Nonlinear fractional partial differential equations are highly applicable for representing a wide variety of features in engineering and research, such as shallow-water, oceanography, fluid dynamics, acoustics, plasma physics, optical fiber system, turbulence, nonlinear biological systems, and control theory. In this research, we chose to construct some new closed form solutions of traveling wave of fractional order nonlinear coupled type Boussinesq–Burger (BB) and coupled type Boussinesq equations. In beachside ocean and coastal engineering, the suggested equations are frequently used to explain the spread of shallow-water waves, depict the propagation of waves through dissipative and nonlinear media, and appears during the investigation of the flow of fluid within a dynamic system. The subsidiary extended tanh-function technique for the suggested equations is solved for achieve new results by conformable derivatives. The fractional order differential transform was used to simplify the solution process by converting fractional differential equations to ordinary type differential equations by using the mentioned method. Using this technique, some applicable wave forms of solitons like bell type, kink type, singular kink, multiple kink, periodic wave, and many other types solution were accomplished, and we express our achieve solutions by 3D, contour, list point, and vector plots by using mathematical software such as MATHEMATICA to express the physical sketch much more clearly. Moreover, we assured that the suggested technique is more reliable, pragmatic, and dependable, that also explore more general exact solutions of close form traveling waves.
Citation: Zaman UHM, Arefin MA, Akbar MA, Uddin MH (2023) Study of the soliton propagation of the fractional nonlinear type evolution equation through a novel technique. PLoS ONE 18(5): e0285178. https://doi.org/10.1371/journal.pone.0285178
Editor: Shou-Fu Tian, China University of Mining and Technology, CHINA
Received: November 24, 2022; Accepted: April 16, 2023; Published: May 22, 2023
Copyright: © 2023 Zaman et al. This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.
Data Availability: All relevant data are within the paper. No external data was used for supporting the research.
Funding: The author(s) received no specific funding for this work.
Competing interests: The authors have declared that no competing interests exist.
1. Commencement & foreword
Fractional calculus (FC) is considered an influential instrument for more precisely describing real-world issues than differential calculus (DC). FC dates to the 1600s, when G.W. Leibnitz asked L’Hospital if an integer-order derivative could be applied to non-integer derivatives . In engineering and mathematics, it would be a valuable method for understanding complex systems. It is a broader version of traditional order integration and differentiation. The usefulness of explicit solutions of the traveling wave for the non-linear fractional order partial type differential equations (NLFPDEs) is notable in the present condition . Recently, NLFPDEs have become popular in optical fiber, geochemistry, chemical physics, fractional dynamics, biophysics, biomechanics, chemical kinematics, relativistic, fluid mechanics, gas dynamics, signal transmission, plasma physics, control theory, earthquakes, solid-state physics, ecosystem, and many other. Fractional order models work on a variety of scales, including the nanoscale, mesoscale, microscale, and macroscale. They are also used in social sciences like dietary supplements, environment, money, and financial concerns. Many additional types of fractional nonlinear evolution equations (FNLEEs) have been used to extract velocity wave propagation, which has piqued the scientific community of curiosity. It is used to simulate wave circulation and heat transfer in physics, as well as numerous types of population models to the atmospheric engineering [3–7].
There are many effective techniques available, like the Hirota-bilinear method [8,9], the exp-function approach , the first-integral technique , the -dressing method , the Riemann-Hilbert approach [13,14], the generalized exponential rational function approach , the Ansatz and sub equation theories , the (G′/G)- expansion approach , the Jacobi elliptic function technique , the modified auxiliary expansion technique , Lie symmetry analysis method ,direct algebraic approach , the -stepest descent method [22,23], improved Bernoulli sub-equation Function technique , the Sine-Gordon expansion method  and residual power series technique , which effectively established for solving the solution to NLFPDEs.
The extended tanh-function approach is the most extensively used of them. By establishing tanh as a new variable for such a dependable treatment of nonlinear wave equations, Malfiet  pioneered the introduction of the practical and popular tanh-function approach. Tanh’s scheme was developed to find the solutions of traveling wave. Wazwaz  presented the expanded tanh technique afterward. In addition, the equations that can be solved using the extended tanh technique have a unique set of solutions known as solitons. Solitons are special traveling waves that maintain their shape even when colliding with others. From hydrodynamics to nonlinear optics, tsunamis to turbulence, plasmas to shock waves, traffic flow to the internet, and tornados to jupiter’s great red spot, this property is applicable in a wide variety of situations. Solitons have gained a great deal of interest recently in quantum fields and nano-technology, particularly in nano-hydrodynamics .
Space-time fractional order coupled type Boussinesq–Burger (BB) and Boussinesq equation are a prominent nonlinear type partial differential equation in this framework.
The field of horizontal velocity is denoted by and represents the altitude of the water’s surface above the bottom-most horizontal level for the above two equations.
These two equations are a fascinating equational model of mathematics which illustrates the transmission of shallow water waves and emerges throughout the study of fluid flow inside a dynamical system. The shallow water equation is often a set of partial differentials (PD) equations which define the fluid flow underneath a surface pressure, flow within vertically well-mixed water bodies, and water body motion. Coastal engineers benefit greatly from a thorough understanding of their solutions to building harbors and coastlines using nonlinear water wave theory .
Various approaches were applied to explain the space-time fractional order coupled type BB equation for example M. H. Heydari and Z. Avazzadeh were solving this equation using the Orthonormal Bernoulli polynomials (OBPs) with derivative in the Caputo form , Kumar et. al.  developed this equation with Caputo derivative by Residual power series, Homotopy analysis transform method, and Homotopy polynomials and Decomposition technique by Caputo derivatives were used to explain the suggested equation by Mahmoud S Alrawashdeh and Shifaa Bani-Issa . Alternatively, the space-time fractional order coupled type Boussinesq equation was solved by modified extended tanh technique with the modified Riemann-Liouville derivative developed the Khalid K. Ali et al.  and Handan C¸erdik Yaslan and Ayse Girgin established this proposed equation using Exp-function method with conformable derivative .
The core objective of the study is to identify innovative exact solutions of the NLFPDEs such as solitons, bells, kinks, and some other kinds of solutions. Those mentioned equations were solved using the extended tanh-function approach. Therefore, solving those proposed equations using the mentioned method with conformable derivatives is completely new. This strategy is used in this research to acquire more current and wide solutions that are easy to implement, configurable, expandable, faster to simulate, and flexible. The solutions are represented in terms of 3D and contour patern additionally with new graphical styles, like vector plot and list point plot, Applying those four sorts of pictographic descriptions, the physical phenomena of our suggested models define more clearly.
The remains of the paper will be rearranging in succeeding manner: In Section 2, the definition and base configuration were presented. Section 3, the execution of methodology for the extended hyperbolic tangent technique has been introduce. Also, in Section 4, we applied previously suggested technique to solving the specific answer for the mentioned equation. Additionally, Section 5 depicted a brief discussion, along with a graphical description and classification, that are given the photographic clarification. And the last section is standing the conclusion.
2 Definition and base configuration
Khalil et al.  have newly suggested the conformable derivative. Take up the function f:[0, ∞)→ℝ and the α-order “conformable derivative” of f, that is determine as: (2.1) for all t>0, α∈(0,1). If f is α-differentiable for some, , and seems to be, then to explain . The following formulas expresses how the use of conformable derivatives to demonstrate a few assumptions.
For the derivatives of quantitative measurements, addition, and products of the differentiable functions, the conformable derivative on time scales is essential, and this is accomplished to using the aforementioned formula . Consistent with the definition supplied by Khalil et al., the theorem given bellow satisfies with conformable derivative, which gives several important qualities.
Taking α∈(0, 1] and let, f, g were α-differentiable at all point t>0.
- , for all
- , for all
- , for all constant function .
- So, if f transforms to differentiable, and then .
Regarding the conformable derivative, Khalil  defines several more properties, like the inequality of Gronwall’s, several integration methods, exponential function, chain law, transformation of Laplace, and expansion of Tailor series.
3 Execution of the methodology
Malfiet  developed the efficient tanh approach for a reliable resolution of nonlinear wave equations in one of his pertinent features. Following that, Wazwaz  improved the extended tanh method. The key idea behind the suggested approach is to represent a polynomial solution in the form of hyperbolic type functions, and then to characterize the variable coefficient of PDE by resolving a combination of arithmetical equations and first-order ODEs.
In this example, there x and t are two independent variables.
The fundamental NLFPEE construction is assumed to be follows: (3.2) Wherever α, β are fractional order derivatives, symbolizes a function of t and x are the temporal derivative and spatial derivative correspondingly, then polynomial of is symbolizes by P and its derivatives where supreme order of linear derivatives also the maximum order nonlinear derivatives are allied. Consider the transformation of waves.(3.3)
Here k and c are independent constants with nonzero values.
If we wanted to use this wave transformation in (3.2), we should rewrite it like this: (3.4)
The Eq (3.4) represents as the ordinary derivative.
Let us take a formal solution of ODE (3.4) to the following framework (3.5) along with (3.6) where ai, bi, μ is any arbitrary constant which are to be determine afterword’s.
Let the homogeneous balance to the derivatives of the linear terms which is maximum order and nonlinear terms which is maximum order appearing in Eq (3.4) to presume n be a constant, which is positive.
Substitute solution (3.5) with (3.6) to Eq (3.4) through the rate of bi derived into phase 2, and we have got polynomials Q. Organizing every coefficient to the corresponding polynomials to zero provides a collection of arithmetical equations for and . Obtain the following series of equations for and by virtue of the figurative computer software, as, for example, Maple.
4 Analyzing causes and proposing solutions
By applying the mentioned technique, we try to find more accurate exact analytic solutions of wave for the essential FNLEEs, that is, the space-time fractional coupled BB and coupled Boussinesq equations via conformable derivative concept.
4.1 The space-time fractional coupled Boussinesq–Burger (BB)
We already introduced the space-time fractional order coupled BB equation by Eqs (1A) and (1B). Then, by applying the consequent nonlinear complex transformation of wave: (4.1.1) where c denoted the speed of traveling wave. Put on Eq (4.1.1), to Eqs (1A) and (1B) reduces to integration of ODE: (4.1.2A) (4.1.2B) where the derivative of ζ is symbolizes by ′ regarding . Integrating the Eqs (4.1.2A) then (4.1.2B) with respect to ζ one time and considering the integrating zero constant and we find (4.1.3A) (4.1.3B)
Equation leads to the following conclusion (4.1.3A): (4.1.4)
Balancing to the maximum order non-linear term and the linear term of maximum order produces the homogeneous balance, which was one. So, the result of Eq (4.1.5) taking the form bellow: (4.1.6) where the evaluated constants are a0, a1 and b1. The left-hand side turns out be a polynomial in Y by replacing (4.1.6) to (4.1.5) with (3.6). Tapping a null value in each of this polynomial’s coefficients produces a set of arithmetical equations of a0, a1, b1 and (which, for the sake of clarity, we do not display.). In this situation, we had to suppose the particular values, which are and we apply the Maple software to estimate those values. Also, we resolve a generalized set of equations using those values, as illustrated in the results below:
The performance parameters in Cluster 1 afford an explicit solution for tanh functions, and it is transformable into another form using the hyperbolic formula and coordinates in space and time (4.1.1).(4.1.7A)(4.1.7B)
The given parameters in Cluster 2 deliver the solution for tanh and coth functions, which can be adjusted with the aid of hyperbolic formulation and space and time coordinates, using transformation of (4.1.1).
For tanh and coth functions, Cluster 3’s parameters afford an explicit solution, and it is transformable into another form by means of the hyperbolic formula and coordinates in space and time (4.1.1).(4.1.9A)(4.1.9B)
For tanh and coth functions, Cluster 4’s parameters offer an explicit solution that may be changed by satisfying hyperbolic formulation, time and space coordinates, and transformation of (4.1.1).(4.1.10A)(4.1.10B)
The parameters of Cluster 5 provide an explicit solution for tanh and coth functions, this can be modified using the hyperbolic formula and the translation (4.1.1) of time and space.(4.1.11A)(4.1.11B)
An explicit solution for the tanh and coth functions is involved in merge 6’s yield parameters, this can be changed by applying the transformation (4.1.1) to time and space coordinates and the hyperbolic formula.(4.1.12A)(4.1.12B)
Cluster 7 yield parameters include an explicit solution to the tanh functions, this can be changed by applying the transformation (4.1.1) to time and space coordinates and the hyperbolic formula.(4.1.13A)(4.1.13B)
It really should be presumed that the solution and of the space-time fractional order coupled BB equation is fresh and this work was the one who eventually detected it, which was mysterious and unexplored in previous works.
The space-time fractional coupled Boussinesq equation
By using Eqs (2A) and (2B), we have previously presented space-time fractional order coupled type Boussinesq equation. Afterwards, we employ simultaneous nonlinear complex transformation of wave: (4.2.1) where c denote the speed of traveling wave. Put on Eq (4.2.1), to Eqs (2A) and (2B) diminish to the next integral order ODE: (4.2.2A) (4.2.2B) where . Integrating the Eq (4.2.2A) and the Eq (4.2.2B) with respect to ζ one time and considering the integrating constant is zero and we find (4.2.3A) (4.2.3B)
From Eq (4.2.3A), we may infer the following: (4.2.4)
Taking the balance to the maximum order of nonlinear term and the linear term of highest order, yields the homogeneous balance is two. Thus, the result of Eq (4.2.5) is taking the form bellow: (4.2.6)
Where the evaluated constants are a0, a1, a2, b1 and b2. The left-hand side turns to be a polynomial in Y by replacing (4.2.6) to (4.2.5) through (3.6). Putting a null value in each of this polynomial’s coefficients produces a set of arithmetical equations for a0, a1, a2, b1, b2, (that we neglect to display for clarity). Here we use software maple to estimate those values, at that time consider the values are So we get by means of those values. Using those values, we explain a generalized group of equations, for example exposed in the table below:
Assemble 1’s elements influencing the coth functions deliver an explicit solution by the transformation of (4.2.1), time-space coordination, and hyperbolic formula, we get, (4.2.7)
The results (4.2.7) can be written as follows using the trigonometric formula: (4.2.8)
Assemble 2’s elements influencing the coth functions provide a clear result that may be changed by the time-space coordination, hyperbolic formula, and translation of (4.2.1).(4.2.9)
The results (4.2.9) can be written as follows using the trigonometric formula: (4.2.10)
Assemble 3’s elements prompting the tanh2 and coth2 functions deliver a clear solution that may be rearranged by applying the time-space coordination, hyperbolic formula, and transformation of (4.2.1).(4.2.11)
Here we take ν = −1. The coth functions of Assemble 4 propose an explicit solution that can be changed using the time-space coordination, hyperbolic method, and transformation of (4.2.1).(4.2.12)
Here we take ν = −1. An explicit result, that can be reorganized by using the time-space coordination, hyperbolic method, and use of the transformation of (4.2.1) for assembly 5 are some of the components that affect the tanh functions.(4.2.13)
Here we take ν = −1. The elements of Assemble 6 that stimulus the tanh and coth functions include an explicit solution that can be reorganized by means of the hyperbolic formulation, time-space coordination, and using the transformation of (4.2.1).(4.2.14)
It is plausible to conclude that the solution of general space-time fractional order coupled Boussinesq equation is revolutionary and that this article was the first to discover it, which had previously been unexplored.
5 Physical description and visual explanation
Through this segment, we will look at the physical explanations for the well-known solutions of traveling wave for the space-time fractional couple BB and couple Boussinesq equations. In a 3D cartesian coordinate system, the plotline is defined as a coiled or planar surface (a). 3D may be exhibited from all angles with a simple camera turn in a photograph. A contour line is a curve that joins sites where a two-variable function has similar values, as shown in the graph (b). We may determine the thickness of two surfaces represented via a list point plot in (c). Finally, we may determine the direction of a wave by the plot of vector shown in (d) . These plots were visualized by Mathematica. We can more explicitly describe the physical sketch using those four types of pictorial descriptions.
The solutions of in Fig 1 and within the duration 0<x<100 and 0<t<2000 are the result of the space-time fractional coupled BB equation, represented kink shape result of traveling wave that have infinite wings on both sides. The results of in (Fig 2) is the singular-kink shape solution of traveling wave and the solutions of within the duration 0<x<10 and 0<t<10 and within the duration 0<x<10 and 0<t<10 are also epitomized exactly equivalent result to the space-time fractional order coupled BB equation with infinite wings or infinite tails.
The wave solution of (4.1.7A), Illustration the kink shape presenting the 3D plotline (a), contour plot (b), 3D listpoint plotline (c) and vector plot (d) of with interval 0<x<10 and 0<t<10.
Structure of singular-kink form wave result for (4.1.7A), presenting the 3D plotline (a), contour plot (b), 3D listpoint plotline (c) and vector plot (d) of with interval 0<x<10 and 0<t<10.
The picture of (Fig 3) represents the single soliton wave form solution of . Singular solitons were the type of solitary wave with a singularity (typically an infinite discontinuity) . Singular solitons are tied to the imaginary center point of a solitary wave. . The solution of with the duration 0<x<6000 and 0<t<2500 are also represented analogous traveling wave solution of the space-time fractional coupled BB equation.
The wave solution (4.1.12B), presenting the single soliton shape and define the 3D plotline (a), contour plot (b), 3D listpoint plotline (c) and vector plot (d) of (x, t) with interval 0<x<1050 and 0<t<100.
Structure of the multiple soliton form solution of traveling wave (4.1.11B), presenting the 3D plotline (a), contour plot (b), 3D listpoint plotline (c) and vector plot (d) of (x, t) with interval 0<x<800 and 0<t<2000.
The wave solution (4.2.8), Illustration of the multiple soliton shape that presenting the 3D plotline (a), contour plot (b), three-dimensional listpoint plotline (c) and vector plot (d) of with interval 0<x<28450 and 0<t<1000.
The profile of the solutions of is show off (Fig 6), which is the solutions of bell shape compaction would be a soliton that is stable and has compact support i.e., does not have an infinite number of wings for the space-time fractional order coupled type BB equation. Singular bell shape solution found by the space-time fractional order coupled type Boussinesq equations of in (Fig 7) also within the duration 0<x<90 and 0<t<20, within the duration 0<x<100 and 0<t<10 and with the duration 0<x<70 and 0<t<60 are construct a familiar wave shape of familiar equation.
Structure of solution (4.1.7B) is the bell shape wave, presenting the 3D plotline (a), contour plot (b), 3D listpoint plotline (c) and vector plot (d) of (x, t) with interval 0<x<10 and 0<t<33.
The diagram shows the singular bell form wave result (4.2.10), presenting the 3D plotline (a), contour plot (b), 3D listpoint plotline (c) and vector plot (d) of (x, t) with interval 0<x<100 and 0<t<25.
Then, the spike of (Fig 8) is like a triple soliton shape solution of traveling wave for of the space-time fractional order coupled type BB equation. Spike of (Fig 9) is like a double soliton shape solution of traveling wave also with duration 0<x<3000 and 0<t<6000 construct a similar wave shape traveling wave solution for space-time fractional order coupled type Boussinesq equation.
Structure of the triple soliton shape wave result (4.1.8B), presenting the 3D plotline (a), contour plot (b), 3D listpoint plotline (c) and vector plot (d) of (x, t) with interval 0<x<350 and 0<t<1500.
The wave solution (4.2.9), Structured of the double soliton shape presenting the 3D plotline (a), contour plot (b), 3D listpoint plotline (c) and vector plot (d) of (x, t) with interval 0<x<10000 and 0<t<100.
At last, (Fig 10) and with the duration 0<x<10 and 0<t<10 illustrate the periodic shape solution for the space-time fractional coupled BB equation. Finally, (Figs 11 and 12) are the imaginary wave shape of the space-time fractional coupled BB equation.
The singular periodic shape wave result of (4.1.10A), presenting the 3D plotline (a), contour plot (b), 3D listpoint plotline (c) and vector plot (d) of (x, t) with interval 0<x<20 and 0<t<10.
Figure of the imaginary wave result (4.1.13A), presenting the 3D plotline (a), contour plot (b), 3D listpoint plotline (c) and vector plot (d) of (x, t) with interval 0<x<10 and 0<t<40.
Structure of the imaginary wave solution (4.1.13B), presenting the 3D plotline (a), contour plot (b), 3D listpoint plotline (c) and vector plot (d) of (x, t) with interval 0<x<20 and 0<t<500.
The graphics for the space-time fractional coupled BB and coupled Boussinesq equation solutions are established here and characterize the different types of pictures which are got by those equations also avoid the similar type wave shape that overlaps to the existing image.
In this study, the recommended extended tanh-function technique and the conformable derivative definition, we have been able to accomplish and symbolically achieve several types of fresh exact solutions of traveling wave to space-time fractional coupled BB and coupled Boussinesq equations. We discovered some closed form solutions for those mentioned equations, as well as kink-shaped, singular-kink shaped, singular periodic shaped, singular bell-shaped, single soliton formed, bell-shaped, multiple solitons shaped, double soliton shaped, triple soliton shaped, imaginary wave-shaped through various free parameters in the feature of 3D, contour, list point, and vector plots. It is important to keep in mind that the value of the unidentified coefficients evaluate with the use of Maple or Mathematica software. It is important to note that all derived solutions are directly replaced with the original equations to ensure their accuracy.
The achieved solutions are applicable to investigate the transmission of shallow water waves, flow within vertically well-mixed water bodies, water body motion, and many more. Coastal engineers gain advantage highly from a well understanding of their particular theory of nonlinear water wave solutions for generating harbors and coastlines. Accordingly, the different developments of exact traveling wave solutions described in this paper may have a big impact on the investigation of fluid flow and wave motion in the ocean. The adopted method is conformable, trustworthy, direct, and effective; besides it offers a variety of unique physical model solutions to NLPFEEs in engineering, applied mathematics, and mathematical physics. This approach can be used to examine additional nonlinear problems that emerge in theoretical physics, applied mathematics, and other sectors of non-linear sciences.
The authors are grateful to the anonymous referees for their constructive criticism and suggestions on how to improve the article. The authors would also like to thank the Research Cell of Jashore University of Science and Technology for supporting the research.
- 1. Topsakal M, TaŞcan F. Exact travelling wave solutions for space-time fractional Klein-Gordon equation and (2+ 1)-Dimensional time-fractional Zoomeron equation via auxiliary equation method. Applied Mathematics and Nonlinear Sciences. 2020;5(1):437–46.
- 2. Ali HS, Habib MA, Miah MM, Akbar MA. Solitary wave solutions to some nonlinear fractional evolution equations in mathematical physics. Heliyon. 2020 Apr 1;6(4):e03727. pmid:32322721
- 3. Uddin MH, Akbar MA, Khan MA, Haque MA. Families of exact traveling wave solutions to the space time fractional modified KdV equation and the fractional Kolmogorov-Petrovskii-Piskunovequation. Journal of Mechanics of Continua and Mathematical Sciences. 2018 Mar;13(1):17–33.
- 4. Uddin MH, Akbar MA, Khan MA, Haque MA. New exact solitary wave solutions to the space-time fractional differential equations with conformable derivative. AIMS Math. 2019 Jan 1;4(2):199–214.
- 5. Uddin MH, Khan MA, Akbar MA, Haque MA. Analytical wave solutions of the space time fractional modified regularized long wave equation involving the conformable fractional derivative. Kerbala International Journal of Modern Science. 2019;5(1):7.
- 6. Uddin MH, Khatun MA, Arefin MA, Akbar MA. Abundant new exact solutions to the fractional nonlinear evolution equation via Riemann-Liouville derivative. Alexandria Engineering Journal. 2021 Dec 1;60(6):5183–91.
- 7. Uddin MH, Arefin MA, Akbar MA, Inc M. New explicit solutions to the fractional-order Burgers’ equation. Mathematical Problems in Engineering. 2021 Jun 11;2021:1–1.
- 8. Yin ZY, Tian SF. Nonlinear wave transitions and their mechanisms of (2+ 1)-dimensional Sawada–Kotera equation. Physica D: Nonlinear Phenomena. 2021 Dec 1; 427:133002.
- 9. Lü X, Chen SJ, Liu GZ, Ma WX. Study on lump behavior for a new (3+ 1)-dimensional generalised Kadomtsev–Petviashvili equation. East Asian J Appl Math. 2021 Jan 1;11(3):594–603.
- 10. Yaslan HÇ, Girgin A. Exp-function method for the conformable space-time fractional STO, ZKBBM and coupled Boussinesq equations. Arab Journal of Basic and applied sciences. 2019 Jan 2;26(1):163–70.
- 11. Eslami M, Rezazadeh H. The first integral method for Wu–Zhang system with conformable time-fractional derivative. Calcolo. 2016 Sep;53:475–85.
- 12. Wang ZY, Tian SF, Cheng J. The ∂̄-dressing method and soliton solutions for the three-component coupled Hirota equations. Journal of Mathematical Physics. 2021 Sep 1;62(9):093510.
- 13. Yang JJ, Tian SF, Li ZQ. Riemann–Hilbert problem for the focusing nonlinear Schrödinger equation with multiple high-order poles under nonzero boundary conditions. Physica D: Nonlinear Phenomena. 2022 Apr 1; 432:133162.
- 14. Li ZQ, Tian SF, Yang JJ. On the soliton resolution and the asymptotic stability of N-soliton solution for the Wadati-Konno-Ichikawa equation with finite density initial data in space-time solitonic regions. Advances in Mathematics. 2022 Nov 19; 409:108639.
- 15. Ghanbari B, Nisar KS, Aldhaifallah M. Abundant solitary wave solutions to an extended nonlinear Schrödinger’s equation with conformable derivative using an efficient integration method. Advances in Difference Equations. 2020 Dec;2020(1):1–25.
- 16. Tian SF, Wang XF, Zhang TT, Qiu WH. Stability analysis, solitary wave and explicit power series solutions of a (2+ 1)-dimensional nonlinear Schrödinger equation in a multicomponent plasma. International Journal of Numerical Methods for Heat & Fluid Flow. 2021 Feb 1;31(5):1732–48.
- 17. Khan H, Barak S, Kumam P, Arif M. Analytical Solutions of Fractional Klein-Gordon and Gas Dynamics Equations, via the (G′/G)-Expansion Method. Symmetry. 2019 Apr 19;11(4):566.
- 18. Kumar VS, Rezazadeh H, Eslami M, Izadi F, Osman MS. Jacobi elliptic function expansion method for solving KdV equation with conformable derivative and dual-power law nonlinearity. International Journal of Applied and Computational Mathematics. 2019 Oct;5:1–0.
- 19. Khater MM, Lu D, Attia RA. Dispersive long wave of nonlinear fractional Wu-Zhang system via a modified auxiliary equation method. AIP Advances. 2019 Feb 5;9(2):025003.
- 20. Guo B, Dong H, Fang Y. Symmetry groups, similarity reductions, and conservation laws of the time-fractional Fujimoto–Watanabe equation using lie symmetry analysis method. Complexity. 2020 Mar 31;2020:1–9.
- 21. Kurt A, Tozar A, Tasbozan O. Applying the new extended direct algebraic method to solve the equation of obliquely interacting waves in shallow waters. Journal of Ocean University of China. 2020 Aug;19:772–80.
- 22. Li ZQ, Tian SF, Yang JJ, Fan E. Soliton resolution for the complex short pulse equation with weighted Sobolev initial data in space-time solitonic regions. Journal of Differential Equations. 2022 Aug 25; 329:31–88.
- 23. Li ZQ, Tian SF, Yang JJ. Soliton resolution for the Wadati–Konno–Ichikawa equation with weighted Sobolev initial data. InAnnales Henri Poincaré 2022 Jul (Vol. 23, No. 7, pp. 2611–2655). Cham: Springer International Publishing.
- 24. Ala V, Demirbilek U, Mamedov KR. An application of improved Bernoulli sub-equation function method to the nonlinear conformable time-fractional SRLW equation. AIMS Mathematics. 2020 Jan 1;5(4):3751–61.
- 25. Yel G, Baskonus HM, Gao W. New dark-bright soliton in the shallow water wave model. Aims Math. 2020 Apr 20;5(4):4027–44.
- 26. Arafa A, Elmahdy G. Application of residual power series method to fractional coupled physical equations arising in fluids flow. International Journal of Differential Equations. 2018 Jan 1;2018.
- 27. Malfliet W, Hereman W. The tanh method: I. Exact solutions of nonlinear evolution and wave equations. Physica Scripta. 1996 Dec 1;54(6):563.
- 28. Wazwaz AM. New solitary wave solutions to the modified forms of Degasperis–Procesi and Camassa–Holm equations. Applied Mathematics and Computation. 2007 Mar 1;186(1):130–41.
- 29. Shukri S, Al-Khaled K. The extended tanh method for solving systems of nonlinear wave equations. Applied Mathematics and Computation. 2010 Nov 1;217(5):1997–2006.
- 30. Khater MM, Kumar D. New exact solutions for the time fractional coupled Boussinesq–Burger equation and approximate long water wave equation in shallow water. Journal of Ocean Engineering and Science. 2017 Sep 1;2(3):223–8.
- 31. Heydari MH, Avazzadeh Z. New formulation of the orthonormal Bernoulli polynomials for solving the variable-order time fractional coupled Boussinesq–Burger’s equations. Engineering with computers. 2021 Oct;37:3509–17.
- 32. Kumar S, Kumar A, Baleanu D. Two analytical methods for time-fractional nonlinear coupled Boussinesq–Burger’s equations arise in propagation of shallow water waves. Nonlinear Dynamics. 2016 Jul;85:699–715.
- 33. Alrawashdeh MS, Bani-Issa S. An efficient technique to solve coupled–time fractional Boussinesq–Burger equation using fractional decomposition method. Advances in Mechanical Engineering. 2021 Jun;13(6):16878140211025424.
- 34. Shallal MA, Jabbar HN, Ali KK. Analytic solution for the space-time fractional Klein-Gordon and coupled conformable Boussinesq equations. Results in physics. 2018 Mar 1;8:372–8.
- 35. Khalil R, Al Horani M, Yousef A, Sababheh M. A new definition of fractional derivative. Journal of computational and applied mathematics. 2014 Jul 1;264:65–70.
- 36. Benkhettou N, Hassani S, Torres DF. A conformable fractional calculus on arbitrary time scales. Journal of King Saud University-Science. 2016 Jan 1;28(1):93–8.
- 37. Zaman UH, Arefin MA, Akbar MA, Uddin MH. Analyzing numerous travelling wave behavior to the fractional-order nonlinear Phi-4 and Allen-Cahn equations throughout a novel technique. Results in Physics. 2022 Jun 1;37:105486.
- 38. Wazwaz AM. Partial differential equations and solitary waves theory. Springer Science & Business Media; 2010 May 28.
- 39. Drazin PG, Johnson RS. Solitons: an introduction. Cambridge university press; 1989 Feb 9.