
==== Front
Sci Rep
Sci Rep
Scientific Reports
2045-2322
Nature Publishing Group UK London

72571
10.1038/s41598-024-72571-0
Article
Soliton solutions of nonlinear coupled Davey–Stewartson Fokas system using modified auxiliary equation method and extended (G′/G2)-expansion method
Khan M. Atta Ullah 1
Sadaf Maasoomah 1
Akram Ghazala 1
Birhanu Asnake asnakeb@hu.edu.et

2
Rehan Kashif 3
Hamed Y. S. 4
1 https://ror.org/011maz450 grid.11173.35 0000 0001 0670 519X Department of Mathematics, University of the Punjab, Lahore, 54590 Pakistan
2 https://ror.org/04r15fz20 grid.192268.6 0000 0000 8953 2273 Department of Mathematics, College of Science, Hawassa University, Hawassa, Ethiopia
3 https://ror.org/05db8zr24 grid.440548.9 0000 0001 0745 4169 Department of Mathematics, University of Engineering and Technology, KSK Campus, Lahore, Pakistan
4 https://ror.org/014g1a453 grid.412895.3 0000 0004 0419 5255 Department of Mathematics and Statistics, College of Science, Taif University, P.O. Box 11099, Taif, 21944 Saudi Arabia
20 9 2024
20 9 2024
2024
14 2194927 4 2024
9 9 2024
© The Author(s) 2024
2024
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The captivating realm of the nonlinear coupled Davey–Stewartson Fokas system is explored in this research paper. As a powerful tool, the proposed system is utilized for the realistic representation of various non-linear dynamical mechanisms in different fields of sciences and engineering including non-linear optical fibers, plasma physics and water waves theory. Two distinct exact methods, namely the modified auxiliary equation method and the extended (G′/G2)-expansion method, are utilized to acquire the exact soliton solutions of the non-linear coupled Davey–Stewartson Fokas system. A plethora of novel soliton solutions containing anti-kink, kink, bright, dark, dark-bright, bright-dark and some other singular soliton solutions, have been obtained using the employed exact methods. The significance of proposed manuscript lies in the novelty of obtained solutions. Kink, dark and bright solitons have wide applications in optical fiber communications, plasma physics and water waves dynamics. The acquired nontrivial exact solutions contain exponential, trigonometric, rational and hyperbolic functions. Some obtained solutions are visually represented through graphical simulations of 3D, 2D-contour and 2D-line plots, providing a comprehensive visualization of the soliton dynamics.The modulation instability of the proposed nonlinear system has been investigated, which ensures the stability of the system.

Keywords

Non-linear coupled Davey–Stewartson Fokas system
Modified auxiliary equation method
Extended (G′G2)-expansion
Exact solutions
Subject terms

Applied mathematics
Computational science
issue-copyright-statement© Springer Nature Limited 2024
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pmcIntroduction

Natural phenomena are characterized by the presence of many nonlinearities and varying factors. A crucial role is played by non-linear partial differential equations (NLPDEs) as mathematical tools for accurately representing a wide range of natural phenomena and their intricate mechanisms. There are many extensive applications of NLPDEs across diverse fields such as optical fibers, mathematical physics, telecommunications, etc.

The NLPDEs including generalized shallow water-like wave (GSWLW) equation1, non-linear Schrödinger equation (NSE)2, LPD model3, Fokas equation4, Kundu–Mukherjee–Naskar (KMN) model5, Biswas–Arshed equation6, etc, are used to model different dynamical mechanisms. The NSE depicts the behavior of particle dynamics in fluids. The weakly non-linear wave propagation theory in the presence of magnetic fields in fluids is discussed using NSE7. The fractional form of NSE is represented in8 which makes it more suitable for modeling of different nonlinear phenomena. The GSWLW equation represents mechanisms of wave phenomena in shallow bodies of water. The application of GSWLW becomes appropriate when the horizontal extent of the fluid exceeds the vertical dimension. The stability analysis and exact soliton solutions of GSWLW are discussed in9. LPD model is used to model dynamics of optical waves in non-linear medium. There are numerous applications of these solutions in physics and other disciplines. For the analysis of optical soliton propagation in fibers, the KMN model can be applied in non-linear optics, fluid dynamics, Bose-Einstein condensates and soliton dynamics for modeling non-linear waves in biological systems10. The (3+1)-dimensional extended Zakharov-Kuznetsov dynamical model has been investigated using two different exact methods11. Solitary wave results that represent the electrostatic potential field, electric and magnetic fields and quantum statistical pressures are also constructed11.

The non-linear coupled Davey–Stewartson Fokas (NCDSF) system has significant applications in plasma physics, fluid dynamics and non-linear optics. In fluid dynamics, this system describes the interaction and evolution of waves, including non-linear and dispersive effects. In plasma physics, the system describes how electrostatic waves interact in a magnetized plasma. The fiber-optic communication and optical signal processing can be modeled using NCDSF system. The integrability of NCDSF system yields valuable insights to understand these phenomena. Recently, NCDSF systems arising in optical fiber has been integrated using the simple equation method and the Sine Gordon-method12. In Ref.13, the non-linearity form of the (2+1)-NCDSF system using adapted parabolic laws has been investigated. It is valuable for estimating surface water waves in finite depth.

NCDSF equation is an extension of NSE, derived by Fokas in 199414. The mathematical description of this system is given, as1 ιAt+Axx+2λA∫-∞y|A|x2dy=0.

Equation (1) is particular case of following generalized NLPDE, given as2 ιAt+(α1-α2)Axx+(α1+α2)Ayy+2λA(α1+α2)∫-∞x|A|y2dx+β1-(α1-α2)∫-∞y|A|x2dy+β2=0.

Equation (1) is retrieved by taking (α1=α2=12) and (β1=β2=0) in Eq. (2). For (α1=0,α2=1), Eq. (2) gives Davey–Stewartson equation15 and (α1=0,α2=0) yields Boiti-model16. From Eq. (1), the NCDSF system can be written, as3 ιAt+α1Axx+α2AU=0,α3Uy-α4Ax2=0,

where A and U are imaginary and real functions in x,  y, t. The functions A(x,  y,  t) and U(x,  y,  t) represent non-linear wave profiles in monomode optical fibers.

The NCDSF system has been investigated by some other researchers17–20. Some optical soliton solutions including lump solitons, bell-shaped solitons and periodic wave solutions are acquired utilizing two exact techniques in17. High-order breather wave solutions and different hybrid solutions of the NCDSF system are retrieved in18. Many rational and semi-rational soliton solutions of the NCDSF system are acquired in19. Some soliton solutions of the NCDSF system are extracted20 that are significant for monomode optical fibers. The novelty of this research work lies in the extraction of some new soliton solutions. Some novel soliton solutions including kink, anti-kink, dark-bright and bright-dark soliton solutions are obtained. These obtained solutions have not been observed in the literature.

The aim of this manuscript is to extract exact soliton solutions of the NCDSF system utilizing two exact techniques, which are the modified auxiliary equation (MAE) method21 and extended (G′G2)-expansion (E(G′G2)E)1 method. There are several reasons for selecting these methods for solving the NCDSF problem. These are both simple and powerful because they have simple algorithms. The MAE method and E(G′G2)E method are powerful tools for solving NLPDEs. These methods are easy to implement as compared to direct solving of NLPDEs for the extraction of exact solutions. The obtained solutions often have physical interpretations and can provide insights into the behavior and dynamics of the underlying system.

Many researchers have employed these methods to solve a lot of NLPDEs and extract exact soliton solutions22–25. The non-linear fractional Wu-Zhang system (NFWZS) has been investigated in22 using the MAE method and many distinct solitary wave solutions are successfully obtained. Three biological models are solved by using the MAE method and obtained many soliton solutions in23. The (3+1)-dimensional potential-YTSF equation24 has been investigated by E(G′G2)E method and many traveling wave solutions are obtained. Different exact soliton solutions are acquired using E(G′G2)E method in25. The MAE method has been applied to fifth order NLPDE and dispersion studies have been conducted involving this equation26. The E(G′G2)E method has been used to extract the solutions of certain non-linear conformable evolution equations27. The Triki–Biswas equation has been solved using E(G′G2)E method and MAE method and extracted exact soliton solutions28. The dynamical soliton model has been solved using three exact method including E(G′G2)E method29.

The motivation behind this research stems from the need to explore and understand the NCDSF system. The NCDSF system is significant in modeling complex phenomena in different scientific fields. The solution of NCDSF system using MAE method and E(G′G2)E method is essential to understand this model. The study of soliton solutions in this framework is beneficial to non-linear optical fibers, plasma physics, and water wave theory. In addition to providing new insights into soliton dynamics, this research aims to enhance the understanding of their applications in various fields by providing new visualizations.

The novelty of this research paper, lies in its distinctive examination of the NCDSF system through the utilization of the MAE method and the E(G′G2)E method. These methodologies are being employed for the first time to extract exact soliton solutions of the NCDSF system. This research paper unveils numerous many novel soliton solutions, including kink, anti-kink, dark, bright, bell-shaped, dark-bright, bright-dark and many other singular soliton solutions, for the NCDSF system. The remarkable variety of these innovative solutions is not observed in the existing literature.

Although significant research has been conducted on NCDSF system, there is a notable gap in understanding the full range of soliton solutions and their practical implications. This paper addresses this gap by applying advanced exact methods including MAE method and the E(G′G2)E method to uncover and analyze a diverse set of novel soliton solutions. The main contribution is the identification and detailed visualization of these solutions, enhancing our understanding of their behavior and applications in various fields.

The research article is organized as follows: “Review of proposed methodologies” presents the distinction between two methods: E(G′G2)E method and the MAE method. Section “Application of exact techniques” focuses on the application of the given methods. In “Graphical illustration of solutions”, the obtained solutions are graphically interpreted. The modulation instability analysis is demonstrated in “Modulation instability”. Section “Results and discussion” offers a thorough discussion and comparison of the results obtained. Finally, “Conclusion” summarizes the findings of this study.

Review of proposed methodologies

The NLPDE of the following form considered, as4 H(A,Ax,Ay,At,Axx,Att,Axt,...)=0,

where A=A(x,y,t) is wave function. The ODE of the following form5 G(Δ′,Δ′′,Δ′′,...)=0,

is obtained utilizing the following traveling wave transformations, given as6 Ax,y,t=Δχeιψ,U(x,y,t)=K(χ),χ=χ1x+χ2y-χ3t,ψ=ψ1x+ψ2y+ψ3t.

The MAE method

In this subsection, description of the MAE method is presented. The general solution of Eq. (5) is presented, as7 Δχ=h0+∑j=1MhjHjg(χ)+qjH-jg(χ),

where g(χ) satisfies the following ODE, given as8 ddχgχ=l1+l2H-gχ+l3HgχlnH.

In Eq. (7), hj′s, and qj′s are unknowns to be extracted. In Eq. (8), l1, l2 and l3 are constant to be determined. The solution of the ODE in Eq. (8) is given, as In case, l12-4l2l3<0 and l3≠0 then9 Hgχ=-l1+-l12+4l2l3tan12-l12+4l2l3χ2l3,

or10 Hgχ=-l1+-l12+4l2l3cot12-l12+4l2l3χ2l3.

In case, l12-4l2l3>0 and l3≠0 then11 Hgχ=-l1+-l12+4l2l3tanh12-l12+4l2l3χ2l3,

or12 Hgχ=-l1+-l12+4l2l3coth12-l12+4l2l3χ2l3.

In case, l12-4l2l3=0 and l3≠0 then13 Hgχ=l1χ+22l3χ.

In Eq. (7), the value of M can be evaluated utilizing the homogeneous balance principle (HBP). Equation (7), along with Eq. (8), will be substituted in Eq. (5), coefficients of Hjg will be collected and equate them zero. In result, a system of homogenous equations will obtained. This can be solved simultaneously using Maple software which yields values of the unknowns hj and qj. By substituting all the values into Eq. (7), the soliton solutions of the proposed system can be retrieved.

E(G′G2)E method

In this subsection, description of the E(G′G2)E method is presented. The general solution of Eq. (5) is assumed, as14 Δχ=h0+∑j=1MhjG′G2j+qjG′G2-j,

where G(χ) satisfies the following ODE, given as15 ddχG′G2=l1+l2G′G22.

where l1≠1 and l2≠0. The solution of Eq. (16) is presented as follows:

In case, l1l2>0, then16 G′G2=l1l2d1cosl1l2χ+d2sinl1l2χd2cosl1l2χ-d1sinl1l2χ.

In case, l1l2<0, then17 G′G2=-l1l2l2d1cosh2l1l2χ+d1sinh2l1l2χ+d2d1cosh2l1l2χ+d1sinh2l1l2χ-d2.

In case, l1=0, l2≠0, then18 G′G2=-2d1l2d2χ+d2.

In Eq. (14), M can be evaluated by utilizing HBP. The Eq. (14), along Eq. (15) is put into Eq. (5). The coefficients of G′G2j are accumulated and equate them to zero. In result system of homogenous equations is acquired. This system solved simultaneously and yields the values of unknowns hj and qj. By substituting all the values into Eq. (14), the soliton solutions of the proposed system can be retrieved.

Application of exact techniques

Utilizing the transformations in Eq. (6), NCDSF system in Eq. (1) is converted into ODE, given as19 2Δ′α1χ1ψ1-χ3Δ′ι-αΔψ12+α1Δ′′χ12+α2ΔK-Δψ3=0,

20 -2α4χ1ΔΔ′+α3χ2K′=0.

The imaginary part of Eq. (20) is eliminated when χ3=2α1χ1ψ1. By integrating Eq. (54) and inserting the value of K=α4χ1Δ2α3χ2+D into Eq. (19), following ODE is obtained, given as21 α2α4χ1Δ3α3χ2+α1Δ′′χ12-ψ3+α1ψ12-α2DΔ=0.

Using the HBP, the degrees of Δ′′ and Δ3 are equated, resulting in the value of M=1. In the following subsections, the proposed exact methods are employed to Eq. (19) to acquire the exact solutions.

Application of MAE method

In this subsection, the ODE from Eq. (21) is solved using MAE method. For M=1, the solution of Eq. (21) from Eq. (6) is given, as22 Δ(χ)=h0+h1Hg(χ)+q1Hg(χ),

where χ=χ1x+χ2y+χ3t and h0,h1 and q1 are constants to be calculated. The formal solutions and their derivatives of Eq. (22) are inserted into Eq. (21). The function Hjg(χ), (j=0,±1,±2,±3,±4), is taken as a common factor, and their coefficients are equated to zero. In result, a system of homogenous equations is obtained. Using Maple software, these equations are solved simultaneously, and the values of unknowns and parameters are obtained. The extracted values of unknowns and parameters are given in the following sets: Set1.h0=h0,q1=±-2α1α3χ1χ2α2α4l2,l1=±α2α4h0-2α1α3χ1χ2α2α4α1α3χ1χ2,ψ1=ψ1,h1=0,ψ3=2χ12l2l3-ψ12α1+Dα2χ2α3+α2α4χ1h02α3χ2.

Set2.h0=h0,q1=0,q1=±-2α1α3χ1χ2α2α4l2,l1=±α2α4h0α1α3χ1χ2-2α1α3χ1χ2α2α4,ψ1=ψ1,ψ3=2α1α3χ12χ2l2l3-α1α3χ2ψ12+α2α4χ1h02+Dα2α3χ2α3χ2.

Set3.23 l1=0,h0=0,ψ3=-4α1α3χ12χ2l2l3-α1α3χ2ψ12+Dα2α3χ2α3χ2,ψ1=ψ1,h1=±-2α1α3χ1χ2α2α4l3,q1=±-2α1α3χ1χ2α2α4l2.

The families of solutions for the NCDSF system can be obtained by putting values from Set 1 to 3 into Eq. (23), along with the transformations described in Eq. (7).

Family  1. For values of Set 1, following solution are obtained:24 Ax,y,t=h0+-2α1α3χ1χ2α2α4l2H-geiψ,U(x,y,t)=α4χ1h0+-2α1α3χ1χ2α2α4l2H-g2α3χ2+D.

In case, l12-4l2l3<0 and l3≠0 then25 A1=h0+2l2l3-2α1α3χ1χ2α2α4-l1+-l12+4l2l3tan12-l12+4l2l3χeiψ,U1=α4χ1α3χ2h0+2l2l3-2α1α3χ1χ2α2α4-l1+-l12+4l2l3tan12-l12+4l2l3χ2+D.

or26 A2=h0+2l2l3-2α1α3χ1χ2α2α4-l1--l12+4l2l3cot12-l12+4l2l3χeiψ,U2=α4χ1α3χ2h0+2l2l3-2α1α3χ1χ2α2α4-l1--l12+4l2l3cot12-l12+4l2l3χ2+D.

In case, l12-4l2l3>0 and l3≠0 then27 A3=h0+2l2l3-2α1α3χ1χ2α2α4-l1--l12+4l2l3tanh12-l12+4l2l3χeiψ,U3=α4χ1α3χ2h0+2l2l3-2α1α3χ1χ2α2α4-l1--l12+4l2l3tanh12-l12+4l2l3χ2+D.

or28 A4=h0+2l2l3-2α1α3χ1χ2α2α4-l1--l12+4l2l3coth12-l12+4l2l3χeiψ,U4=α4χ1α3χ2h0+2l2l3-2α1α3χ1χ2α2α4-l1--l12+4l2l3coth12-l12+4l2l3χ2+D.

In case, l12-4l2l3=0 and l3≠0 then29 A5=h0+2l2l3χl1χ+2-2α1α3χ1χ2α2α4eiψ,U5=α4χ1α3χ2h0+2l2l3χl1χ+2-2α1α3χ1χ2α2α42+D.

Family2. For values of Set 2, following solution are obtained:30 Ax,y,t=h0+-2α1α3χ1χ2α2α4l3Hgeiψ,Ux,y,t=α4χ1α3χ2h0+-2α1α3χ1χ2α2α4l3Hg2+D.

In case, l12-4l2l3<0 and l3≠0 then31 A6=h0+-l1+-l12+4l2l3tan1/2-l12+4l2l3χ2-2α1α3χ1χ2α2α4-1eiψ,U6=α4χ1α3χ2h0+-l1+-l12+4l2l3tan1/2-l12+4l2l3χ2-2α1α3χ1χ2α2α4-12+D.

or32 A7=h0+-l1+-l12+4l2l3cot1/2-l12+4l2l3χ2-2α1α3χ1χ2α2α4-1eiψ,U7=α4χ1α3χ2h0+-l1--l12+4l2l3cot1/2-l12+4l2l3χ2-2α1α3χ1χ2α2α4-12+D.

In case, l12-4l2l3>0 and l3≠0 then33 A8=h0+-l1--l12+4l2l3tanh1/2-l12+4l2l3χ2-2α1α3χ1χ2α2α4-1eiψ,U8=α4χ1α3χ2h0+-l1--l12+4l2l3tanh1/2-l12+4l2l3χ2-2α1α3χ1χ2α2α4-12+D.

or34 A9=h0+-l1--l12+4l2l3coth1/2-l12+4l2l3χ2-2α1α3χ1χ2α2α4-1eiψ,U9=α4χ1α3χ2h0+-l1--l12+4l2l3coth1/2-l12+4l2l3χ2-2α1α3χ1χ2α2α4-12+D.

In case, l12-4l2l3=0 and l3≠0 then35 A10=h0+12l1χ+2χ-2α1α3χ1χ2α2α4eiψ,U10=α4χ1α3χ2h0+14l1χ+22l3χ2-2α1α3χ1χ2α2α4+D.

Family3. For values of Set 3, following solution are obtained:36 Ax,y,t=-2α1α3χ1χ2α2α4l3Hg+-2α1α3χ1χ2α2α4l2H-geiψ,Ux,y,t=α4χ1α3χ2-2α1α3χ1χ2α2α4l3Hg+-2α1α3χ1χ2α2α4l2H-g2+D.

In case, l12-4l2l3<0 and l3≠0 then37 A11=-l1+-l12+4l2l3tan12-l12+4l2l3χ2-2α1α3χ1χ2α2α4-12eiψ+2l2l3-2α1α3χ1χ2α2α4-l1+-l12+4l2l3tan12-l12+4l2l3χeiψ,U11=α4χ1α3χ2-l1+-l12+4l2l3tan12-l12+4l2l3χ2-2α1α3χ1χ2α2α4-12+2l2l3-2α1α3χ1χ2α2α4-l1+-l12+4l2l3tan12-l12+4l2l3χ2+D.

or 38 A12=-l1--l12+4l2l3cot12-l12+4l2l3χ2-2α1α3χ1χ2α2α4-12eiψ+2l2l3-2α1α3χ1χ2α2α4-l1--l12+4l2l3cot12-l12+4l2l3χeiψ,U12=α4χ1α3χ2-l1--l12+4l2l3cot12-l12+4l2l3χ2-2α1α3χ1χ2α2α4-12+2l2l3-2α1α3χ1χ2α2α4-l1--l12+4l2l3cot12-l12+4l2l3χ2+D.

In case, l12-4l2l3>0 and l3≠0 then39 A13=-l1--l12+4l2l3tanh12-l12+4l2l3χ2-2α1α3χ1χ2α2α4-12eiψ+2l2l3-2α1α3χ1χ2α2α4-l1--l12+4l2l3tanh12-l12+4l2l3χeiψ,U13=α4χ1α3χ2-l1--l12+4l2l3tanh12-l12+4l2l3χ2-2α1α3χ1χ2α2α4-12+2l2l3-2α1α3χ1χ2α2α4-l1--l12+4l2l3tanh12-l12+4l2l3χ2+D.

or40 A14=-l1--l12+4l2l3coth12-l12+4l2l3χ2-2α1α3χ1χ2α2α4-12eiψ+2l2l3-2α1α3χ1χ2α2α4-l1--l12+4l2l3coth12-l12+4l2l3χeiψ,U14=α4χ1α3χ2-l1--l12+4l2l3coth12-l12+4l2l3χ2-2α1α3χ1χ2α2α4-12+2l2l3-2α1α3χ1χ2α2α4-l1--l12+4l2l3coth12-l12+4l2l3χ2+D.

In case, l12-4l2l3=0 and l3≠0 then41 A15=l1χ+22χ-2α1α3χ1χ2α2α4+2l2l3χl1χ+2-2α1α3χ1χ2α2α4eiψ,U15=α4χ1α3χ2l1χ+22χ-2α1α3χ1χ2α2α4+2l2l3χl1χ+2-2α1α3χ1χ2α2α42+D.

The E(G′G2)E method has been applied to obtain the solutions of proposed equation in the following subsection.

Application of E(G′G2)E method

In this subsection, the ODE from Eq. (21) is solved using the E(G′G2)E method. For M=1, the solution of Eq. (21) from Eq. (14) is given, as42 Δχ=h0+h1G′G2+q1G′G2-1,

where χ=χ1x+χ2y+χ3t and h0,h1 and q1 are constants to be calculated. The formal solutions and its derivatives of Eq. (22) are inserted into Eq. (21). The function G′G2j, (j=0,±1,±2,±3,±4), is taken as a common factor, and their coefficients are set equal to zero. As a result, a system of equations is obtained. Using Maple software, these equations are solved simultaneously, and the values of unknowns and parameters are obtained. The extracted values of unknowns and parameters are given in the following sets:

Set 1.χ3=χ3,h0=0,h1=±-2α1α3χ1χ2α2α4l2,ψ3=2α1χ12l1l2-α1ψ12+α2D,q1=0.

Set 2.h0=0,ψ3=-4α1α3χ12χ2l1l2-α1α3χ2ψ12+Dα2α3χ2α3χ2,χ3=χ3,q1=±-2α1α3χ1χ2α2α4l1,h1=±-2α1α3χ1χ2α2α4l2.

Set 3.χ3=χ3,h0=0,h1=0,ψ3=2α1χ12l1l2-α1ψ12+α2D,q1=±-2α1α3χ1χ2α2α4l1.

The families of solutions for the NCDSF system can be obtained by substituting the values from Set 1 to 3 into Eq. (23), along with the transformations described in Eq. (6).

Family  1.

For values of Set 1, following solution are obtained:43 Ax,y,t=-2α1α3χ1χ2α2α4l2G′G2eiψ,Ux,y,t=α4χ1-2α1α3χ1χ2α2α4l2G′G22α3χ2+D.

In case, 0<l1l2 then44 A16=-2α1α3χ1χ2α2α4l2d1cosl1l2χ+d2sinl1l2χd2cosl1l2χ-d1sinl1l2χl1l2eiψ,U16=α4χ1-2α1α3χ1χ2α2α4l2d1cosl1l2χ+d2sinl1l2χd2cosl1l2χ-d1sinl1l2χl1l22α3χ2+D.

In case, 0>l1l2 then45 A17=-2α1α3χ1χ2α2α4l2l1l2d1cosh2l1l2χ+d1sinh2l1l2χ+d2d1cosh2l1l2χ+d1sinh2l1l2χ-d2eiψ,U17=α4χ1-2α1α3χ1χ2α2α4l2l1l2d1cosh2l1l2χ+d1sinh2l1l2χ+d2d1cosh2l1l2χ+d1sinh2l1l2χ-d22α3χ2+D.

In case, l2≠0,l1=0 then46 A18=-2α1α3χ1χ2α2α4l2-2d1l2d2χ+d2eiψ,U18=α4χ1-2d1l2d2χ+d22α3χ2+D.

Family2.

For values of Set 2, following solution are obtained:47 Ax,y,t=-2α1α3χ1χ2α2α4l2G′G2+l1-2α1α3χ1χ2α2α4G′G2-1eiψ,Ux,y,t=α4χ1α3χ2-2α1α3χ1χ2α2α4l2G′G2+l1-2α1α3χ1χ2α2α4G′G2-12+D.

In case, l12-4l2l3>0 and l3≠0 then48 A19=-2α1α3χ1χ2α2α4l2d1cosl1l2χ+d2sinl1l2χd2cosl1l2χ-d1sinl1l2χl1l2eiψ+-2l2α1α3χ1χ2l1α2α4l1d2cosl1l2χ-d1sinl1l2χd1cosl1l2χ+d2sinl1l2χeiψ,U19=α4χ1α3χ2-2α1α3χ1χ2α2α4l2d1cosl1l2χ+d2sinl1l2χd2cosl1l2χ-d1sinl1l2χl1l2+-2l2α1α3χ1χ2l1α2α4l1d2cosl1l2χ-d1sinl1l2χd1cosl1l2χ+d2sinl1l2χ2+D.

In case, l12-4l2l3<0 and l3≠0 then49 A20=-2l1l2α1α3χ1χ2α2α4l2d1cosh2l1l2χ+d1sinh2l1l2χ+d2d1cosh2l1l2χ+d1sinh2l1l2χ-d2eiψ+-2l2α1α3χ1χ2l1α2l1l2α4l1d1cosh2l1l2χ+d1sinh2l1l2χ-d2d1cosh2l1l2χ+d1sinh2l1l2χ+d2eiψ,U20=α4χ1α3χ2l2l1l2d1cosh2l1l2χ+d1sinh2l1l2χ+d2-2α1α3χ1χ2α2α4-12d1cosh2l1l2χ+d1sinh2l1l2χ-d2+-2l2α1α3χ1χ2l1l2l1α2α4l1d1cosh2l1l2χ+d1sinh2l1l2χ-d2d1cosh2l1l2χ+d1sinh2l1l2χ+d22+D.

In case, l1=0 and l2≠0 then50 A21=±-2α1α3χ1χ2α2α4-2l2d1l2d2χ+1-l1l2d2χ+12d1eiψ,U21=α4χ1α3χ2±-2α1α3χ1χ2α2α4-2l2d1l2d2χ+1-l1l2d2χ+12d12.

Family3. For values of Set 2, following solution are obtained:51 Ax,y,t=Ax,y,t=±-2α1α3χ1χ2α2α4l1G′G2-1eiψ,Ux,y,t=α4χ1α3χ2±-2α1α3χ1χ2α2α4l1G′G2-12.

In case, l12-4l2l3>0 and l3≠0 then52 A22=±-2l2α1α3χ1χ2α2l1α4l1d2cosl1l2χ-d1sinl1l2χd1cosl1l2χ+d2sinl1l2χeiψ,U22=α4χ1α3χ2±-2l2α1α3χ1χ2α2l1α4l1d2cosl1l2χ-d1sinl1l2χd1cosl1l2χ+d2sinl1l2χ2.

In case, l12-4l2l3<0 and l3≠0 then53 A23=±-2l2α1α3χ1χ2α2|l1l2|l1α4d1cosh2l1l2χ+d1sinh2l1l2χ-d2d1cosh2l1l2χ+d1sinh2l1l2χ+d2eiψ,U23=α4χ1α3χ2±-2l2α1α3χ1χ2α2|l1l2|l1α4d1cosh2l1l2χ+d1sinh2l1l2χ-d2d1cosh2l1l2χ+d1sinh2l1l2χ+d22.

In case, l2≠0,l1=0 then54 A24=±-2l2α1α3χ1χ2α2|l1l2|l1α4d1cosh2l1l2χ+d1sinh2l1l2χ-d2d1cosh2l1l2χ+d1sinh2l1l2χ+d2eiψ,U24=α4χ1α3χ2±-2l2α1α3χ1χ2α2|l1l2|l1α4d1cosh2l1l2χ+d1sinh2l1l2χ-d2d1cosh2l1l2χ+d1sinh2l1l2χ+d22.

The functions Ui=Ui(x,y,t) and Ai=Ai(x,y,t) satisfy the governing system from Eq. (3) which implies theses are solutions of proposed system, where i=1,2,3,...,24.

Graphical illustration of solutions

In this section, the elegant graphical representations of some obtained soliton and solitary wave solutions are presented. These graphical illustrations portray the significant soliton and solitary waves patterns. The 3D graphs, 2D contour graphs and 2D line graphs of are drawn for the graphical simulations of some obtained solutions. All graphs are plotted for absolute values of imaginary functions |A(x,  y,  t)| and for real function U(x,  y,  t).

A wide range of soliton patterns including kink, anti-kink, bright, dark, dark-bright, bright-dark and singular solitons, are exhibited in the graphs of the obtained solutions. The particular numerical values are given to free parameters to plot these graphs. In 3D graphs, the functions |A(x,  y,  t)| and U(x,  y,  t) is represented on the vertical axis. To draw, line plots, the value of t=1 has been assigned in all figures. The graphs of the some acquired soliton and solitary wave solutions are shown in Fig. 1, 2, 3, 4 and 5. Different soliton solutions are presented in the following figures. The graphs of obtained solutions are plotted using the Maple 2022 software.

Modulation instability

In this section, the modulation instability (MI) is evaluated. The linear stability analysis method has been utilized to evaluate the MI of the proposed nonlinear system. The phenomenon of MI, also known as self-modulation in nonlinear waves, has been investigated in30. Various types of instabilities that arise in different types of waves have been evaluated in30. The MI of some other nonlinear systems has been discussed in30–32. It is considered that Eq. (3) has solution of following form55 Ax,y,t=P0+χPx,y,te-IP0t,Ux,y,t=Φ0+χΦx,y,t,

where P0 and Φ0 are depicted as normalized optical powers, serving as reference levels for the respective fields. Eq. (55) is inserted into equations of the system in Eq. (3). The system is linearized with respect to P and Φ. Following reduced form of the proposed system is obtained, given as56 IχPt+α2Φ0χ+χP0P+α1χPxx+α2χΦP0=0,-2χP0Pxα4+α3χΦy=0.

The functions P(x, y, t) and Φ(x,y,t) have the solutions of the following form57 Px,y,t=θ1eIκ3t+κ1x+κ2y+θ2e-Iκ3t+κ1x+κ2y,Φx,y,t=μ1eIκ3t+κ1x+κ2y+μ2e-Iκ3t+κ1x+κ2y.

Equation (57) is inserted into equations of the system in Eq. (56). The coefficients of eIκ3t+κ1x+κ2y and e-Iκ3t+κ1x+κ2y are collected and transform into matrix form, given as-α1χκ12+α2Φ0χ+χP0-χκ30α2χP000-α1χκ12+α2Φ0χ+χP0+χκ30α2χP0-2χP0Iκ1α40α3χIκ200-2χP0Iκ1α40α3χIκ2θ1θ2μ1μ2=0.

By extracting the solution of determinant of the left matrix in term of temporal variable κ3, following relation is obtained,κ3=2P02α2α4κ1±α1κ12-Φ0α2-P0.

Since, there is no imaginary component in Eq. (58). This means that modulation instability does not occur and that the solutions of Eq. (3) are stable. Therefore, MI does not occur and the solutions of Eq. (3) are stable.Figure 1 The 3D, 2D contour, line graphs of |A1| and U1 are depicted in (a–c) and (d–f), respectively for numeric values of y=h0=χ2=-1,χ1=2,α1=0.01,α2=ψ2=0.1 and rest of arbitrary parameters are assigned 1.

Figure 2 The 3D, 2D contour, line graphs of |A13| and U13 are depicted in (a–c) and (d–f), respectively for numeric values of h0=α1=α2=ψ2=0.1,χ1=2,ϕ2=ϕ3=-2,y=-1 and rest of arbitrary parameters are assigned 1.

Figure 3 The 3D, 2D contour, line graphs of |A5| and U5 are depicted in (a–c) and (d–f), respectively for numeric values of χ1=2,χ2=χ3=y=χ2=-1,α2=-0.01,ψ2=0.1 and rest of arbitrary parameters are assigned 1.

Figure 4 The 3D, 2D contour, line graphs of |A17| and U17 are depicted in (a–c) and (d–f), respectively for numeric values of d2=3,ψ2=α1=α2=0.1,χ1=2,χ2=χ3=-2,α2=ψ2=0.1 and rest of arbitrary parameters are assigned 1.

Figure 5 The 3D, 2D contour, line graphs of |A16| and U16 are depicted in (a–c) and (d–f), respectively for numeric values of y=-1,d2=-3,ψ2=α1=α2=0.1,χ1=2,χ2=χ3=-2,α2=ψ2=0.1 and rest of arbitrary parameters are assigned 1.

Results and discussion

The NCDSF system has been investigated by utilizing exact methods namely MAE method and E(G′G2)E method. Some soliton and solitary wave solutions have been acquired by utilizing proposed methods. The acquired solutions contain hyperbolic, trigonometric, rational functions and exponential functions. Some novel soliton and solitary wave solutions are attained using MAE method and E(G′G2)E method. Some interesting soliton and solitary wave patterns have been observed by the simulations of graphs.

The obtained soliton solutions of NCDSF system are important in comprehending different mechanisms governed by this system. 3D, 2D contour and 2D-line graphs are drawn in Sect. 4. These graphs are used to visualize and understand the obtained complex mathematical functions. The numerical values are assigned to arbitrary parameters intelligently to obtain the some significant soliton and solitary wave wave patterns including kink, anti-kink, bright, dark, dark-bright, bright-dark, and many singular solitons.

In Fig. 1, the kink and anti-kink soliton solutions are depicted. Figure 2 showcases the bright-dark soliton solution of functions |A13| and U13. Figure 3 represents the dark-bright soliton solutions. The dark soliton and bright soliton solutions are illustrated in Fig. 4. Lastly, Fig. 5 displays the singular solution with a periodic-like nature. In comparing the results with the existing literature, the following similarities and differences are observed: In21, NCDSF system is investigated in which singular dark, singular bright and periodic wave solutions are obtained but no kink, anti-kink, dark-bright and bright-dark solutions are observed. In17, NCDSF system is investigated in which dark and bright soliton solutions are obtained but no kink, anti-kink, dark-bright, bright-dark and periodic wave solutions are observed. Davey–Stewartson system has been solved in13 and obtained dark, bright, kink soliton and periodic wave solutions, however, no anti-kink, dark-bright, bright-dark soliton solutions are identified.

The obtained soliton and solitary wave solutions have many physical applications. Kink and anti-kink soliton solutions have some significant applications in plasma physics, optical fibers and mechanical systems. Kink, anti-kink soliton solutions are crucial in signal processing and noise reduction in optical communications that these provide stable reverse transitions in light intensity. Fusion reactors and space plasmas use Kink and anti-kink soliton solutions in plasma physics. Kink, anti-kink soliton solutions appear in mechanical systems as reverse stress waves, enabling durable materials to recover from high stresses. Bright-dark solitons, with a high-intensity core surrounded by low intensity, are essential in non-linear optics and Bose-Einstein condensates. In order to understand complex wave interactions, dark-bright solitons combine features of both. It is essential for optical transmissions, fluid dynamics, and materials science to have bright and dark solitons with high-intensity and low-intensity regions, respectively. Solitons offer valuable insights and can be applied to a variety of engineering and scientific fields.

All figures have been generated by using the software Maple 2022 version 2022 which is available on https://www.maplesoft.com/products/Maple/.

Conclusion

The NCDSF system has been investigated utilizing two exact methods: the MAE method and the E(G′G2)E method. The proposed methods are used to acquire some novel exact soliton solutions. The paper provides a comprehensive description and explanation of these methods, which greatly contributes to understanding the solutions of system. Mainly, nontrivial solutions are considered in applications of proposed methods and trivial solutions are neglected. The 3D, 2D-contour and 2D-line plots have been plotted for the visualizations of some obtained solutions. These graphs play a important role in effectively depicting the behaviors of the soliton solutions. The plotted graphs represents different patterns of soliton solutions including kink solitons, anti-kink solitons, bright solitons, dark solitons, dark-bright solitons, bright-dark solitons, and various other singular soliton solutions.

The employed methods are effective for certain types of NLPDEs but their application can be complex and challenging, especially when dealing with high-order or coupled NLPDEs. Both the solutions and the auxiliary functions are assumed to have specific forms. Methods may not be applicable or may yield incomplete results if actual solutions differ from these forms. In future, some studies can be made on NCDSF system including bifurcation analysis, sensitivity analysis, instability and Chaos studies, etc. Some other power analytical and numerical methods can be applied to explore the diversity of this system. NCDSF system has the potential to improve mathematical models for solving different complex mechanisms. This model still needs to undergo a significant amount of development.

Acknowledgements

The authors extend their appreciation to Taif University, Saudi Arabia, for supporting this work through project number (TU-DSPP-2024-47). The authors are also grateful to anonymous referees for their valuable suggestions, which significantly improved this manuscript.

Author contributions

MAK participated in the data curation, formal analysis, methodology, software and writing of the original draft. MS participated in the formal analysis, methodology, investigation, validation, supervision, review and editing of the manuscript. GA participated in the conceptualization, methodology, administration, validation, supervision, visualization and writing of the manuscript. AB participated in the conceptualization, methodology, validation, visualization and writing of the manuscript. KR participated in the data curation, formal analysis, methodology, visualization and writing of the original draft. All authors read and approved the final manuscript. YSH participated in the formal analysis, methodology, software, visualization, investigation, review and editing of the manuscript. All authors read and approved the final manuscript.

Funding

This research was funded by Taif University, Saudi Arabia, Project No. (TU-DSPP-2024-47).

Data availability

All data generated or analyzed during this study are included in this manuscript.

Competing interests

The authors declare no competing interests.

Publisher's note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
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