Advances in Mathematical Physics
Volume 2019 (2019), Article ID 9801638, 7 pages
https://doi.org/10.1155/2019/9801638

Research Article

New Exact Solutions of Nonlinear (3 + 1)-Dimensional Boiti-Leon-Manna-Pempinelli Equation

Mohamed R. Ali¹ and Wen-Xiu Ma2,3,4,5,6

¹Department of Mathematics, Benha Faculty of Engineering, Benha University, Benha, Egypt
²Department of Mathematics, Zhejiang Normal University, Jinhua, Zhejiang 321004, China
³Department of Mathematics, King Abdulaziz University, Jeddah, Saudi Arabia
⁴Department of Mathematics and Statistics, University of South Florida, Tampa, FL 33620, USA
⁵College of Mathematics and Systems Science, Shandong University of Science and Technology, Qingdao, Shandong 266590, China
⁶Department of Mathematical Sciences, International Institute for Symmetry Analysis and Mathematical Modelling, North-West University, Mafikeng Campus, Mmabatho 2735, South Africa

Correspondence should be addressed to Mohamed R. Ali; mohamed.reda@bhit.bu.edu.eg and Wen-Xiu Ma; wma3@usf.edu

Received 16 February 2019; Accepted 6 May 2019; Published 3 June 2019

Academic Editor: Andrei D. Mironov

Copyright © 2019 Mohamed R. Ali and Wen-Xiu Ma. 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.

Abstract.

Based on Hirota’s bilinear structure, we evolute a new protuberance type arrangement of the (3+1)-dimensional Boiti-Boiti-Leon-Manna-Pempinelli equation, which depicts nonlinear wave spreads in incompressible fluid. New lump arrangement is built by applying the bilinear strategy and picking appropriate polynomial. Under various parameter settings, this lump arrangement has three sorts of numerous irregularity waves, blended arrangements including lump waves and solitons are additionally developed. Association practices are seen between lump soliton and soliton. Research demonstrates that soliton can somewhat swallow or release lump waves. The shape and highlights for these subsequent arrangements are portrayed by exploiting the three-dimensional plots and comparing shape plots by picking suitable parameters. The physical significance of these charts is given.

1. Introduction

Numerous analysts in the ongoing years considered numerous kinds of advancement equations portraying distinctive cases in liquid and plasma fields. A wide range of techniques are utilized to examine development equations in (3+1) measurements, for example, Hirota’s strategy [1, 2] exponential function method [3, 4], tanh-coth methods and sine-cosine [5, 6], and numerous different techniques. One of the notable equations is the (3+1)-dimensional Boiti-Leon-Manna-Pempinelli (BLMP) equation which depicts the liquid engendering and can be considered as a model for incompressible liquid [7–12]. This equation composes as;

This equation presented by Darvishi et al. [7] as an augmentation of the (2+1)-dimensional equation depicts the (2+1)-dimensional connection of the Riemann wave along the y-axis with a long wave proliferated along the x-axis. BLMP equation concentrated by alternate points of view, for example, the utilization of the exponential function method [7, 8]. The precise three-wave arrangements and various soliton arrangements were found by applying the Hirota bilinear technique [9–12]. Lump waves, as exceptional nonlinear wave wonders, have been seen in numerous fields [13–18]. It is huge to most likely find and anticipate protuberance-like waves in applications [19–21]. As of late, exploration on the lump arrangements has pulled in an ever-increasing number of considerations [22–29]. Therefore, theoretically, researches on lump waves are helpful to better understand and predict possible extremes for nonlinear evolution systems. Bilinear technique is a powerful representative calculation to develop numerous solitons. Very as of late, Ma et al. stretched out this technique to look for lump arrangements, where its basic thought is to pick appropriate polynomial capacities in bilinear frames [30–38]. We are roused to investigate new dynamical properties through examining irregularity answer for nonlinear frameworks. Here, we examine some of irregularity soliton arrangements, their elements, and the vulnerability of communication with different sorts of arrangements utilizing Hirota strategy for (1). By utilizing the complex strategy with two-term truncated arrangement, gathering the coefficients of the eigenfunction and comparing them to zero, we infer the equivalent ansatz in [39, 40]:

That is called Cole-Hopf change, where is an assistant or test work that will be assumed later. Beginning by substituting from (2) in (1),

The change expands the nonlinearity; however, it approves us to expect the test work. In [13], Zhang utilized the Hirota bilinearity with Bell polynomials hypotheses to create some lump soliton, lump kink arrangements, and irregularity with one-stripe soliton and with two-stripe lump solitons for (1).

2. Lump Soliton Solutions for BLMP Equation

To create lump arrangement, we deem that

where , , are genuine obscure that will be resolved consequently. By direct substitution from (4) in (3) and gathering the coefficients of polynomials in , and , we acquire a nonlinear algebraic system in ; by solving those equations with aid of Maple, we get some sets of solutions as follows:

So,

Utilizing (2), the solution of (1) has the form

Substituting (5) and (6) in (7) forms lump-kink solution as shown in Figure 1 with , and .

Figure 1: Top: 3D plots for (7) at (a) , (b) at , and (c) at . Nethermost: contour plots for (7) at (d) , (e) at , and (f) at

3. Interaction Solutions

3.1. Lump Soliton with One-Stripe Wave

Assume that the test work is a confederation of quadratic function with exponential function as follows:

where and , are r genuine obscure constants that will be resolved later on. Utilizing the ansatz in (2),

Complicated algebraic system is driven by substituting from (8) in (3) and collecting the coefficients of polynomials in , and . We solve the constructed system utilizing Maple and snaffle the following assortment of solutions:

To provide the singularity and promote the wave to localize in all directions, the following stipulation must be possessed in consideration:

Substituting from (10) in (8),

From (12) and (9), we produce lump arrangement with stripe (solitary wave) arrangement. By taking an estimations of arbitrary constants as , and , we plot the outcomes in Figure 2 for various estimations of .

Figure 2: Upper plots: 3D plots for (9) at (a) , at (b) , and at (c) and . Contour plots (nethermost plots) for (9) at (d) , at (e) , and at (f) and .

3.2. Interaction of Lump Solution with Rough Wave (Two-Stripe Solitons)

We presume that the new ansatz is a collection of quadratic function and hyperbolic function as follows:

Switching from (13) in (2), we embezzle a variety arrangement of answers for (1):

More intense computations were finished utilizing Maple software to get the obscure constants in accordance with representation form (13) in (3) and emulate the coefficients of , and to zero. Settling the subsequent nonlinear framework produces a few instances of the compelled parameters. For each situation, we back substitute in (13) as follows:

Substituting from (15) into (13),

Through a similar system, we get the arrangements of (1) and plot in Figure 3 for .

Figure 3: Upper plots: 3D plots for (14) at (a) , at (b) , and at (c) and . Contour plots (nethermost plots) for (14) at (d) , at (e) , and at (f) and

4. Conclusions

In this work, we constructed lump solutions and mixed solution involving lump waves and solitons for the incompressible fluid system (1) via bilinear method and symbolic computation. Starting form Cole-Hopf transformation which is that investigated by Singular Manifold method with two-term truncated series, we drive some of the new and novel lump-solitons: lump-kink, lump interacted with one-stripe soliton or kink and interacted lump with two-stripe soliton, or kink wave after many complicated calculations utilizing Maple software. The three-dimensional plots and contour plots are presented for all solutions that we obtained. Via our intensive search, there is no one to investigate these types of solutions for (1). Our work is important for an interaction between lumps and to better get these frameworks.

Data Availability

The data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

We would thank the editing board and reviewers for their valuable response and fast reply that enhance the obtained results.

References

A.-M. Wazwaz, “Soliton solutions for two (3+1) -dimensional non-integrable KdV-type equations,” Mathematical and Computer Modelling, vol. 55, no. 5-6, pp. 1845–1848, 2012.

A.-M. Wazwaz, “New (3+1)-Dimensional nonlinear evolution equations with burgers and Sharma-Tasso-Olver equations constituting the main parts,” Proceedings of the Romanian Academy Series A - Mathematics Physics Technical Sciences Information Science, vol. 16, no. 1, pp. 32–40, 2015.

K. Khan and M. Ali Akbar, “Traveling wave solutions of the (2 + 1)-dimensional Zoomeron equation and the Burgers equations via the MSE method and the Exp-function method,” Ain Shams Engineering Journal, vol. 5, no. 1, pp. 247–256, 2014.

C. Chun, “New solitary wave solutions to nonlinear evolution equations by the Exp-function method,” Computers & Mathematics with Applications, vol. 61, no. 8, pp. 2107–2110, 2011.

A. M. Wazwaz, “The tanh-coth and the sine-cosine methods for kinks, solitons, and periodic solutions for the Pochhammer-Chree equations,” Applied Mathematics and Computation, vol. 195, no. 1, pp. 24–33, 2008.

A.-M. Wazwaz, “The sine-cosine and the tanh methods: reliable tools for analytic treatment of nonlinear dispersive equations,” Applied Mathematics and Computation, vol. 173, no. 1, pp. 150–164, 2006.

M. T. Darvishi, M. Najafi, L. Kavitha, and M. Venkatesh, “Stair and step soliton solutions of the integrable (2+1) and (3+1)-dimensional Boiti - Leon - Manna - Pempinelli equations,” Communications in Theoretical Physics, vol. 58, no. 6, pp. 785–794, 2012.

Y. Tang and W. Zai, “Multiple-soliton solutions for nonlinear partial differential equations,” Journal of Mathematics Research, vol. 7, no. 3, p. 75, 2015.

H. Ma, Y. Bai, and A. Deng, “Exact three-wave solutions for the (3+1)-dimensional Boiti-Leon-Manna-Pempinelli equation,” Advances in Difference Equations, vol. 2013, no. 1, p. 321, 2013.

H. Ma and Y. Bai, “Wronskian determinant solutions for the (3 + 1)-dimensional Boiti-Leon-Manna-Pempinelli equation,” Journal of Applied Mathematics and Physics, vol. 01, no. 05, pp. 18–24, 2013.

M. N. S. Arbabi and M. Najafi, “New soliton solutions of (3+1)-dimensional Boiti-Leon-Manna-Pempinelli equation,” International Journal of Modern Applied Physics, vol. 2, no. 3, pp. 120–129, 2013.

Y. Tang and W. Zai, “New periodic-wave solutions for (2+1)- and (3+1)-dimensional Boiti-Leon-Manna-Pempinelli equations,” Nonlinear Dynamics, vol. 81, no. 1-2, pp. 249–255, 2015.

S. Kumar, Q. Zhou, and W. Liu, “Invariant traveling wave solutions of parity-time-symmetric mixed linear-nonlinear optical lattices with three types of nonlinearity,” Laser Physics, vol. 29, no. 4, Article ID 045401, 2019.

J. Liu, M. Eslami, H. Rezazadeh, and M. Mirzazadeh, “Rational solutions and lump solutions to a non-isospectral and generalized variable-coefficient Kadomtsev–Petviashvili equation,” Nonlinear Dynamics, vol. 95, no. 2, pp. 1027–1033, 2019.

Y. Zhang, C. Yang, W. Yu, M. Mirzazadeh, Q. Zhou, and W. Liu, “Interactions of vector anti-dark solitons for the coupled nonlinear Schrödinger equation in inhomogeneous fibers,” Nonlinear Dynamics, vol. 94, no. 2, pp. 1351–1360, 2018.

H. Triki, Q. Zhou, and W. Liu, “W-shaped solitons in inhomogeneous cigar-shaped Bose–Einstein condensates with repulsive interatomic interactions,” Laser Physics, vol. 29, no. 5, Article ID 055401, 2019.

H. Triki, C. Bensalem, A. Biswas et al., “W-shaped and bright optical solitons in negative indexed materials,” Chaos, Solitons & Fractals, vol. 123, pp. 101–107, 2019.

D. R. Solli, C. Ropers, and B. Jalali, “Active control of rogue waves for stimulated supercontinuum generation,” Physical Review Letters, vol. 101, no. 23, Article ID 233902, 2008.

C. Bayindir, “Early detection of rogue waves by the wavelet transforms,” Physics Letters A, vol. 380, no. 1-2, pp. 156–161, 2016.

M. Erkintalo, “Predicting the unpredictable?” Nature Photonics, vol. 9, no. 9, pp. 560–562, 2015.

F. Fedele, C. Lugni, and A. Chawla, “The sinking of the El Faro: Predicting real world rogue waves during Hurricane Joaquin,” Scientific Reports, vol. 7, no. 1, Article ID 11188, 2017.

M. E. Yahia, R. E. Tolba, N. A. El-Bedwehy, S. K. El-Labany, and W. M. Moslem, “Rogue waves lead to the instability in GaN semiconductors,” Scientific Reports, vol. 5, no. 1, Article ID 12245, 2015.

W. X. Ma, Z. Qin, and X. Lu, “Lump solutions to dimensionally reduced p-gkp and p-gbkp equations,” Nonlinear Dynamics, vol. 84, no. 2, pp. 923–931, 2016.

X. Lv and W.-X. Ma, “Study of lump dynamics based on a dimensionally reduced Hirota bilinear equation,” Nonlinear Dynamics, vol. 85, no. 2, pp. 1217–1222, 2016.

X. Lü, W. X. Ma, Y. Zhou, and C. M. Khalique, “Rational solutions to an extended Kadomtsev-Petviashvili-like equation with symbolic computation,” Computers & Mathematics with Applications, vol. 71, no. 8, pp. 1560–1567, 2016.

J.-B. Zhang and W.-X. Ma, “Mixed lump-kink solutions to the BKP equation,” Computers & Mathematics with Applications, vol. 74, no. 3, pp. 591–596, 2017.

B. Li and Y. Ma, “Gaussian rogue waves for a nonlinear variable coefficient Schrödinger system in inhomogeneous optical nanofibers,” Journal of Nanoelectronics and Optoelectronics, vol. 12, no. 12, pp. 1397–1401, 2017.

B.-Q. Li and Y.-L. Ma, “Rogue waves for the optical fiber system with variable coefficients,” Optik - International Journal for Light and Electron Optics, vol. 158, pp. 177–184, 2018.

J. Yang, W. Ma, and Z. Qin, “Lump and lump-soliton solutions to the (2+1)-dimensional Ito equation,” Analysis and Mathematical Physics, vol. 8, no. 3, pp. 427–436, 2018.

W.-X. Ma, X. Yong, and H.-Q. Zhang, “Diversity of interaction solutions to the (2+1)-dimensional Ito equation,” Computers & Mathematics with Applications, vol. 75, no. 1, pp. 289–295, 2018.

W.-X. Ma and Y. Zhou, “Lump solutions to nonlinear partial differential equations via Hirota bilinear forms,” Journal of Differential Equations, vol. 264, no. 4, pp. 2633–2659, 2018.

Z. Yan, Z. Wen, and C. Hang, “Spatial solitons and stability in self-focusing and defocusing Kerr nonlinear media with generalized parity-time-symmetric Scarff-II potentials,” Physical Review E: Statistical, Nonlinear, and Soft Matter Physics, vol. 92, no. 2, Article ID 022913, 2015.

Z. Yan, V. V. Konotop, A. V. Yulin, and W. M. Liu, “Two-dimensional superfluid flows in inhomogeneous Bose-Einstein condensates,” Physical Review E: Statistical, Nonlinear, and Soft Matter Physics, vol. 85, no. 1-2, Article ID 016601, 2012.

Y. Zhenya and V. V. Konotop, “Exact solutions to three-dimensional generalized nonlinear Schrödinger equations with varying potential and nonlinearities,” Physical Review E: Statistical, Nonlinear, and Soft Matter Physics, vol. 80, Article ID 036607, 2009.

Z. Yan, V. V. Konotop, and N. Akhmediev, “Three-dimensional rogue waves in nonstationary parabolic potentials,” Physical Review E: Statistical, Nonlinear, and Soft Matter Physics, vol. 82, no. 3, Article ID 036610, 2010.

M. R. Ali and A. R. Hadhoud, “Hybrid orthonormal bernstein and block-pulse functions wavelet scheme for solving the 2D bratu problem,” Results in Physics, vol. 12, pp. 525–530, 2019.

M. R. Ali, M. M. Mousa, and W.-X. Ma, “Solution of nonlinear Volterra integral equations with weakly singular Kernel by using the HOBW method,” Advances in Mathematical Physics, vol. 2019, Article ID 1705651, 10 pages, 2019.

R. Sadat, M. Kassem, and W.-X. Ma, “Abundant lump-type solutions and interaction solutions for a nonlinear (3+1) dimensional model,” Advances in Mathematical Physics, vol. 2018, Article ID 9178480, 8 pages, 2018.

M. R. Ali, “A truncation method for solving the time-fractional benjamin-ono equation,” Journal of Applied Mathematics, vol. 2019, Article ID 3456848, 7 pages, 2019.

Y.-N. Tang, W.-X. Ma, and W. Xu, “Grammian and Pfaffian solutions as well as Pfaffianization for a (3+1)-dimensional generalized shallow water equation,” Chinese Physics B, vol. 21, no. 7, Article ID 070212, 2012.