Abundant soliton solutions of the modified KdV-KP equation

Results in Physics 58 (2024) 107478 Available online 13 February 2024 2211-3797/© 2024 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). Contents lists available at ScienceDirect Results in Physics journal homepage: www.elsevier.com/locate/rinp Abundant soliton solutions of the modified KdV-KP equation Choon Kit Chan a, Ghazala Akram b, Muhammad Bilal Riazc,d, Maasoomah Sadaf b, Iqra Zainab b, Ahmed S.M. Alzaidi e, Muhammad Abbas f,∗ aFaculty of Engineering and Quantity Surveying, INTI International University, 71800 Negeri Sembilan, Malaysia bDepartment of Mathematics, University of the Punjab, Lahore 54590, Pakistan cIT4Innovations, VSB – Technical University of Ostrava, Ostrava, Czech Republic dDepartment of Computer Science and Mathematics, Lebanese American University, Byblos, Lebanon eDepartment of Mathematics and Statistics, College of Science, Taif University, P.O. Box 11099, Taif 21944, Saudi Arabia fDepartment of Mathematics, University of Sargodha, Sargodha 40100, Pakistan ARTICLE INFO Keywords: Modified KdV-KP equation Soliton solutions Modified auxiliary equation method Extended rational sine–cosine method Extended rational sinh–cosh method ABSTRACT The precise solutions of the (2+1)-dimensional modified KdV-KP equation are examined in this study. Three reliable methods, namely, the modified auxiliary equation scheme, extended rational sine–cosine methodology, and extended rational sinh–cosh technique, are utilized for the first time to extract multiple soliton solutions for the governing model. These types of solutions have applications in plasma, solid state, neuronal, biological production, and diffusion processes. Graphical simulations of the obtained solutions are included in the form of 3 dimensional surface plots to reveal their physical structure and dynamical features. Introduction In recent years, there has been an increasing interest in the direct exploration of precise solutions for nonlinear partial differential equations (NLPDEs) within the fields of physical science and nonlinear science. The examination of traveling wave solutions for such equations holds great significance in the analysis of nonlinear physical phenomena. Nonlinear phenomena are prevalent across various scientific domains, including solid state physics, plasma physics and fluid dynamics. Mathematicians and scientists in the physical realm are dedicated to discover more accurate solutions to gain deeper comprehension of these nonlinear phenomena. Numerous effective approaches have been put forth to procure exact solutions for nonlinear evolution equations. Among traveling waves a great importance is being given to the solitary wave dynamics of NLPDEs, especially solitons. There are numerous NLPDEs that posses soliton solutions [1–3]. A variety of exact techniques are available to explore solitary wave dynamics of NLPDEs [4–8]. The presented effort targets the investigation of the modified Korteweg–de Vries–Kadomtsev–Petviashvili (mKdV-KP) equation for traveling wave and soliton solutions using three proficient techniques, namely, the modified auxiliary equation (MAE) methodology, extended rational sine–cosine (ERSC) scheme and extended rational sinh–cosh (ERShCh) technique. The mathematical form of mKdV-KP equation is ℑ𝑥𝑡 −3 2ℑ𝑥𝑥 + 12ℑℑ2 𝑥+ 6ℑ2ℑ𝑥𝑥 +ℑ𝑥𝑥𝑥𝑥 +ℑ𝑦𝑦 = 0,(1) ∗Corresponding author. E-mail addresses: [email protected] (M.B. Riaz), [email protected] (M. Abbas). where ℑ(𝑥, 𝑦, 𝑡)is velocity of the wave profile [9]. Eq. (1) expresses the phenomenon of small surface tension in comparison to the gravitational force in fluid dynamics. Malek et al. studied the governing equation as an initial-value-problem using a modified version of the linearized perturbation expansion methodology. They verified that their outcomes are compatible with those earned by the method of Lie groups [10]. Taghizadeh et al. employed a method having roots in theory of commutative algebra, named the first integral method, to explore the model for sake of soliton solutions [11]. Al-Fhaid constructed several closed form solutions for the leading equation via F-expansion technique. He obtained triangular periodic and soliton solutions as the limiting case of doubly periodic Jacobi wave solutions [12]. Khan et al. derived traveling periodic wave and solitary wave solutions for the equation under consideration by utilizing an ansatz technique, named the enhanced (G’/G)-expansion method [13]. It is for the first time that the mKdV-KP equation is being studied using the MAE method, ERSC method and ERShCh method. The suggested methodologies are proficient, easy to proceed and reliable to extract analytic exact solutions of NLPDEs. These methods can generate a variety of closed form solutions including trigonometric, hyperbolic and rational functions. Theoretically various types of traveling wave solutions can be obtained by assigning arbitrary values to free parameters appearing in aforementioned techniques for considered model. In laboratory these solutions can be used as prior knowledge to generate desired possible soliton pulses in fluids. https://doi.org/10.1016/j.rinp.2024.107478 Received 18 November 2023; Received in revised form 11 January 2024; Accepted 12 February 2024 Results in Physics 58 (2024) 107478 2 C.K. Chan et al. The work is organized as follows: In Section ‘‘Proceeding steps of proposed techniques’’ the proceeding steps of adopted methods are explained. Sections ‘‘Construction of solution via modified auxiliary equation method’’, ‘‘Construction of solution via extended rational sine–cosine method’’ and ‘‘Construction of solution via extended rational sinh–cosh method’’ elaborate the results of governing model earned via MAE method, ERSC method and ERShCh method, respectively. Section ‘‘Graphical explanation’’ provides graphical explanation about earned solutions. Section ‘‘Conclusion’’ contains concluding remarks. Proceeding steps of proposed techniques The NLPDE has the form, as 𝑃(ℑ,ℑ𝑥,ℑ𝑦,ℑ𝑡,ℑ𝑥𝑥,ℑ𝑥𝑡,…) = 0,(2) where ℑ=ℑ(𝑥, 𝑦, 𝑡)satisfies the NLPDE (2). Substituting the relation ℑ(𝑥, 𝑦, 𝑡) = 𝜙(𝛬),(3) where 𝛬=𝑥+𝑦−𝜔𝑡, converts Eq. (2) into an ODE, as 𝑄(𝜙, 𝜙′, 𝜙′′,…) = 0,(4) where 𝜔is a constant and 𝜙′=𝑑𝜙 𝑑𝛬 . The modified auxiliary equation method [14] Suppose the solution of Eq. (4) has the form, as 𝜙(𝛬) = 𝛺0+ N ∑ 𝑖=1 [𝛺𝑖(𝑔ℎ)𝑖+𝜆𝑖(𝑔ℎ)−𝑖],(5) where 𝛺𝑖s, 𝜆𝑖s are unknowns to be evaluated and ℎ(𝛬)satisfies the ODE ℎ′(𝛬) = 𝜅2+𝜅1𝑔−ℎ+𝜅3𝑔ℎ ln𝑔.(6) Here 𝜅1, 𝜅2, 𝜅3and gare arbitrary constants with 𝑔 > 0, 𝑔 ≠1. The concept of homogeneous balancing (HB) can be used to calculate positive integer N[2]. Further, 𝛺𝑖s and 𝜆𝑖s cannot be zero simultaneously. The solutions of Eq. (6) are as follows : ∙If 𝜅2 2− 4𝜅1𝜅3<0and 𝜅3≠0, 𝑔ℎ(𝛬)= −𝜅2+√4𝜅1𝜅3−𝜅2 2tan( √4𝜅1𝜅3−𝜅2 2𝛬 2) 2𝜅3 or 𝑔ℎ(𝛬)= − 𝜅2+√4𝜅1𝜅3−𝜅2 2cot( √4𝜅1𝜅3−𝜅2 2𝛬 2) 2𝜅3 . ∙If 𝜅2 2− 4𝜅1𝜅3>0and 𝜅3≠0, 𝑔ℎ(𝛬)= − 𝜅2+√𝜅2 2− 4𝜅1𝜅3tanh( √𝜅2 2−4𝜅1𝜅3𝛬 2) 2𝜅3 or 𝑔ℎ(𝛬)= − 𝜅2+√𝜅2 2− 4𝜅1𝜅3coth( √𝜅2 2−4𝜅1𝜅3𝛬 2) 2𝜅3 . ∙If 𝜅2 2− 4𝜅1𝜅3= 0 and 𝜅3≠0, 𝑔ℎ(𝛬)= − 2 + 𝜅2𝛬 2𝜅3𝛬. Extended rational sine–cosine method The general solution of Eq. (4) by ERSC method is given, as 𝜙(𝛬) = 𝜛0sin (𝑚𝛬) 𝜛2+𝜛1cos (𝑚𝛬),cos (𝑚𝛬)≠−𝜛2 𝜛1 (7) or 𝜙(𝛬) = 𝜛0cos (𝑚𝛬) 𝜛2+𝜛1sin (𝑚𝛬),sin (𝑚𝛬)≠−𝜛2 𝜛1 ,(8) where mis the wave number. Substitute Eq. (7) or Eq. (8) into Eq. (4). Once the terms with same cos (𝑚𝛬)𝑖or sin (𝑚𝛬)𝑖powers are gathered, they are all equated to zero which results in a collection of algebraic equations that can be used to find unknown constants 𝜛𝑖’s. Calculations using the Maple software can be used to derive the solutions to the algebraic problems. Extended rational sinh–cosh method The general solution of Eq. (4) by ERShCh method is given, as 𝜙(𝛬) = 𝜛0sinh (𝑚𝛬) 𝜛2+𝜛1cosh (𝑚𝛬),cosh (𝑚𝛬)≠−𝜛2 𝜛1 (9) or 𝜙(𝛬) = 𝜛0cosh (𝑚𝛬) 𝜛2+𝜛1sinh (𝑚𝛬),sinh (𝑚𝛬)≠−𝜛2 𝜛1 ,(10) where mis the wave number. Substitute Eq. (9) or Eq. (10) into Eq. (4). Once the terms with same cosh (𝑚𝛬)𝑖or sinh (𝑚𝛬)𝑖powers are gathered, they are all equated to zero which results in a collection of algebraic equations that can be used to find unknown constants 𝜛𝑖’s. Calculations using the Maple software can be used to derive the solutions to the algebraic problems. Construction of solution via modified auxiliary equation method Considering the transformation (3), where 𝜔represents speed of wave, the nonlinear PDE (1) will be converted to a nonlinear ODE, as 2𝜙′′ − (2𝜔+ 1)𝜙+ 4𝜙3= 0.(11) Consider the solution of Eq. (11) according to the MAE method, as 𝜙(𝛬) = 𝛺0+ N ∑ 𝑖=1 [𝛺𝑖(𝑔ℎ)𝑖+𝜆𝑖(𝑔ℎ)−𝑖].(12) Maintaining a balance between nonlinear terms and highest order derivative terms of Eq. (11) by following HB principle, gives N= 1. Then the corresponding form of Eq. (12) is 𝜙(𝛬) = 𝛺0+𝛺1𝑔ℎ+𝜆1𝑔−ℎ.(13) By employing Eq. (13) in conjunction with Eq. (6) in Eq. (11) and subsequently setting the coefficients of each power of 𝑔ℎto zero, a set of algebraic equations is obtained. The potential solutions for this system are determined using the assistance of Maple software, as 𝐅𝐚𝐦𝐢𝐥𝐲𝟏 ∶ {𝜔= − 1 2𝜅22+ 2 𝜅1𝜅3−1 2, 𝛺0=𝑖 2𝜅2, 𝛺1=𝑖𝜅3, 𝜆1= 0}. 𝐅𝐚𝐦𝐢𝐥𝐲𝟐 ∶ {𝜔= − 1 2𝜅22+ 2 𝜅1𝜅3−1 2, 𝛺0=𝑖 2𝜅2, 𝛺1= 0, 𝜆1=𝑖𝜅1}. The solutions corresponding to each family for mKdV-KP equation are, as 𝐅𝐚𝐦𝐢𝐥𝐲𝟏 ∙𝜅2 2− 4𝜅1𝜅3<0and 𝜅3≠0, provide ℑ1,1(𝑥, 𝑦, 𝑡) = 𝑖 2𝜅2+𝑖 2(−𝜅2+√4𝜅1𝜅3−𝜅22tan (1 2√4𝜅1𝜅3−𝜅22𝛬)) or ℑ∗ 1,1(𝑥, 𝑦, 𝑡) = 𝑖 2𝜅2−𝑖 2(𝜅2+√4𝜅1𝜅3−𝜅22cot (1 2√4𝜅1𝜅3−𝜅22𝛬)). ∙𝜅2 2− 4𝜅1𝜅3>0and 𝜅3≠0, provide ℑ1,2(𝑥, 𝑦, 𝑡) = 𝑖 2𝜅2−𝑖 2(𝜅2+√−4 𝜅1𝜅3+𝜅22tanh (1 2√−4 𝜅1𝜅3+𝜅22𝛬)) Results in Physics 58 (2024) 107478 3 C.K. Chan et al. Fig. 1. Graphical simulation of |ℑ1,3(𝑥, 𝑦, 𝑡)|: 3 dimensional surface plot obtained by taking 𝜅1= 1, 𝜅2= 2, 𝜅3= 1, 𝑡 = 1. Fig. 2. Graphical simulation of |ℑ∗ 2,1(𝑥, 𝑦, 𝑡)|: 3 dimensional surface plot obtained by taking 𝜅1= 0.2, 𝜅2= 0.1, 𝜅3= 0.3, 𝑡 = 1. Fig. 3. Graphical simulation of |ℑ2,2(𝑥, 𝑦, 𝑡)|: 3 dimensional surface plot obtained by taking 𝜅1= 0.2, 𝜅2= 2, 𝜅3= 0.3, 𝑡 = 1. Fig. 4. Graphical simulation of |ℑ3(𝑥, 𝑦, 𝑡)|: 3 dimensional surface plot obtained by taking 𝜛1= 1, 𝑚 = 1, 𝑡 = 0.1. or ℑ∗ 1,2(𝑥, 𝑦, 𝑡) = 𝑖 2𝜅2−𝑖 2(𝜅2+√−4 𝜅1𝜅3+𝜅22coth (1 2√−4 𝜅1𝜅3+𝜅22𝛬)). ∙𝜅2 2− 4𝜅1𝜅3= 0 and 𝜅3≠0, yield ℑ1,3(𝑥, 𝑦, 𝑡) = 𝑖 2𝜅2−𝑖(𝛬 𝜅2+ 2) 2𝛬. 𝐅𝐚𝐦𝐢𝐥𝐲𝟐 ∙𝜅2 2− 4𝜅1𝜅3<0and 𝜅3≠0, yield ℑ2,1(𝑥, 𝑦, 𝑡) = 𝑖 2𝜅2+2𝑖𝜅1𝜅3 −𝜅2+√4𝜅1𝜅3−𝜅22tan (1 2√4𝜅1𝜅3−𝜅22𝛬) or ℑ∗ 2,1(𝑥, 𝑦, 𝑡) = 𝑖 2𝜅2−2𝑖𝜅1𝜅3 𝜅2+√4𝜅1𝜅3−𝜅22cot (1 2√4𝜅1𝜅3−𝜅22𝛬). ∙𝜅2 2− 4𝜅1𝜅3>0and 𝜅3≠0, provide ℑ2,2(𝑥, 𝑦, 𝑡) = 𝑖 2𝜅2−2𝑖𝜅1𝜅3 𝜅2+√−4 𝜅1𝜅3+𝜅22tanh (1 2√−4 𝜅1𝜅3+𝜅22𝛬) or ℑ∗ 2,2(𝑥, 𝑦, 𝑡) = 𝑖 2𝜅2−2𝑖𝜅1𝜅3 𝜅2+√−4 𝜅1𝜅3+𝜅22coth (1 2√−4 𝜅1𝜅3+𝜅22𝛬). ∙𝜅2 2− 4𝜅1𝜅3= 0 and 𝜅3≠0, give ℑ2,3(𝑥, 𝑦, 𝑡) = 𝑖 2𝜅2−2𝑖𝜅1𝜅3𝛬 𝛬 𝜅2+ 2 . Construction of solution via extended rational sine–cosine method In accordance with ERSC method the solution of Eq. (11) takes the form of Eq. (7). Inserting Eq. (7) in Eq. (11) and taking coefficients of all powers of cos(𝑚𝛬)equal to zero, yields the following system: cos(𝑚𝛬)2∶ −2 𝜔 𝜛12− 4 𝜛02−𝜛12= 0, cos(𝑚𝛬)1∶ 2 𝑚2𝜛1𝜛2− 4 𝜔 𝜛1𝜛2− 2 𝜛1𝜛2= 0, cos(𝑚𝛬)0∶ 4 𝑚2𝜛12− 2 𝑚2𝜛22− 2 𝜔 𝜛22+ 4 𝜛02−𝜛22= 0. (14) The possible solutions of system (14) are, as 𝐅𝐚𝐦𝐢𝐥𝐲 𝟑 ∶ {𝜔=1 2𝑚2−1 2, 𝜛0= ± 𝑖 2𝑚𝜛1, 𝜛2= ±𝜛1}. 𝐅𝐚𝐦𝐢𝐥𝐲 𝟒 ∶ {𝜔= 2 𝑚2−1 2, 𝜛0= ±𝑖𝑚𝜛1, 𝜛2= 0}. Results in Physics 58 (2024) 107478 4 C.K. Chan et al. Fig. 5. Graphical simulation of |ℑ5(𝑥, 𝑦, 𝑡)|: 3 dimensional surface plot obtained by taking 𝜛1= 1, 𝑚 = 1, 𝑡 = 0.1. Fig. 6. Graphical simulation of |ℑ6(𝑥, 𝑦, 𝑡)|: 3 dimensional surface plot obtained by taking 𝜛1= 1, 𝑚 = 1, 𝑡 = 0.1. The solutions corresponding to Family 3 for mKdV-KP equation are, as ℑ3(𝑥, 𝑦, 𝑡)=± 𝑖𝑚 sin (𝑚𝛬) 2(cos (𝑚𝛬)− 1) . The solutions corresponding to Family 4 for mKdV-KP equation are, as ℑ4(𝑥, 𝑦, 𝑡)=±𝑖𝑚 sin (𝑚𝛬) cos (𝑚𝛬). Construction of solution via extended rational sinh–cosh method In accordance with ERShCh method the solution of Eq. (11) takes the form of Eq. (9). Inserting Eq. (9) in Eq. (11) and taking coefficients of all powers of cosh(𝑚𝛬)equal to zero, yields the following system: cosh(𝑚𝛬)2∶ −2 𝜔 𝜛12+ 4 𝜛02−𝜛12= 0, cosh(𝑚𝛬)1∶ −2 𝑚2𝜛1𝜛2− 4 𝜔 𝜛1𝜛2− 2 𝜛1𝜛2= 0, cosh(𝑚𝛬)0∶ −4 𝑚2𝜛12+ 2 𝑚2𝜛22− 2 𝜔 𝜛22− 4 𝜛02−𝜛22= 0. (15) The possible solutions of system (15) are, as 𝐅𝐚𝐦𝐢𝐥𝐲 𝟓 ∶ {𝜔= − 1 2𝑚2−1 2, 𝜛0= ± 𝑖 2𝑚𝜛1, 𝜛2= ±𝜛1}. 𝐅𝐚𝐦𝐢𝐥𝐲 𝟔 ∶ {𝜔= −2 𝑚2−1 2, 𝜛0= ±𝑖𝑚𝜛1, 𝜛2= 0}. The solutions corresponding to Family 5 for mKdV-KP equation are, as ℑ5(𝑥, 𝑦, 𝑡)=± 𝑖𝑚 sinh (𝑚𝛬) 2(cosh (𝑚𝛬)− 1) . The solutions corresponding to Family 6 for mKdV-KP equation are, as ℑ6(𝑥, 𝑦, 𝑡)=±𝑖𝑚 sinh (𝑚𝛬) cosh (𝑚𝛬). Graphical explanation The traveling behavior of earned solutions are graphically elaborated using 3 dimensional surface plots. The values of free parameters, for which suitable dynamical structures are obtained, are mentioned in the caption of each figure. Figs. 1,2and 3are obtained for solutions derived from MAE method. Fig. 1 corresponds to solution ℑ1,3(𝑥, 𝑦, 𝑡). Its traveling structure shows that it represents bright soliton for specified values of free parameters. Fig. 2 corresponds to solution ℑ∗ 2,1(𝑥, 𝑦, 𝑡). Its traveling structure shows that it has damped periodic behavior for specified values of free parameters. Fig. 3 corresponds to solution ℑ2,2(𝑥, 𝑦, 𝑡). Its traveling structure shows that it represents dark soliton for specified values of free parameters. Fig. 4 is obtained for solution ℑ3(𝑥, 𝑦, 𝑡)derived from ERSC method. The corresponding traveling structure represents damped periodic wave pattern. Figs. 5 and 6are obtained for solutions derived from ERShCh method. Fig. 5 corresponds to solution ℑ5(𝑥, 𝑦, 𝑡)and shows bright solitonic behavior. Fig. 6 corresponds to solution ℑ6(𝑥, 𝑦, 𝑡)and shows dark solitonic behavior. Dark solitons are obtained from the earned solutions. Dark solitons are more important in the sense that they are less prone to loss and more stable in disturbing conditions than bright solitons. Dark solitons have thus found extensive use in different fields where loss of energy matters a lot. Conclusion The modified auxiliary equation method, extended rational sine– cosine method and extended rational sinh–cosh approach are successfully employed for the first time to construct soliton solutions to the (2+1)-dimensional mKdV-KP problem. Several sets of graphs are used to demonstrate the wave paths and structure of the solutions. It is observed from the graphs that the three mentioned techniques are reliable and proficient as all have provided same traveling structures for considered model. The obtained solutions showed periodic, bright and dark traveling behaviors. The obtained results showed that from graphical point of view the aforementioned techniques provided same results but from mathematical point of view the modified auxiliary equation methodology has the capability to provide more solutions. The 3 dimensional surface plots are included for better understanding of the dynamical wave profiles of the earned results. There are numerous analytical exact techniques which can be utilized in future to calculate results different from those produced in the proposed article. Moreover, fractional form of the considered model can be studied using latest definitions of fractional derivative to understand fractional impacts on the solutions. However, the earned results can be used as a prior knowledge for experimental setups to generate desired possible wave profiles in laboratory. The exact solutions of the (2+1)-dimensional mKdV-KP equation in the literature will be substantially increased by these solutions. CRediT authorship contribution statement Choon Kit Chan: Writing – review & editing, Writing – original draft, Methodology, Funding acquisition. Ghazala Akram: Writing – review & editing, Writing – original draft, Methodology. Muhammad Bilal Riaz: Writing – review & editing, Writing – original draft, Methodology, Conceptlization, Software. Maasoomah Sadaf: Writing – review & editing, Writing – original draft, Methodology. Iqra Zainab: Writing – review & editing, Writing – original draft, Methodology. Ahmed S.M. Alzaidi: Writing – review & editing, Writing – original draft, Methodology, Funding acquisition. Muhammad Abbas: Writing – review & editing, Writing – original draft, Methodology. Results in Physics 58 (2024) 107478 5 C.K. Chan et al. Declaration of competing interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. Data availability No data was used for the research described in the article. Acknowledgments The researcher Ahmed SM Alzaidi would like to acknowledge Deanship of Scientific Research, Taif University for funding this work. The Author Muhammad Bilal Riaz is highly thankful to Ministry of Education, Youth and Sports of the Czech Republic. Funding This work was supported by the Ministry of Education, Youth and Sports of the Czech Republic through the e-INFRA CZ (ID:90254). References [1] Akram G, Sadaf M, Zainab I. Observations of fractional effects of 𝛽-derivative and M-truncated derivative for space time fractional Phi-4 equation via two analytical techniques. Chaos Solitons Fractals 2022;154:111645. [2] Sirisubtawee S, Koonprasert. S. Exact traveling wave solutions of certain nonlinear partial differential equations using the (𝐺′ 𝐺2)-expansion method. Adv Math Phys 2018;2018:1–15. [3] Akram G, Sadaf M, Mariyam H. 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