Soliton Solutions to the BA Model and (3 + 1)-Dimensional KP Equation Using Advanced <span class="nowrap"><svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-4.5268pt" id="M1" height="17.3386pt" version="1.1" viewBox="-0.0657574 -12.8118 76.0048 17.3386" width="76.0048pt"><g transform="matrix(.017,0,0,-0.017,0,0)"><path id="g190-102" d="M380 106C343 72 306 56 265 56C195 56 116 112 115 248C235 252 361 262 377 265C396 269 400 277 400 297C400 374 333 449 250 449H249C198 449 144 421 103 376S37 269 37 201C37 88 109 -12 232 -12C263 -12 332 6 395 84L380 106ZM225 412C281 412 315 364 314 312C314 297 308 292 290 292C232 290 176 289 120 289C135 370 180 412 225 412Z"/></g><g transform="matrix(.017,0,0,-0.017,7.293,0)"><path id="g190-121" d="M474 0V26C414 34 401 43 364 100L267 248C300 297 324 332 345 358C381 400 394 405 455 411V437H272V411C316 406 323 401 305 370C287 337 267 306 247 276L188 369C169 397 173 405 215 411V437H16V411C71 404 83 396 114 348L201 212C171 167 144 127 116 92C77 42 66 34 4 26V0H190V26C139 34 136 43 156 77C175 113 198 150 220 183L294 66C311 39 302 31 260 26V0H474Z"/></g><g transform="matrix(.017,0,0,-0.017,15.393,0)"><path id="g190-113" d="M169 380V459C122 440 66 423 24 416V392C86 384 90 382 90 317V-135C90 -201 81 -207 17 -213V-240H253V-213C176 -207 169 -201 169 -125V6C182 -1 208 -11 238 -12C368 12 487 109 487 260C487 358 421 449 310 449C298 449 279 444 261 433L169 380ZM169 346C196 367 237 389 269 389C341 389 403 329 403 221C403 109 347 37 263 37C228 37 191 53 169 76V346Z"/></g><g transform="matrix(.017,0,0,-0.017,24.475,0)"><path id="g113-41" d="M300 -147C201 -63 143 98 143 270S200 602 300 686L282 710C136 610 70 450 70 271V270C70 89 136 -72 282 -170L300 -147Z"/></g><g transform="matrix(.017,0,0,-0.017,30.413,0)"><path id="g117-33" d="M535 230V280H52V230H535Z"/></g><g transform="matrix(.017,0,0,-0.017,40.486,0)"><path id="g113-243" d="M542 242C542 369 469 429 379 468C370 472 372 496 376 513L416 681L400 691L344 648L219 51C167 69 113 128 113 217C113 324 166 387 266 428L252 459C124 418 23 336 23 202C23 95 90 13 205 -8L182 -132C165 -222 191 -254 209 -261L215 -258L270 -7C415 -4 542 98 542 242ZM469 232C469 126 403 33 279 39L355 409C399 394 469 331 469 232Z"/></g><g transform="matrix(.017,0,0,-0.017,50.184,0)"><use xlink:href="#g113-41"/></g><g transform="matrix(.017,0,0,-0.017,56.121,0)"><path id="g113-236" d="M425 682C425 693 422 699 407 699C369 699 308 677 260 646C213 651 191 674 191 701C191 715 195 726 200 734L183 740C153 730 118 709 118 681C118 644 163 619 218 615C168 582 115 534 115 463C115 418 147 382 180 369C102 329 23 261 23 163C23 74 74 38 158 14C183 7 201 2 221 -2C279 -15 290 -33 290 -58C290 -94 251 -145 230 -169L252 -189C280 -171 367 -103 367 -25C367 20 333 51 266 68C245 73 218 80 191 89C139 106 107 135 107 192C107 261 161 329 221 355C248 350 281 347 314 347C330 347 370 388 370 406C370 418 366 422 352 422C321 422 280 413 255 407C218 413 192 448 192 493C192 544 229 588 266 610C309 610 354 615 376 623C397 631 425 651 425 682Z"/></g><g transform="matrix(.017,0,0,-0.017,63.81,0)"><path id="g113-42" d="M275 270C275 450 212 609 64 710L45 686C145 604 203 442 203 270S147 -63 45 -147L64 -170C213 -68 275 89 275 270Z"/></g><g transform="matrix(.017,0,0,-0.017,69.747,0)"><use xlink:href="#g113-42"/></g></svg>-</span>Expansion Scheme in Mathematical Physics

Mawa, HZ; Islam, S. M. Rayhanul; Bashar, Md. Habibul; Roshid, Md. Mamunur; Islam, Jahedul; Akhter, Sadia

doi:https://doi.org/10.1155/2023/5564509

Mathematical Problems in Engineering

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Research Article | Open Access

Volume 2023 | Article ID 5564509 | https://doi.org/10.1155/2023/5564509

Soliton Solutions to the BA Model and (3 + 1)-Dimensional KP Equation Using Advanced -Expansion Scheme in Mathematical Physics

HZ Mawa,¹S. M. Rayhanul Islam,²Md. Habibul Bashar,³Md. Mamunur Roshid,⁴Jahedul Islam,¹and Sadia Akhter⁵

Academic Editor: Zhao Li

Received06 Oct 2023

Revised24 Oct 2023

Accepted28 Oct 2023

Published21 Nov 2023

Abstract

In this manuscript, the primary motivation is the implementation of the advanced -expansion method to construct the soliton solution, which contains some controlling parameters of two distinct equations via the Biswas–Arshed model and the (3 + 1)-dimensional Kadomtsev–Petviashvili equation. Here, the solutions’ behaviors are presented graphically under some conditions on those parameters. The height of the wave, wave direction, and angle of the obtained wave is formed by substituting the particular values of the considerations over showing figures with the control plot. With the collaboration of the advanced -expansion method, we construct entirely the solitary wave results as well as rogue type soliton, combined singular soliton, kink, singular kink, bright and dark soliton, periodic shape, double periodic shape soliton, etc. Therefore, it is remarkable to perceive that the advanced -expansion technique is a simple, viable, and numerical solid apparatus for clarifying careful outcomes to the other nonstraight equivalences.

1. Introduction

Nonlinear partial differential conditions (NLPDEs) are a critical subject and have spread broadly all over the planet in a wide range of dynamic designs. Numerous mathematicians and physicists are dissecting dynamic designs. Electrical conduction, plasma physics, mathematical natural sciences, fluid mechanics, optical fiber, solid-state physics, shallow water wave propagation, mathematical dynamics, and many other fields use dynamic structures as significant components of nonlinear physical simulations [1–10]. Recently, many experts looked into the optical soliton solutions of the NLPDEs. These solutions are essential for seeing how integrated physical phenomena work inside. For obtaining optical solutions for NLPDEs, numerous significant strategies have been proposed, including the modified polynomial expansion technique [11], the enhanced -expansion approach [12], the -expansion technique [13], the generalized Kudryashov approach [14], the new auxiliary equation technique [15], the lie symmetry approach [16], the extended Fan subequation technique [17], the complex technique [18], the improved Bernoulli subequation approach [19], and so on.

Moreover, strength and bifurcation examination plays a significant role in figuring out how a nonlinear robust framework behaves. These days, numerous researchers have concentrated on the bifurcation examination of nonlinear differential conditions [20–29] to acquire significant knowledge of how nonlinear models behave and steadiness.

We have considered the two NLPDEs via the Biswas and Arshed (BA) and (3 + 1)-dimensional Kadomtsev–Petviashvili (KP) model in this manuscript. Many scholars have studied the BA and the (3 + 1)-dimensional KP models in the last few decades and found many optical solutions. In its continuity, using two distinct schemes in [30, 31], they found the exact soliton solutions and the singular and dark solitons of the BA model with Kerr and power law in nonlinearity. The newly -model expansion technique was applied to the BA model and attained the optical soliton solutions, representing the dark, bright, singular, rational, and periodic wave profile in [32]. In addition, it has been observed that some scholars have found the optical solution of the BA model using the trial solution technique [33], the modified simple equation approach [34], the mapping technique [35], and the extended trial function approach [36], which are dark, bright, singular, and periodic type wave profiles.

On the other hand, the (3 + 1)-dimensional KP model was first introduced in 1970 by Soviet physicists Kadomtsev and Petviashvili [37] which narrates the evolution of semi-one-dimensional shallow water waves while the effect of surface tension and viscosity is negligible. After that, many authors have studied the different forms of the KP model [38, 39]. Recently, one soliton and one resonant soliton solution have been found from the (3 + 1)-dimensional KP model using consistent tanh expansion [40]. Using the Bilinear method in [41], the KP model has explored the multiple lump solutions via 1-lump wave, 3-lump wave, 6-lump wave, and 8-lump waves. In addition, the simplified homogeneous balance method has been applied to the KP model and found the one single soliton and one double soliton solution in [42]. The Hirota bilinear transformation has been applied to the KP equation and obtained the one and two rough wave solutions in [43].

The purpose of the manuscript is to apply the advanced -expansion approach [44, 45] to the BA model and the (3 + 1)-dimensional KP model, and to find some optical soliton solutions, namely w-shape, kink shape, periodic soliton solution shape, double periodic shape, dark soliton shape, combined singular soliton, and rogue wave profiles. Based on the above discussion in the previous literature, we can say that some wave profiles of the BA and (3 + 1)-dimensional KP models are new. Finally, it can perfect water rollers of extended wavelength with softly nonlinear repairing forces and regularity distribution. It can also be used to model waves in ferromagnetic media, nonlinear optics, optical fiber, and plasma physics.

The novelty of this paper is that, interestingly, the advanced -extension approach is utilized for our concerned models concerning my insight. The impact of various norms of wave number on the got arrangements is likewise made sense of graphically. The acquired outcomes are helpful for the ultrashort light heartbeats in optical filaments. Our review model has much significance in quantum optics and liquid mechanics for making sense of the optical qualities of the femtosecond lasers and femtochemistry objects. Optical soliton annoyance is the foundation of the broadcast communications industry. This industry stays in business due to the wonder of soliton transmission innovation.

We have divided this article into follows: the literature review, objectives, and background are discussed in Section 1. We talked about the description of the tactic in Section 2. The governing equation is represented in Section 3. Section 4 applied the proposed method to the (3 + 1)-dimensional KP and BA model. Graphical and physical explanations have been discussed in Section 5. Finally, the conclusion is given in Section 6.

2. The Advanced -Expansion Method

Section 2 consists of the summary of the advance -expansion method [44, 45]. We consider the NLPDEs, which is of the formwhere is the wave function to be determined, R is a polynomial of , and its partial derivatives. Step-1. First, we take a conversion variable to change all independent variables into a single variable, such as

The wave variable mentioned in Equation (2) turn the NLPDE Equation (1) into an ODE as follows: Step-2. According to the advanced -expanssion method, the exact solution of Equation (3) is assumed to bewhere are constants to be determinted. The derivative of satisfies the ODE in the succeeding system

Then, the obtained results of ODE Equation (5) are of the hyperbolic, trigonometric, and the following forms: Case I: hyperbolic function solution (when ):and Case II: trigonometric function solution (when ):and Case III: when and Case IV: when and where is assimilating constants and or depends on sign of . Step-3. It is concerning the transformation of Equation (4) into Equation (3) and by combining all of the similar orders of with the Equation (5). We obtain a polynomial form of . A collection of algebraic systems can be obtained by equating every coefficient of this polynomial to zero. Step-4. Take up the approximation of the constants can be changed by measuring the mathematical terms come to be in Step 4. Replacing the approximations of the constants organized with the preparations of Equation (5), we will get new and extensive exact voyaging wave courses of action of the nonlinear advancement of Equation (1).

3. Governing Model

3.1. The BA Model

Recently, Biswas and Arshed [46] proposed a model with Kerr law nonlinearity, namely BA model is given as follows:

In Equation (12), the dependent variable q(x, t) signifies the wave velocity that depends upon spatial (x) and temporal (t) variables. The first term portrays temporal evolution. and stand for the coefficient of GVD and spatiotemporal dispersion (STD); and represent third-order STD and third-order dispersion; is the effect of self-steepening, and are the effect of dispersions.

To start integration process, letwhere , and are denoted by the amplitude portion of the wave, soliton speed, phase component, frequency, wave number, and phase constant, respectively. Next, put Equation (13) into Equation (12), the real part of Equation (12) has the following form:and the imaginary part becomes

3.2. The (3 + 1)-Dimensional KP Equation

Let us take into account the (3 + 1)-dimensional KP equation is in the following form:

The dependent variable represents the wave velocity.

Using traveling wave variable to reduce the Equation (16) becomes

Equation (17) is an assimilated equation. Then assimilate two times with the help of and we pursue the assimilating constant to zero. Then, we obtainwhere

4. Applications

4.1. For BA Model

In this segment, we applied the advanced -expansion method for Equations (14) and (15). Balancing the nonlinear terms and highest order derivative terms, we obtain the balance number for Equations (14) and (15). So, the solution of the Equations (14) and (15) takes the following form:

Differentiating the Equation (19) with respect to and putting the values of in Equations (14) and (15), and equating the coefficient of equal to zero.

Solving those systems of equivalences, we obtain the results for real part that is Equation (14) are as follows: Set-1: Case-I: we get the following hyperbolic solutions for yields Family-1:where , , and . Case-II: we get the following trigonometric solutions for , yields Family-2:where , , and . Case-III and Case IV: the values of and are not specified when . As a result, the outcome cannot be determined. This case is, therefore, dismissed. Essentially, when B = 0, the executing worth of , and are undefined. So, they cannot be determined. So, this case was also discarded.

Again, we obtain the solutions for imaginary part that is Equation (15) we get following set Set-2: Case-I: we get the following hyperbolic solutions when Family-3:where

and Case-II: we get following trigonometric solution when Family-4:where and Case-III: when the calculated value of are undefined. So, the result cannot be determined. For this reason, this case is discarded. Case-IV: when and Family-5:where and

4.2. For (3 + 1)-Dimensional KP Equation

In this segment, we apply the advance -expansion approach for Equation (18) and since here the nonlinear term is and the highest order derivative is . So, the balance number is . So, the solution of the Equation (18) takes Equation (19) and differentiates Equation (19) w. r. t. ξ and putting the values of and in Equation (18) and equating the coefficient of equal to zero. Solving those systems of equations, we obtain the solutions for Equation (18), which are Set-1: Set-2: Case-I: we get following hyperbolic solutions when Family-6:where and . Family-7:where and . Case-II: we get following trigonometric solutions when Family-8:where and . Family-9:where and . Case-III and Case IV: the values of are not specified when . As a result, the outcome cannot be determined. This case is therefore dismissed. Essentially, when the executing worth of are undefined. So, the result cannot be determined. For this reason, this case is discarded.

5. Physical and Graphical Explanations

This section will discuss the physical interpretation and graphical presentation of the (3 + 1)-dimensional KP and BA models that obtained exact and single-wave results. The precise traveling wave solutions for the (3 + 1)-dimensional KP equation and BA models can be obtained by utilizing the advanced -expansion method. The arrangements are all hyperbolic function arrangements. The arrangements , and are all trigonometric function results, and the rational function arrangements being .

According to the condition , the soliton solution represents the w-shape wave profile for selecting the free parameters within the displacement . The 3D plot with density plot wave features of the solution depicted in Figure 1(a)–1(c) for the value of , respectively. It can be seen that the wave propagates along the x- and t-axes. Figure 1(d) represents the 2D line plot of the within displacement . We need to observe the concave up and concave down of our desired sketch for the inflection point. By the observation, we find that at that point, the sketch shows the concave up to the concave down by the definition of inflection point we define that point.

(a)

(b)

(c)

(d)

According to the condition , imaginary form the of solution which represents kink-shape with within the displacements . Figure 2(a)–2(c) represents a 3D plot with density plot for the value of , respectively. Figure 2(d) indicates the 2D line plot of the within displacement .

(a)

(b)

(c)

(d)

In the same way the solution is a normal form and the sketch indicates in normal system, which represents in Figure 3. It indications the periodic soliton solution-shape type exact traveling wave solution within the displacements and . Figure 3(a)–3(c) represents 3D plot with density plot for the value of , respectively. Figure 3(d) shows the 2D line plot of the within displacement .

(a)

(b)

(c)

(d)

The solution is a complex form and the figure represents an imaginary form which represents in Figure 4. It spectacles the singular kink-shape type exact traveling wave solution with within the displacements Figure 4(a)–4(c) represents 3D plot with density plot for the value of , respectively. Figure 4(d) shows the 2D line plot of the within displacement .

(a)

(b)

(c)

(d)

And the solution is a complex form and the figure indicates in absolute system which represents in Figure 5. It shows the dark soliton-shape kind exact traveling wave solution with within the displacements . Figure 5(a)–5(c) represents 3D plot with density plot for the value of , respectively. Figure 5(d) indicates the 2D line plot of the within displacement .

(a)

(b)

(c)

(d)

Again the solution is a complex form and the figure in imaginary form which represents in Figure 6. Its expressions the double periodic-shape kind exact traveling wave solution with within the displacements and . Figure 6(a)–6(c) represents 3D plot with density plot for the value of respectively. Figure 6(d) indicates the 2D line plot of the within displacement .

(a)

(b)

(c)

(d)

Also the solution is a complex form and sketch of absolute form and complex form represented in Figures 7 and 8. Figure 7 represents the combined singular soliton—shape. Figure 8 represents rouge kind shape with within the displacements and also represents 2D line plot within displacement .

(a)

(b)

(c)

(d)

(a)

(b)

(c)

(d)

6. Conclusion

This work explores the advanced -expansion method successfully, and the significant shape solution is constructed with the controlling parameters. These solutions are elaborated systematically and graphically with 3D and 2D plots. Finally, it is found that the advanced -expansion method to BA model and KP equation and such typical solutions might be beneficial to analyze and characterize many nonlinear phenomena in nonlinear optic, quantum field theory, solid state physics, and order to explain some intricate nonlinear physical phenomena, this method provides solutions with free parameters. This paper’s solutions demonstrate that the approach is highly effective and adaptable.

Data Availability

No underlying data were collected or produced in this study.

Conflicts of Interest

The authors declare no conflict of interest.

Authors’ Contributions

This paper was written with equal and significant contributions from all authors. All writers read and supported the last original copy.

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Copyright © 2023 HZ Mawa et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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