 ( − β ( a 2 + b 2 ) − 12 sec 2 ( a y + b z + c t + x ) − c + 4 ) ,

when k → 0 .

Case 3. If α 0 = k 2 − 1 , α 1 = 2 − k 2 , and α 2 = − 1 , in this case Eq. (15) has a solution in the following form:

(24) u ( ζ ) = 1 6 − β ( a 2 + b 2 ) − 12 ( k 2 − 1 ) dn ( ζ , k ) 2 − c + 4 k 2 − 8 .

Case 4. If α 0 = k 2 , α 1 = − ( 1 + k 2 ) , and α 2 = 1 , then Eq. (15) satisfies the following solution:

(25) u ( ζ ) = 1 6 − β ( a 2 + b 2 ) − 12 k 2 ns ( ζ , k ) 2 − c + 4 k 2 + 4 .

This solution reduces to:

(26) u ( x , y , z , t ) = 1 6 ( − β ( a 2 + b 2 ) − 12 tanh 2 ( a y + b z + c t + x ) − c + 8 ) ,

when k → 1 . Figure 2 represents the solution given in Eq. (26).

$Figure 2 3D plots for solutions reported in Eq. (26) when z = 0 z=0 and a = b = c = β = 1 a=b=c=\beta =1 , at (a) t = 0 t=0 and (b) y = 0 . y=0.$

Figure 2

3D plots for solutions reported in Eq. (26) when z = 0 and a = b = c = β = 1 , at (a) t = 0 and (b) y = 0 .

Case 5. If α 0 = 1 − k 2 , α 1 = 2 − k 2 , and α 2 = 1 , then Eq. (15) has a solution:

(27) u ( ζ ) = 1 6 − β ( a 2 + b 2 ) − 12 ( 1 − k 2 ) cs ( ζ , k ) 2 − c + 4 k 2 − 8 .

This solution reduces to:

(28) u ( x , y , z , t ) = 1 6 ( − β ( a 2 + b 2 ) − 12 tan 2 ( a y + b z + c t + x ) − c − 8 ) ,

when k → 0 .

Case 6. If α 0 = 1 , α 1 = 2 k 2 − 1 , and α 2 = k 2 ( k 2 − 1 ) , then Eq. (15) satisfies the following solution:

(29) u ( ζ ) = 1 6 − β ( a 2 + b 2 ) − 12 sd ( ζ , k ) 2 − c − 8 k 2 + 4 .

Therefore, the aforementioned solution reduces to:

(30) u ( x , y , z , t ) = 1 6 ( − β ( a 2 + b 2 ) − 12 csch 2 ( a y + b z + c t + x ) − c − 4 ) ,

when k → 1 .

Set 2.

(31) μ 0 = 1 6 ( a 2 ( − β ) − 4 α 1 − b 2 β − c ) , μ 1 = 0 , μ − 2 = 0 , μ − 1 = 0 , μ 2 = − 2 α 2 .

Using the result in Eq. (31), we obtain the following

Case 1. If α 0 = 1 , α 1 = − ( 1 + k 2 ) , and α 2 = k 2 , then the KP equation in Eq. (15) has a solution in the form:

(32) u ( ζ ) = 1 6 ( − β ( a 2 + b 2 ) − 12 k 2 sn ( ζ , k ) 2 − c + 4 k 2 + 4 ) .

Case 2. If α 0 = 1 − k 2 , α 1 = 2 k 2 − 1 , and α 2 = − k 2 , then Eq. (15) admits the following solution:

(33) u ( ζ ) = 1 6 ( − β ( a 2 + b 2 ) + 12 k 2 cn ( ζ , k ) 2 − c − 8 k 2 + 4 ) .

Then, we obtain the following explicit solution:

(34) u ( x , y , z , t ) = 1 6 ( − β ( a 2 + b 2 ) + 12 sech 2 ( a y + b z + c t + x ) − c − 4 ) ,

when k → 1 .

Case 3. If α 0 = k 2 − 1 , α 1 = 2 − k 2 , and α 2 = − 1 , then Eq. (15) has a solution in the following form:

(35) u ( ζ ) = 1 6 ( − β ( a 2 + b 2 ) + 12 dn ( ζ , k ) 2 − c + 4 k 2 − 8 ) .

Case 4. If α 0 = k 2 , α 1 = − ( 1 + k 2 ) , and α 2 = 1 , then Eq. (15) satisfies the following solution:

(36) u ( ζ ) = 1 6 ( − β ( a 2 + b 2 ) − 12 ns ( ζ , k ) 2 − c + 4 k 2 + 4 ) .

Case 5. If α 0 = 1 − k 2 , α 1 = 2 − k 2 , and α 2 = 1 , then Eq. (15) has a solution:

(37) u ( ζ ) = 1 6 ( − β ( a 2 + b 2 ) − 12 cs ( ζ , k ) 2 − c + 4 k 2 − 8 ) .

Hence, the aforementioned solution transforms to the following:

(38) u ( x , y , z , t ) = 1 6 ( − β ( a 2 + b 2 ) − 12 cot 2 ( a y + b z + c t + x ) − c − 8 ) ,

when k → 0 .

Case 6. If α 0 = 1 , α 1 = 2 k 2 − 1 , and α 2 = k 2 ( k 2 − 1 ) , then Eq. (15) satisfies the following solution:

(39) u ( ζ ) = 1 6 ( − β ( a 2 + b 2 ) − 12 k 2 ( k 2 − 1 ) sd ( ζ , k ) 2 − c − 8 k 2 + 4 ) .

Set 3.

(40) μ 0 = 1 6 ( a 2 ( − β ) − 4 α 1 − b 2 β − c ) , μ 1 = 0 , μ − 2 = − 2 α 0 , μ − 1 = 0 , μ 2 = − 2 α 2 .

Using the result in Eq. (40), we obtain the following.

Case 1. If α 0 = 1 , α 1 = − ( 1 + k 2 ) , and α 2 = k 2 , then the KP equation in Eq. (15) has a solution in the form:

(41) u ( ζ ) = 1 6 − β ( a 2 + b 2 ) − 12 sn ( ζ , k ) 2 − 12 k 2 sn ( ζ , k ) 2 − c + 4 k 2 + 4 .

Therefore, from the aforementioned transformed solution, the following explicit solution is attained:

(42) u ( x , y , z , t ) = 1 6 ( − β ( a 2 + b 2 ) − 12 coth 2 ( a y + b z + c t + x ) − 12 tanh 2 ( a y + b z + c t + x ) − c + 8 ) ,

when k m → 1 .

Case 2. If α 0 = 1 − k 2 , α 1 = 2 k 2 − 1 , and α 2 = − k 2 , then Eq. (15) admits the following solution:

(43) u ( ζ ) = 1 6 − β ( a 2 + b 2 ) − 12 ( 1 − k 2 ) cn ( ζ , k ) 2 + 12 k 2 cn ( ζ , k ) 2 − c − 8 k 2 + 4 .

Case 3. If α 0 = k 2 − 1 , α 1 = 2 − k 2 , and α 2 = − 1 , then Eq. (15) has a solution in the following form:

(44) u ( ζ ) = 1 6 − β ( a 2 + b 2 ) − 12 ( k 2 − 1 ) dn ( ζ , k ) 2 + 12 dn ( ζ , k ) 2 − c + 4 k 2 − 8 .

Case 4. If α 0 = m 2 , α 1 = − ( 1 + k 2 ) , and α 2 = 1 , then Eq. (15) satisfies the following solution:

(45) u ( ζ ) = 1 6 − β ( a 2 + b 2 ) − 12 k 2 ns ( ζ , k ) 2 − 12 ns ( ζ , k ) 2 − c + 4 k 2 + 4 .

Case 5. If α 0 = 1 − k 2 , α 1 = 2 − k 2 , and α 2 = 1 , then Eq. (15) has a solution:

(46) u ( ζ ) = 1 6 − β ( a 2 + b 2 ) − 12 ( 1 − k 2 ) cs ( ζ , k ) 2 − 12 cs ( ζ , k ) 2 − c + 4 k 2 − 8 .

Finally, the aforementioned transformed solution reduces to the following solution for the governing model:

(47) u ( x , y , z , t ) = 1 6 ( − β ( a 2 + b 2 ) − 12 cot 2 ( a y + b z + c t + x ) − 12 tan 2 ( a y + b z + c t + x ) − c − 8 ) ,

when k → 0 . Figure 3 represents the solutions given in Eq. (47).

$Figure 3 3D plot for solutions reported in Eq. (47) when t = z = 0 t=z=0 and a = b = c = β = 1 . a=b=c=\beta =1.$

Figure 3

3D plot for solutions reported in Eq. (47) when t = z = 0 and a = b = c = β = 1 .

Case 6. If α 0 = 1 , α 1 = 2 k 2 − 1 , and α 2 = k 2 ( k 2 − 1 ) , then Eq. (15) satisfies the following solution:

(48) u ( ζ ) = 1 6 − β ( a 2 + b 2 ) − 12 sd ( ζ , k ) 2 − 12 k 2 ( k 2 − 1 ) sd ( ζ , k ) 2 − c − 8 k 2 + 4 .

3.2 Exact solutions via Kudryashov technique

In the same manner, the KP equation admits the following solution form, via the application of the Kudryashov technique as follows:

(49) U ( ζ ) = μ 0 + μ 1 Φ ( ζ ) + μ 2 Φ 2 ( ζ ) ,

where Φ ( ζ ) is said to satisfy the differential equation earlier mentioned in Eq. (13). Then, on substituting Eq. (49) with the help of Eq. (13) into Eq. (16), and further equating the resulting coefficients of Φ ( ζ ) to zero, we obtain a system of algebraic equations. Furthermore, solving the resultant system for μ 0 , μ 1 , and μ 2 , we obtain the following solution case:

(50) μ 0 = 1 6 ( − β ( a 2 + b 2 ) − c − 1 ) , μ 1 = 2 , μ 2 = − 2 .

Using the aforementioned values into Eq. (49), we obtain

(51) u ( ζ ) = 1 6 ( − β ( a 2 + b 2 ) − c − 1 ) + 2 h e ζ + 1 − 2 ( h e ζ + 1 ) 2 ,

or more explicitly,

(52) u ( x , y , z , t ) = 1 6 ( − β ( a 2 + b 2 ) − c − 1 ) + 2 h e a y + b z + c t + x + 1 − 2 ( h e a y + b z + c t + x + 1 ) 2 ,

where h is an arbitrary constant. Moreover, the aforementioned exponential solution (Eq. (52)) obtained via the application Kudryashov approach is shown in Figure 4 for different values of h .

$Figure 4 3D plots for solutions reported in Eq. (52) when t = z = 0 t=z=0 and a = b = c = β = 1 , a=b=c=\beta =1, at (a) h = 1 h=1 , (b) h = − 1 h=-1 , and (c) h = 10 h=10 .$

Figure 4

3D plots for solutions reported in Eq. (52) when t = z = 0 and a = b = c = β = 1 , at (a) h = 1 , (b) h = − 1 , and (c) h = 10 .

4 Discussion

This study examines the extended KP equation by constructing diverse exact solutions with the use of two analytical approaches. The approaches of interest in this study are the Jacobi elliptic function method and the Kudryashov method, which are chosen owing to their successful applicability in treating classes of both real and complex-valued evolution equations – arising from dissimilar nonlinear processes, and the general nonlinear sciences. As expected, the approaches of choice revealed a variety of traveling wave solutions of different types. Starting with the Jacobi elliptic function technique, a lot of Jacobi elliptic function solutions generalizing related exact traveling wave solutions in the literature are obtained; moreover, these functions/solutions are recast to a series of periodic and hyperbolic structures upon playing with the generalized Jacobi parameter k . In addition, the Kudryashov technique gives only one exact traveling wave solution featuring exponential function; this is, however, known for the Kudryashov technique in disclosing pretty few exact solutions – see the application of the method in refs [15] and [20], among others, where few solutions are similarly disclosed by the approach. Finally, the current study also gives the 3D plots of some of the acquired solutions in Figures 1–4. Figure 1 shows the dark solution as in (a) and the periodic soliton solution as in (b), while Figure 2 represents the bright soliton solution. Figure 3 depicts the shape of periodic solution for Eq. (47). The solution in Eq. (52) is plotted for different values of h .

5 Conclusion

As a concluding note, this study has examined a universal nonlinear wave model called the extended KP equation, through the construction of diverse exact traveling wave solutions. The obtained exact solution was carried out via the application of two promising analytical approaches by the names the Jacobi elliptic function technique (also called the modified auxiliary equation technique) and the Kudryashov technique. The choice of these approaches was motivated by their noticeable efficacy in dealing with diverse nonlinear evolution and Schrodinger equations. As expected, the used approaches revealed different traveling wave solution cases involving a range of Jacobi functions using Jacobi elliptic techniques; however, an exponential traveling wave solution was obtained using the Kudryashov technique. It is important to state here that Jacobi elliptic function solutions reduce to hyperbolic and periodic solutions by varying the parameter k . Finally, we have graphically illustrated some of the obtained solutions to further make sense of their depictions. The used techniques are recommended to solve nonlinear equations.

Acknowledgments

The researchers would like to thank Deanship of Scientific Research, Taif University, for funding this work.

Funding information: This work was funded by the Deanship of Scientific Research, Taif University.
Author contributions: All authors have accepted responsibility for the entire content of this manuscript and approved its submission.
Conflict of interest: The authors state no conflict of interest.

References

[1] Althobaiti A, Althobaiti S, El-Rashidy K, Seadawy AR. Exact solutions for the nonlinear extended KdV equation in a stratified shear flow using modified exponential rational method. Results Phys. 2021;29:104723. 10.1016/j.rinp.2021.104723Search in Google Scholar

[2] Nuruddeen RI. Multiple soliton solutions for the (3+1) conformable space-time fractional modified Korteweg–de-Vries equations. J Ocean Eng Sci. 2018;3:11–8. 10.1016/j.joes.2017.11.004Search in Google Scholar

[3] Seadawy AR. Fractional solitary wave solutions of the nonlinear higher-order extended KdV equation in a stratified shear flow: PartI. Comp Math Appl. 2015;70:345–52. 10.1016/j.camwa.2015.04.015Search in Google Scholar

[4] Seadawy AR, Rehman SU, Younis M, Rizvi ST, Althobaiti A. On solitons: Propagation of shallow water waves for the fifth-order KdV hierarchy integrable equation. Open Phys. 2021;19(1):828–42. 10.1515/phys-2021-0089Search in Google Scholar

[5] Rizvi ST, Seadawy AR, Akram U, Younis M, Althobaiti A. Solitary wave solutions along with Painleve analysis for the Ablowitz–Kaup–Newell–Segur water waves equation. Modern Phys Lett B. 2022;36(2):2150548. 10.1142/S0217984921505485Search in Google Scholar

[6] Akram U, Seadawy AR, Rizvi ST, Younis M, Althobaiti A. Some new dispersive dromions and integrability analysis for the Davey–Stewartson (DS-II) model in fluid dynamics. Modern Phys Lett B. 2022;36(2):2150539. 10.1142/S0217984921505394Search in Google Scholar

[7] Khalfallah M. New exact traveling wave solutions of the (3+1) dimensional Kadomtsev–Petviashvili (KP) equation. Commun Nonlinear Sci Numer Simul. 2009;14(4):1169–75. 10.1016/j.cnsns.2007.11.010Search in Google Scholar

[8] Althobaiti A. Travelling waves solutions of the KP equation in weakly dispersive media. Open Phys. 2022;20(1):715–23. 10.1515/phys-2022-0053Search in Google Scholar

[9] Sinelshchikov DI. Comment on: new exact traveling wave solutions of the (3. 1)-dimensional Kadomtsev–Petviashvili (KP) equation. Commun Nonlinear Sci Numer Simul. 2010;15(11):3235–6. 10.1016/j.cnsns.2009.11.028Search in Google Scholar

[10] Ma YL, Li BQ. Mixed lump and soliton solutions for a generalized (3+1)-dimensional Kadomtsev–Petviashvili equation. AIMS Math. 2020;5(2):1162–76. 10.3934/math.2020080Search in Google Scholar

[11] Bulut H, Aksan EN, Kayhan M, Sulaıman TA. New solitary wave structures to the (3+1) dimensional Kadomtsev–Petviashvili and Schrödinger equation. J Ocean Eng Sci. 2019;4(4):373–8. 10.1016/j.joes.2019.06.002Search in Google Scholar

[12] Kadomtsev BB, Petviashvili VI. On the stability of solitary waves in weakly dispersive media. Sov Phys Dokl. 1970;15:539–41. Search in Google Scholar

[13] Ablowitz MJ, Segur H. On the evolution of packets of water waves. J Fluid Mech. 1979;92(4):691–715. 10.1017/S0022112079000835Search in Google Scholar

[14] Hereman W. Shallow water waves and solitary waves, Encyclopedia of complexity and systems. Heibelberg, Germany: Springer Verlag; 2009. 10.1007/978-0-387-30440-3_480Search in Google Scholar

[15] Khalid KA, Nuruddeen RI, Adel RH. New exact solitary wave solutions for the extended (3+1)-dimensional Jimbo-Miwa equations. Results Phys. 2018;9:12–6. 10.1016/j.rinp.2018.01.073Search in Google Scholar

[16] Alotaibi T, Althobaiti A. Exact wave solutions of the nonlinear Rosenau equation using an analytical method. Open Phys. 2021;19(1):889–96. 10.1515/phys-2021-0103Search in Google Scholar

[17] Alotaibi T, Althobaiti A. Exact solutions of the nonlinear modified Benjamin-Bona-Mahony equation by an analytical method. Fractal Fractional. 2022;6(7):399. 10.3390/fractalfract6070399Search in Google Scholar

[18] Rabie WB, Ahmed HM, Seadawy AR, Althobaiti A. The higher-order nonlinear Schrödinger’s dynamical equation with fourth-order dispersion and cubic-quintic nonlinearity via dispersive analytical soliton wave solutions. Opt Quantum Electron. 2021;53(12):1–25. 10.1007/s11082-021-03278-zSearch in Google Scholar

[19] Gepreel KA, Nofal TA, Althobaiti AA. The modified rational Jacobi elliptic functions method for nonlinear differential difference equations. J Appl Math. 2012;2012:427479. 10.1155/2012/427479Search in Google Scholar

[20] Nuruddeen RI, Nass AM. Exact solitary wave solution for the fractional and classical GEW-Burgers equations: an application of Kudryashov method. Taibah Uni J Sci. 2018;12:309–14. 10.1080/16583655.2018.1469283Search in Google Scholar

[21] Gurefe Y, Pandir Y, Akturk T. On the nonlinear mathematical model representing the Coriolis effect. Math Problems Eng. 2022;2022:2504907. 10.1155/2022/2504907Search in Google Scholar

[22] Gepreel KA, Althobaiti AA. Exact solutions of nonlinear partial fractional differential equations using fractional sub-equations method. Indian J Phys. 2014;88(3):293–300. 10.1007/s12648-013-0407-0Search in Google Scholar

[23] Kaewta S, Sirisubtaweee S, Koonprasert S, Sungnul S. Applications of the G-š/G2-Expansion Method for Solving Certain Nonlinear Conformable Evolution Equations. Fractal Fract. 2021;5:88. 10.3390/fractalfract5030088Search in Google Scholar

[24] Ashraf R, Rashid S, Jarad F, Althobaiti A. Numerical solutions of fuzzy equal width models via generalized fuzzy fractional derivative operators. AIMS Math. 2022 Jan 1;7(2):2695–728. 10.3934/math.2022152Search in Google Scholar

[25] Seadawy AR, Rizvi ST, Ali I, Younis M, Ali K, Makhlouf MM, Althobaiti A. Conservation laws, optical molecules, modulation instability and Painlevé analysis for the Chen-Lee-Liu model. Opt Quantum Electron. 2021 Apr;53(4):1–5. 10.1007/s11082-021-02823-0Search in Google Scholar

[26] Raddadi MH, Younis M, Seadawy AR, Rehman SU, Bilal M, Rizvi ST, et al. Dynamical behaviour of shallow water waves and solitary wave solutions of the Dullin-Gottwald-Holm dynamical system. J King Saud Univ-Sci. 2021;33(8):101627. 10.1016/j.jksus.2021.101627Search in Google Scholar

[27] Ahmed S, Ashraf R, Seadawy AR, Rizvi ST, Younis M, Althobaiti A, et al. Lump, multi-wave, kinky breathers, interactional solutions and stability analysis for general (2+1)-rth dispersionless Dym equation. Results Phys. 2021;25:104160. 10.1016/j.rinp.2021.104160Search in Google Scholar

[28] Seadawy AR, Younis M, Althobaiti A. Various forms of M-shaped rational, periodic cross kink waves and breathers for Bose-Einstien condensate model. Opt Quantum Electron. 2022;54(3):1–7. 10.1007/s11082-021-03498-3Search in Google Scholar

[29] Rizvi ST, Seadawy AR, Hanif M, Younis M, Ali K, Althobaiti A. Investigation of chirp-free dromions to higher-order nonlinear Schrödinger equation with non-Kerr terms. Int J Modern Phys B. 2022;36(5):2250043. 10.1142/S0217979222500436Search in Google Scholar

[30] Rehman HU, Seadawy AR, Younis M, Rizvi ST, Anwar I, Baber MZ, et al. Weakly nonlinear electron-acoustic waves in the fluid ions propagated via a (3+1)-dimensional generalized Korteweg–de-Vries-Zakharov-Kuznetsov equation in plasma physics. Results Phys. 2022;33:105069. 10.1016/j.rinp.2021.105069Search in Google Scholar

[31] Almatrafi MB, Alharbi AR, Seadawy AR. Structure of analytical and numerical wave solutions for the Ito integro-differential equation arising in shallow water waves. J King Saud Univ-Sci. 2021;33(3):101375. 10.1016/j.jksus.2021.101375Search in Google Scholar

[32] Gepreel KA, Nofal TA, Althobaiti AA. Numerical solutions of the nonlinear partial fractional Zakharov-Kuznetsov equations with time and space fractional. Sci Res Essay. 2014;9(11):471–82. 10.5897/SRE2013.5769Search in Google Scholar

[33] Shakhanda R, Goswami P, He J-H, Althobaiti A. An approximate solution of the time-fractional two-mode coupled Burgers equations. Fract Fract. 2021;5(4):196. 10.3390/fractalfract5040196Search in Google Scholar

[34] Banaja M, AlQarni AA, Bakodah HO, Biswas A. Bright and dark solitons in cascaded system by improved Adomian decomposition scheme. Optik. 2016;130:1107–14. 10.1016/j.ijleo.2016.11.125Search in Google Scholar

[35] Agarwal P, Nieto JJ. Some fractional integral formulas for the Mittag-Leffler type function with four parameters. Open Math. 2015 Sep;13(1):000010151520150051. 10.1515/math-2015-0051Search in Google Scholar

[36] Agarwal P, Choi J, Paris RB. Extended Riemann-Liouville fractional derivative operator and its applications. J Nonlinear Sci Appl. 2015;8(5):451–66. 10.22436/jnsa.008.05.01Search in Google Scholar

Received: 2023-04-14

Revised: 2023-07-03

Accepted: 2023-08-04

Published Online: 2023-08-31

This work is licensed under the Creative Commons Attribution 4.0 International License.

Articles in the same Issue

https://doi.org/10.1515/phys-2023-0106

Keywords for this article

traveling waves; exact solutions; Jacobi functions; exponential solution; KP equation

Creative Commons

BY 4.0