Exact solutions of Shynaray-IIA equation (S-IIAE) using the improved modified Sardar sub-equation method

The nonlinear partial differential equations (NPDEs) are widely used for the representation of physical problems arising in various branches of science and engineering, and often they describe real phenomena of complex nature in physics, fluid dynamics, and many other fields. For example, the Korteweg and de Vries equation (1895), Fisher’s equation (1951), the equation governing wave propagation in low-pass electrical transmission line (Houwe et al. 2020), the Biswas–Arshed equation (Sabi’u et al. 2019b), the Boussinesq-like equations (Shi et al. 2015; Mirhosseini-Alizamini et al. 2021), the incompressible Navier–Stokes equations in streamfunction‐vorticity formulation (Raza et al. 2021), the Sine–Gordon equation (Ben-Yu et al. 1986), and the nonlinear Schrodinger equation (Kato 1987). The Shynaray-IIA Equation (S-IIAE) are coupled system of nonlinear PDEs, whereas, it involves significant challenges of a serious nature, which are associated with intricate and nontrivial behaviors of the system (Myrzakulova et al. 2022). The well-known mathematical form of S-IIAE (Fahim et al. 2022) is expressed below:where \(q(x,t)\), \(r(x,t)\) and \(v(x,t)\) represent the unknown variables and they naturally depend on independent variables \(x\) and \(t\). Note that \(m\) and \(n\) are constants. The integrable property of this equation via the inverse scattering transform, other properties such as geometrical and gauge equivalence, and space curves integrable motion are extensively studied, see Sagidullayeva et al. (2022) and the reference therein.

The solution of NPDEs became a challenging problem in mathematics and engineering, especially to get the stability and consistency of such systems while finding numerical solutions (Shah et al. 2010; Hussain et al. 2019). The most commonly used numerical methods for solving the NPDE are the finite difference methods, spectral methods, finite element and finite volume methods (Ma and Yan 2006; Sod 1978; Fallah et al. 2000; Meuris et al. 2023). The non-numerical solutions of such equations are not an easy task all the time, therefore, the most serious concerns of researchers regarding the exact solutions to such equations have been addressed positively which is strictly based on the motivation of physical insights of the problem. The solitary wave methods are among the convincing methods for finding the exact solution to NPDEs. Previously, many methods have been used to find the exact solutions of NPDEs, such as the tanh method (Almatrafi 2023), the homogeneous balance method (Jafari et al. 2014), the sub-equation method (Akinyemi et al. 2021; Senol et al. 2021), the Kudryashov method and its modifications (Kudryashov 2012, 2020), the sine-Gordon methods (Baskonus et al. 2019; Kumar et al. 2017a; Fahim et al. 2022), the generalized algebraic and Q-expansion methods (Almatrafi and Alharbi 2023; Alharbi and Almatrafi 2022), the Jacobi elliptic function expansion method (Khan et al. 2022), and so on, see Iqbal (2018), Bilal et al. (2021), Seadawy et al. (2021, 2019), Iqbal et al. (2023), Khatri et al. (2019), Kumar et al. (2017b, 2021), Dahiya et al. (2021), Sabi’u et al. (2019a, 2022, 2023) for more details and the references therein. In addition, some of the most accurate and efficient techniques were designed with extreme efforts of researchers for the solutions of NLPDEs. However, some of these methods do not provide reasonable solutions to the system of NLPDEs adequately. Therefore, this prompted us to apply IMSSEM (Akinyemi 2021) for the solutions of S-IIAE. More precisely, the IMSSEM is a novel approach that combines and utilizes the strengths of existing methodologies to offer robust and versatile solutions for nonlinear NPDEs.

The primary objective of this research paper is to present the exact solution to the challenging S-IIAE for the first time in the literature using the IMSSEM (Akinyemi 2021), by employing this innovative method, we aim to overcome the limitations of traditional approaches and provide concise and elegant solutions to this complex equation.

The structure of this work is as follows: we give a brief introduction to the importance of NPDEs in science and engineering and some available methods for solving them in Sect. 1. Section 2 contains the methodology of the suggested technique for the solution of S-IIAE. In Sect. 3, we used the IMSSEM to solve the S-IIAE precisely. Section 4 addresses the graphical depiction of the solutions. Section 5 of the study contains its conclusion.

This section will elucidate the procedures of the IMSSEM (Akinyemi 2021) for solving NPDEs. Now, Let us consider the NLPDEwhere the function \(u=u(x,t)\) represents an unknown function in the given context. In order to proceed, we have introduced a wave transformation as follows:by using the transformation in Eq. (3) into the nonlinear PDE in Eq. (2), where \(c\ne 0\), we can reduce the PDE into an ODE of integer order

We have solved the ODE (4) by using the IMSSEM, the technique has the standard form:where \({a}_{j}\) \((j=\mathrm{0,1},\mathrm{2,3},\cdots N)\).

The value of \(N\) can be determined by homogeneous balancing procedure (HBP), by balancing the highest order nonlinear and highest order derivative term in Eq. (4). Therefore, the highest degree of \(\frac{{{\text{d}}}^{{\text{r}}}{\text{u}}}{{\text{d}}{\upxi }^{{\text{r}}}}\) is classified as:

In this section, we present the exact solutions of S-IIAE (1) by IMSSEM (Akinyemi 2021)

In case when \(r=\epsilon \overline{q }\) \((\epsilon =\pm 1)\), the S-IIAE takes the following form

In the above equation \(m\), \(n\) and \(\epsilon \) are constants. By using the traveling wave transformation, Eq. (26) is reduced into the following ODEwhere \(v, \theta , \omega , \delta \) characterize the frequency, phase constant, wave number and velocity of soliton, respectively. Substituting Eq. (27) into the first part of the system (26) and separating the real and imaginary parts, we have the real part of the form

Equation (28) is integrated, and we get

Substitute Eq. (29) into the first part of (28) and separate the real and imaginary parts aswhere the imaginary part is given by

By using the HBP, by balancing the highest order derivative and highest order nonlinear term, we obtained \(N=1\). The determined value of \(N\) is substituted in Eq. (5), we obtain the simple form of the solution as:

This section provides the result discussions on the derived solutions for the system of S-IIAE using IMSSEM. The derived solutions are rational, exponential, trigonometric and hyperbolic function solutions. for example, the solution Eq. (42) is a rational function solution, Eqs. (43) and (44) correspond to the exponential function solutions, Eqs. (45) to (51) represent the hyperbolic function solutions, and Eqs. (52) to (58) are trigonometric functions solutions. Moreover, Fig. 1 gives the graphical representations for some of the derived solitary wave solutions to Fig. 6. All the shapes in Figs. 1, 2, 3, 4, 5 and 6 are recovered using \(\upepsilon =1,\updelta =-0.5,\upomega =1,\mathrm{ a}=0,\upeta =1, {{\text{a}}}_{1}=2, c=1, k=1, C=1, y=0,\) and \(m=1\). Moreover, all the 2D plots are recovered at \(t=0.2\). It is important to note that for the sake of demonstration of some of the solitary wave structures in Figs. 1, 2, 3, 4, 5 and 6, we plotted the solutions \({\text{U}}(\upeta )\) and \({\text{G}}(\upeta )\) from the derived solutions Eqs. (42) to (58). Among the important recovered structure are the dark, bright, kink and multiple wave solitons. For example, Figs. 1, 2, 5b and 6b give the multiple wave solitons, Figs. 3a and 5a represent the kink soliton waves. Figures 3b and 5b represent the bright soliton whereas Figs. 4a and 6a represent the dark solitons.

In this research paper, we presented a novel approach especially for obtaining the exact solutions to the S-IIAE, by employing the IMSSEM. The study successfully derived a family of exact solutions for these equations. These solutions provide valuable insights into the dynamics and behavior of S-IIAE and can be utilized in various fields of physics, and applied mathematics. The behavior of the solutions for the direct study is presented in two and three-dimensional graphs. The proposed technique offers a promising avenue for tackling other complex nonlinear equations and warrants further exploration in future research. Future studies of S-IIAE can be considered on the analytical, semi-analytical and numerical solutions to investigate a variety of interesting results related to the indicated model, including the modulation instability and consistency of the solutions, their physical feasibility, and lie symmetry analysis.