The new extended rational SGEEM for construction of optical solitons to the (2+1)– dimensional Kundu–Mukherjee–Naskar model

This work proposes the new extended rational sinh-Gordon equation expansion technique (SGEEM). The computational approach is formulated based on the well-known sinh-Gordon equation. The proposed technique generalizes the sineGordon/sinh-Gordon expansion methods in a rational format. The efficiency of the suggested technique is tested on the (2+1)=imensional Kundu=iukherjee=jaskar (KMN) model. Various of optical soliton solutions have been obtained using this new method. The conditions which guarantee the existence of valid solitons are given.

However, a novel extended rational sinh-Gordon equation expansion technique is developed in this research. The new approach is based on the well-known sine-Gordon and sinh-Gordon equations. We employ the new approach to the (2+1)=imensional Kundu=iukherjee=jaskar model [40] in generating various optical solitons.
The (2+1)=imensional KMN model is given by [40] In Eq. (1.1), the unknown function Θ(x, y,t) stands for the nonlinear wave envelope. The nonzero constants a and b are the coefficients of the dispersion term and the term that is different from conventional Kerr law nonlinearity or any known non-Kerr law media. The first term Θ t represents the temporal evolution of the wave. Eq. (1.1) describes the oceanic rogue waves as well as hole waves. It may also be used in describing optical wave propagation through coherently excited resonant wave guides that is doped with Erbium atoms [40,41].

Analysis of the Method
In this section, we give the description of the novel extended rational sinh-Gordon equation expansion technique.
Consider the nonlinear partial differential equation where the subscript represents the partial derivative of Θ with respect to x.
Substituting the travelling wave transformation into Eq. (2.6), the following nonlinear ordinary differential (NODE) is obtained: where the superscript indicates the derivative of the function Θ with respect to ∆.
The general steps of the new generalized rational sinh-Gordon equation expansion method are given as follows: Step-I: Suppose that Eq. (2.6) adopts the following form of rational solution: Step II: The unknown parameters involved are obtained by substituting Eq. (2.9) along with Ω = sinh(Ω) and/or Ω = cosh(Ω) into Eq. (2.8). This produces a polynomial in powers of hyperbolic functions. Summing the coefficients of these hyperbolic functions of the same power, provides a group of algebraic equations after equating each summation to zero. Step and

Applications
In this section, we give the applications of the new extended rational sinh-Gordon equation expansion method.
Consider the complex wave transformation In Eq. (3.1), γ 1 and γ 2 are the frequencies of the solitons in x− and y-directions, respectively. The constant µ stands for the velocity of the soliton. The parameter κ is the phase constant. The parameters σ 1 and σ 2 stand for the inverse width of the soliton along x− and y−directions, respectively [40].
Substituting Eq. (3.1) into (1.1), gives from the real part, and from the real part.

Set-1: When
, we have the following mixed dark-bright soliton: and the mixed singular soliton

6)
Set-2: When we get the following dark soliton and the singular soliton

Set-4: When
we get the following trigonometric function solution: (3.10)

Set-6:
When we get the following trigonometric functions solution:

Results and Discussion
In this study, We succeeded in formulating the extended rational sinh-Gordon equation expansion technique. The developed method is employed to the (2+1)=imensional Kundu=iukherjee=jaskar model to test its efficiency. Mixed dark-bright, singular solitons and trigonometric functions solutions are successfully constructed.
Recently, Yamgoue et al. [44] introduced the rational sine-Gordon expansion method. The authors introduced the following trial solution which was generated from the sine-Gordon equation [45,46]: In this study, we come up with the following sets of trial solutions that were generated from the sinh-Gordon equation [42]: and These newly introduced trial solutions generalize all the kind of solutions that may be obtained by using the sine-and sinh-Gordon expansion methods in rational format.

Conclusion
In this research, the extended rational sinh-Gordon equation expansion method is developed. The newly developed technique gives variety of wave solutions when tested on the (2+1)= imensional Kundu= iukherjee=jaskar model. Dark, mixed dark-bright, singular, mixed singular solitons and trigonometric functions solutions are successfully constructed. The conditions which guarantee the existence of the valid solutions to this model are given. The 2-, 3-dimensional and contour graphs to this model are plotted. It is known that dark soliton describes the solitary waves with lower intensity than the background, bright soliton describes the solitary waves whose peak intensity is larger than the background [?]. The singular soliton solutions is a solitary wave with discontinuous derivatives; instances of such solitary waves are compactions, which have finite (compact) assistance, and peakons, whose peaks have a discontinuous first derivative [48,49]. The generalized sinh-Gordon equation expansion method is efficient and powerful mathematical tool which may be used in generating varieties of wave solutions to different kind of nonlinear wave equations.
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