New exact traveling wave solutions to the (2+1)-dimensional Chiral nonlinear Schr(cid:127)odinger equation

In this research work, we successfully construct various kinds of exact traveling wave solutions such as trigonometric like, singular and periodic wave solutions as well as hyperbolic solutions to the (2+1)-dimensional Chiral nonlinear Schr(cid:127)odinger equation (CNLSE) which is used as a governing equation to discuss the wave in the quantum (cid:12)eld theory. The mechanisms used to obtain these solutions are extended rational sine-cosine/sinh-cosh and the constraint conditions for the existence of valid solutions are also given. The attained results exhibit that the proposed techniques are a signi(cid:12)cant addition for exploring several types of nonlinear partial diﬀerential equations in applied sciences. Moreover, 3D, 2D-polar, and contour pro(cid:12)les are depicted for showing the physical behavior of the reported solutions by setting suitable values of unknown parameters.


Introduction
During the past few decades, nonlinear partial differential equations (NLPDEs) have gained great attention to scientific societies. Diverse complex nonlinear physical phenomena such as fluid mechanics, plasma physics, solid-state physics, chemical physics, geochemistry, optical fiber, quantum field theory, mathematical biology, etc., are modeled in the form of NLPDEs. To see the behavior of these intricate phenomena, it is necessary to find the solution to the governing equation. For this, we need to extract the exact solution of the NLPDEs. Finding the analytic solutions of NLPDEs has the importance to discuss the stability of numerical solutions and also the development of a wide range of new mathematical solvers to simplify the routine calculation. Therefore, the foremost concern for the researchers is to find the exact solutions of NLPDEs. For this sake, various powerful techniques have been developed for finding the analytic solutions of NLPDEs by using different symbolic computation like Mathematica, Matlab, and Maple . The study of quantum field theory is still booming, as the uses of its mechanism to many physical problems. The quantum hall effect is a quantum-mechanical version of the hall effect that can be observed in a two-dimensional electron system subject to strong magnetic fields and low temperatures. The hall conductance undergoes quantum hall transitions to take quantized values at a certain level. The prominent feature of the quantum hall effect is the persistence of the quantization when the electron density is varied. In two dimensions, when electrons are subjected to a magnetic field then they follow circular cyclotron orbits. When the system is treated quantum mechanically, these orbits are quantized. Schrödingers equation works as a practical model to determine the values of these energy levels. Quantum field theory remains one of the most dynamic areas of theoretical physics today, providing a common language to several other branches of physics. Solitary waves play an important role in the non-perturbative developments in the quantum field theory. Recently, the CNLSE has been analyzed by a number of effective approaches [28][29][30][31][32][33][34][35][36][37][38][39] which provide effective outcomes in diverse areas of nonlinear sciences.
Therefore, In the present work, the main goal is to seek exact traveling wave solutions of (2+1)-dimensional CNLSE employing new extended rational sine-cosine method and extended rational sinh-cosh method [40,41]. The acquired solutions exhibit that the extended rational sine-cosine and rational sinh-cosh approaches are simple and efficient mathematical tools for investigating several complex nonlinear models in the fields of nonlinear sciences.
The (2+1)-dimensional CNLSE with constant coefficients is given by [39] iΘ t + a(Θ xx + Θ yy ) where Θ is the complex function of x, y and t, respectively denote the scaled spatial and time coordinates, while a indicates the coefficient of the dispersion term. Also b 1 ,b 2 are nonlinear coupling constant terms. This type of nonlinearity is called current density. The inverse scattering transform method cannot integrate the Eq. (1) as it fails the Painleve test of integrability. It is to be emphasized that the above equation is not invariant under the Galilean transformation. The basic feature of applied techniques is to establish some elementary relationships between NLPDEs and other simple NLODEs. It has been examined that with the aid of simple solutions and solvable ODEs, different kinds of exact traveling wave solutions of some complicated NLPDEs can be easily obtained. This is the key concept of the above techniques, while the primary benefit of applying techniques is that we have achieved in a single move to get various types of new analytic solutions and provide us a guideline that how to organize these solutions. This method is very, consistent, efficient, reliable and much more practical, to obtain the wide range of traveling wave solutions for complex NLPDEs that appear in many fields like, telecommunication engineering, mathematical biology, mathematical physics, optical fiber and vice versa.
This piece of article is segregated as follow: In section 2, key points of the proposed methods are presented. In section 3, applications of proposed methods are presented. Graphical representation is depicted in section 4. Results and discussion are demonstrated in section 5. Lastly, the conclusive closure is revealed in section 6.

Key points of the proposed methods
In this section we describe the first step of the extended rational methods for seeking the analytic solutions of PDEs where u = u(x, y, t) is an unknown function and F is a polynomial in u and its partial derivatives. Suppose that where x, y and t are spatial domains and time, respectively. Then, by using (3), Eq. (2) can be turned into following ODE w.r.t. ξ In next we discuss the exact solutions of Eq. (4) by using the extended rational techniques.

Extended rational sin-cos method
According to this method, we assume that the analytic solution can be expressed in the following forms where a 0 ,a 1 and a 2 are parameters to be found in terms of the other parameters. The non-zero constant η is the wave number. The derivatives of the predicted solutions are in the first form and in the second form. We substitute Eqs. (7) or (10) into the reduced form of the governing equation obtained above in Eq. (4). On collecting the same power coefficients of the cos m (ηξ) or sin m (ηξ) and equating to zero, we have a set of algebraic expression. The obtained algebraic polynomial produce the values of the coefficients involved. After determine a 0 , a 1 , a 2 , c and η in terms of other parameters and substitute into Eq. (5) and Eq. (6), one gets solutions for Eq. (4) in rational sin-cos forms.

Extended rational sinh-cosh method
According to this method, which was introduced by Darvishiet. al., [40], we suppose that the exact solution can be written in the following forms where a 0 , a 1 and a 2 are parameters to be found in terms of the other parameters. The non-zero constant η is the wave number. The derivatives of the predicted solutions are in the first form and in the second form. We substitute Eqs. (15) or (18) into the reduced form of the governing equation obtained above in Eq. (4). On collecting the same power coefficients of the cosh m (ηξ) or sinh m (ηξ) and equating to zero, we have a set of algebraic expression. The obtained algebraic polynomial produce the values of the coefficients involved. After determine a 0 , a 1 , a 2 , c and η in terms of other parameters and substitute into Eq. (13) and Eq. (14), one gets solutions for Eq. (4) in rational sinh-cosh forms.

Applications
For Eq. (1), we start with the complex wave transformation: where c, ν, φ, ω and k are parameters, which represent the amplitude component of the soliton, velocity of soliton, phase constant, frequency and wave number respectively. Substituting (21) into Eq. (1), we get the real part and the relation from the imaginary part.

Solutions via extended rational sine-cosine method
Suppose that (22) has the solutions in the form Substituting Eq. (24) and its derivative into Eq. (22) and the coefficients with the same power of cos(ητ ) m equating to zero and resultantly, we get the following set of algebraic equations by using Mathematica: On solving these algebraic equations, we get the following solution sets as: For set 1 the solutions of Eq. (1) can be written as : Similarly, for set 2 the solutions of Eq. (1) can be written as : ] e i(px+qy+ωt+φ) .
OR Consider the Eq. (22) has solutions in the form, as Substituting Eq. (31) and its derivative into Eq. (22) and the coefficients with the same power of sin(ητ ) m equating to zero and resultantly, we get the following set of algebraic equations by using Mathematica: On solving these algebraic equations, we get the following solution sets as: For set 3 the solutions of Eq. (1) can be written as : Similarly, for set 4 the solutions of Eq. (1) can be written as :
For set 7 the solutions of Eq. (1) can be written as : Similarly, for set 8 the solutions of Eq. (1) can be written as: ] e i(px+qy+ωt+φ) .

Results and discussion
In this section we compare our results with some existing results in the literature. In this study, we have applied two innovative integrations norms to the (2+1)-dimensional CNLSE. Different kinds of solutions like trigonometric like, singular and periodic wave solutions as well as hyperbolic solutions solutions to the (2+1)-dimensional CNLSE are extracted. In published work, we have observed that, N. Raza and S. Arshad [28] discussed the different types of chiral solitons for the proposed model by using two robust and efficient integration tools. The chiral solitons are constructed for governing model via exp(−φ(ρ))-expansion and modified simple function methods by A. Javid and N. Raza [29]. Moreover N. Raza and A. Javid [36] derived optical dark and dark-singular soliton solutions of (2+1)-dimensional CNLSE through extended direct algebraic and extended trial equation methods. We observe that the retrieved solutions are new and to the best of our knowledge the applications of these techniques to the (2+1)-dimensional CNLSE have not been reported in the literature beforehand and could be beneficial to understand the different physical behaviors. For instance, hyperbolic functions such as the hyperbolic tangent emerge in the calculation and rapidity of special relativity while, the hyperbolic cotangent arises in the Langevin function for magnetic polarization [42]. The graphical description of derived solutions have been articulated in the mentioned figures by assigning the different values of the involved parameters.

Conclusion
In this work, we have investigated diverse exact traveling wave solutions like hyperbolic and trigonometric solutions to (2+1)-dimensional CNLSE with extended rational sine-cosine and rational extended sinh-cosh schemes. These various kinds of solutions are favorable for explaining diverse nonlinear physical phenomena. Our results exhibited that the proposed methods are powerful and can be used to extract exact solutions for other NLPDEs that appear in many fields like engineering, mathematical biology, physics, chemistry, and vice versa. Furthermore, for the physical illustration of the acquired results, we plot 3D, 2D-polar and contour profiles by setting suitable values of involved parameters.