New optical soliton solutions for coupled resonant Davey-Stewartson system with conformable operator

This paper investigates the novel soliton solutions of the coupled fractional system of the resonant Davey-Stewartson equations, which is a notable and significant model in dynamics of fluids for characterizing the 3-dimensional wave packet evolution of finite depth on water within weak nonlinearity. The fractional derivatives are considered in terms of conformable sense. Accordingly, we utilize a complex traveling wave transformation to reduce the proposed system to an integer-order system of ordinary differential equations. The phase portrait and the equilibria of the obtained integer-order ordinary differential system will be studied. Using ansatz method, the new types of bright, singular, and dark soliton solutions are derived and established in view of the hyperbolic, trigonometric, and rational functions of the governing system. To achieve this, illustrative examples of the fractional Davey-Stewartson system are provided to demonstrate the feasibility and reliability of the procedure used in this study. The trajectory solutions of the traveling waves are shown explicitly and graphically. The effect of conformable derivatives on behavior of acquired solutions for different fractional orders is also discussed. By comparing the proposed method with the other existing methods, the results show that the execute of this method is concise, simple, and straightforward. The results are useful for obtaining and explaining some new soliton phenomena.


Introduction
Since the advent of the concept of fractional calculus, fractional differential equations (FDEs) have attracted the interest of many researchers due to their importance in accurately demonstrating the dynamics of abundant real-world systems in various fields of sciences such as physics, diffusion, biology, chaos theory, chemistry, engineering, economics, commerce, and many others (Ross 1977;Mainardi 1997;Baleanu et al. 2012;Choo et al. 2017;Al-Smadi et al. 2021a;Formato et al. 2017;Al-Smadi 2021;Alwreikat and Rjoub 2020;Ozyurt et al. 2021;Da'ie 2021;Mikhailov and Chachkov 2020;Noori et al. 2021). In particular, more and much attention has been paid to constructing exact and approximate solutions to Des in fractional sense. Anyhow, in contemporary literature, various analytical and numerical methods have been developed and implemented to obtain approximate, exact, traveling wave, and soliton solutions to FDEs. For instance, the variational iteration method, residual power series method, reproducing kernel Hilbert space method, multistep generalized differential transform method, homotopy perturbation method, homotopy analysis method, extended trial equation method, G � ∕G -expansion method, undetermined of coefficients method, Riccati-Bernoulli Sub-ODE method, and among others (Al-Smadi et al. 2017, 2020a, 2021bAl-Smadi and Abu Arqub 2019;He 2003;Bulut et al. 2014;Wang et al. 2008;Alabedalhadi et al. 2020;Hasan et al. 2022;Jawad et al. 2019;Koç et al. 2021;Darvishi et al. 2018;Baskonus et al. 2021). Moreover, some of the important aspects of FDEs have been studied in many studies including the Cauchy problems and solution stability (Baskonus et al. 2021;Saadatmandi and Dehghan 2010). On the other hand, not like the differential derivatives of integer order, there are many different fractional operators that are used in dealing with biological, physical, and engineering systems. Riesz-Caputo, Caputo-Hadamard, Grünwald-Letnikov, Erdélyi-Kober, Sonin-Letnikov, Caputo-Fabrizio, Marchaud, Weyl, and Atangana-Baleanu, are some of these fractional operators (Zhou et al. 2009;Al-Smadi et al. 2020b, c;Abu Arqub and Al-Smadi 2020;Hasan et al. 2021;Zhou 2018;Babaoglu 2008).
Davey and Stewartson introduced the Davey-Stewartson (DS) equation in 1974, that is a notable and significant model in dynamics of fluids to characterize 3-dimensional wave packet evolution of finite depth on water within weak nonlinearity. Furthermore, the DS equation is a model for short and long-waves resonances, and other types of propagating waves. The DS system originates from multi-scale test respecting propagate of surface waves along horizontally sea surface. A coupled pair of nonlinear DS equations in two dependent variables can be reduced to the (1 + 1)-dimensional nonlinear Schrodinger (NLS) equation via a suitable dimensional reduction (Al-Smadi et al. 2021c). There are several studies which have been introduced to investigate the exact and approximate solutions for DS system. In particular, we have the inverse scattering transform (Sung 1994), Exp-function method (Zhang et al. 2021), Jacobi elliptic function method (Yan 2003), direct algebraic method (Gilson and Nimmo 1991), homotopy perturbation method (Zedan and Tantawy 2009), (G � ∕G)-expansion method (Zedan and Tantawy 2009), spectral transform (Ebadi and Biswas 2011), Darboux transformation (Boiti et al. 1989), and among others.
This paper investigates new optical soliton solutions for a conformable coupled resonant Davey-Stewartson (CCRDS) system. To reach this end, we apply the ansatz method that depends on a sutiable mathematical assumptions in terms of important functions, like hyperbolic, rational and trigonometric functions. We seek to explore two types of bright soliton solutions using two different formulas in terms of hyperbolic and rational functions.
In addition, we establish singular and dark soliton solutions in terms of trigonometric and hyperbolic, respectively. The novelty of this paper lie in all inferred soliton solutions are new and diverse in three types of solitons that constructed through concise, simple, and straightforward method. We consider the CCRDS system in the form (Aghdaei and Manafian 2016): where ≡ (x, y, t) represents amplitude of a surface wave packet and real quantity ≡ (x, y, t) may be regarded the velocity potential of the mean flow interacting with the surface wave, while the independents variables x, y and t is the dimensionless variable, propagation coordinate and time, respectively. Also, 2 = −1 dictates the hyperbolic nature, and 2 = 1 dictates the elliptic nature of the system (1), while is a constant coefficient. It is worthy that the system (2) at integer derivative orders, = 1, reduces to the following resonant NLS equation (Pashaev and Lee 2002): The main contributions of the paper are to investigate the optical soliton solutions for coupled resonant fractional Davey-Stewartson system via ansatz method and to design algorithm for proposed technique. Motivated by these ideas, the rest of the paper will be organized as follows: Sect. 2 presents the definition of the conformable derivative and their properties. The analysis of the governing system via a complex transformation is formulated in Sect. 3. In the same section, the dynamical system and the classification of its equilibria are studied. In Sect. 4, the bright, singular, and dark soliton solutions are established for the CCRDS system. Some conclusions that be gained throughout the paper have been presented in Sect. 5.

Conformable fractional derivative overview
The conformable fractional derivative is a new simple definition of a derivative of fractional order analogous to the definition of the ordinary derivative introduced by Khalil et al. (2014). Given a function f (t) ∶ (0, +∞) → ℝ . Then the conformable fractional derivative of order of f is given as for all t > 0, ∈ (0, 1] . The function f is -conformable differentiable at a point t if the limit in (3) exists. The conformable derivative accomplishes many renowned required properties (Abdeljawad 2015).
. Then, we have

Complex traveling wave transformation and equilibria classifications
In this section we seek to utilize a suitable complex traveling wave transformation for the CCRDS system (2) to translate it into integer-order ordinary differential equation (ODE) system. Furthermore, we study the planner dynamical system that corresponds to the integerorder ODE, then determine its equilibrium points and classification. The governing system (2) can be reduced into integer-order differential system via the following complex traveling wave transformation: where Φ( ) and Ψ( ) gives the structure of the wave profile with where refers to the velocity of soliton, and are the soliton frequency, and represents the wave number of the soliton. Substitute the transformation (5) with (6) into the CCRDS system (1) together using the above properties of the conformable derivative, then simplify the result to obtain By integrating twice the second equation in (7) with respect to and considering the integration constants to be zero, we get the following relation (5) (x, y, t) = Φ( (x, y, t))e iΣ (x,y,t) , (x, y, t) = Ψ( (x, y, t)), Using the relation (8) in the first equation of (7), we reach to the following integerorder ODE In the context of the classification of equilibria, we introduce the assumption Φ = dΦ d to get the planner dynamical system that corresponds to the integer-order ODE (9) in the form The system (10) can be written as Cross the multiplication of (11) and integrate both sides. The Hamiltonian function of the system (10) becomes The investigation of the equilibrium points, Φ,Φ , for the dynamical system (10) can be listed as follows: case I There are unique equilibrium point, E 1 (0, 0) , if one of the conditions is satisfied: | | = 1 , or | | > 1 and ≥ 2 2 + 2 , or | | < 1 and ≤ 2 2 + 2 .

Exact traveling wave solutions for CCRDS system
In this section, we look to deducing exact traveling wave solutions for the governing system (2) using direct method via appropriateness assumptions to solutions for the integer-order ODE (9). We will construct two types of bright soliton solutions with aid two various formulas in terms of hyperbolic and rational functions. Furthermore, singular, and dark soliton have will obtain using suitable formulas.

Bright-I soliton solutions
Assume that the integer-order ODE (9) has a solution in terms of hyperbolic functions as where Π 1 , Π 2 and are constants to be determined. Substitute the formal solution (16) into the integer-order ODE (9) and simplify the result, then group the coefficients of the similar terms in the numerator and set it are to zero, we get an algebraic system. Solve the obtained algebraic system to get: Case 1 Case 2 Using the given values in the solutions sets (17) and (18) in the formal solution (16) along the relation (9), then inserting the results into (5) with aid of (6), we get the following bright-I soliton solutions for the governing system (2) that correspond to case 1.: Consequently, the bright-I soliton solutions for the governing system (2) that correspond to case 2 are given as: To illustrate the physical naturality for the deduced solutions in (19) and (20), we depicted the profile of surfaces for the gotten solutions. Figures 2 and 3 presents the 3D and 2D of | | 5 (x, y, t) | | and | | 5 (x, y, t) | | , respectively, at − and Π − , where the derivatives are considered at integer and fractional orders.
To more illustrates, Fig. 4 shows the 2D plots of | | 8 (x, 0, 0) | | and | | 8 (x, 0, 0) | | at + and Π + , where the derivatives orders considered in various fractional sense, that give more understand of their effect on the behavior of bright-I solitons.
(23)   Accordingly, we have the following bright-II soliton solution Figure 5 and 6 present the graphical representation of the bright-II soliton solutions 1 (x, y, t) and 1 (x, y, t) in (23), respectively, at selected parameters in 3D and 2D plots where the derivatives are considered in integer and fractional orders. Moreover, the profile of the obtained bright-II soliton solutions in (25) for the CCRDS system (2) are plotted in 3D and 2D in Figs. 7 and 8, respectively, at various fractional derivatives orders.