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

various complicated nonlinear physical phenomena can be expressed in the form of nonlinear partial dierential equations (NPDEs). Nonlinear Schrödinger’s type equations (NLSEs) are unique types of nonlinear partial differential equations which are complex in nature. Such equations can be used to define various nonlinear physical aspects such as plasma physics, fluid dynamics, photonics, quantum electronics, and electromagnetism [1, 2, 3, 4, 5]. One of the important and hot subjects for investigating soliton propagation through nonlinear optical fibers is the theory of optical solitons [6]. The spread of ultrashort pulses of electromagnetic radiation in a nonlinear medium is a multidimensional phenomenon. The interaction between various physical elements such as material dispersion, diffraction and nonlinear reaction affect the dynamics of the pulse [6]. This area has drawn the attention of many scientists for more than two decades. Different computational methods have been used to reveal solutions of various type of NLEEs such as the modified exp(–Ψ(η))-expansion function method [7, 8, 9], the first integral method [10, 11], the improved Bernoulli sub-equation function method [12, 13], the trial solution method [14, 15], the new auxiliary equation method [16], the extended simple equation method [17], the solitary wave ansatz method [18], the functional variable method [19] and several others [20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39].

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 Kunduukherjeeaskar model [40] in generating various optical solitons.

The (2+1)imensional KMN model is given by [40]

iΘ<!-- T -->t+α<!-- a -->Θ<!-- T -->xy+iβ<!-- ? -->Θ<!-- T -->(Θ<!-- T -->Θ<!-- T -->x∗<!-- * -->−<!-- - -->Θ<!-- T -->∗<!-- * -->Θ<!-- T -->x)=0. $$\begin{array}{} \displaystyle i\Theta_{t}+\alpha\Theta_{xy}+i\beta\Theta(\Theta\Theta^{*}_{x}-\Theta^{*}\Theta_{x})=0. \end{array}$$(1.1)

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]

In this section, we give the description of the novel extended rational sinh-Gordon equation expansion technique.

Consider the following sinh-Gordon equation [42]

Θ<!-- T -->xt=γ<!-- ? -->sinh⁡<!-- ? -->(Θ<!-- T -->). $$\begin{array}{} \displaystyle \Theta_{xt}=\gamma \sinh(\Theta). \end{array}$$(2.1)

One may recall the following as the solutions of Eq. (2.1) [43]:

sinh⁡<!-- ? -->(Ω<!-- O -->)=±<!-- ? -->csch(Δ<!-- ? -->)orsinh⁡<!-- ? -->(Ω<!-- O -->)=±<!-- ? -->isech(Δ<!-- ? -->), $$\begin{array}{} \displaystyle \sinh(\Omega)=\pm \;\mbox{csch}(\Delta)\;\;\;\mbox{or}\;\;\;\sinh(\Omega)=\pm\;i\; \mbox{sech}(\Delta), \end{array}$$(2.2)

cosh⁡<!-- ? -->(Ω<!-- O -->)=±<!-- ? -->coth⁡<!-- ? -->(Δ<!-- ? -->)orcosh⁡<!-- ? -->(Ω<!-- O -->)=±<!-- ? -->tanh⁡<!-- ? -->(Δ<!-- ? -->), $$\begin{array}{} \displaystyle \cosh(\Omega)=\pm \coth(\Delta)\;\;\;\mbox{or}\;\;\;\cosh(\Omega)=\pm\;\; \tanh(\Delta), \end{array}$$(2.3)

sinh⁡<!-- ? -->(Ω<!-- O -->)=tan⁡<!-- ? -->(Δ<!-- ? -->)orsinh⁡<!-- ? -->(Ω<!-- O -->)=−<!-- - -->cot⁡<!-- ? -->(Δ<!-- ? -->) $$\begin{array}{} \displaystyle \sinh(\Omega)=\tan(\Delta)\;\;\;\mbox{or}\;\;\;\sinh(\Omega)=-\cot(\Delta) \end{array}$$(2.4)

and

cosh⁡<!-- ? -->(Ω<!-- O -->)=±<!-- ? -->sec⁡<!-- ? -->(Δ<!-- ? -->)orcosh⁡<!-- ? -->(Ω<!-- O -->)=±<!-- ? -->tan⁡<!-- ? -->(Δ<!-- ? -->), $$\begin{array}{} \displaystyle \cosh(\Omega)=\pm \sec(\Delta)\;\;\;\mbox{or}\;\;\;\cosh(\Omega)=\pm\;\; \tan(\Delta), \end{array}$$(2.5)

where i=−<!-- - -->1 $\begin{array}{} \displaystyle i=\sqrt{-1} \end{array}$ and Ω′ = sinh(Ω), or Ω′ = cosh(Ω). For details, see [43].

Consider the nonlinear partial differential equation

P(Θ<!-- T -->x,Θ<!-- T -->2Θ<!-- T -->xx,Θ<!-- T -->xxx…<!-- ? -->)=0, $$\begin{array}{} \displaystyle P(\Theta_{x},\;\Theta^{2}\Theta_{xx},\;\Theta_{xxx}\;\ldots)=0, \end{array}$$(2.6)

where the subscript represents the partial derivative of Θ with respect to x.

Substituting the travelling wave transformation

Θ<!-- T -->(x,t)=Θ<!-- T -->(Δ<!-- ? -->),Δ<!-- ? -->=ω<!-- ? -->(x−<!-- - -->μ<!-- ? -->t) $$\begin{array}{} \displaystyle \Theta(x,t)=\Theta(\Delta),\;\;\; \Delta=\omega(x-\mu t) \end{array}$$(2.7)

into Eq. (2.6), the following nonlinear ordinary differential (NODE) is obtained:

D(Θ<!-- T -->,Θ<!-- T -->′,Θ<!-- T -->″,Θ<!-- T -->2Θ<!-- T -->′,…<!-- ? -->)=0, $$\begin{array}{} \displaystyle D(\Theta,\;\Theta^{'},\;\Theta^{''},\;\Theta^{2}\Theta^{'},\; \ldots)=0, \end{array}$$(2.8)

where the superscript indicates the derivative of the function Θ with respect to Δ.

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:

Θ<!-- T -->(Ω<!-- O -->)=∑<!-- ? -->j=1m[bjsinh⁡<!-- ? -->(Ω<!-- O -->)+ajcosh⁡<!-- ? -->(Ω<!-- O -->)]j+a0∑<!-- ? -->j=1m[cjsinh⁡<!-- ? -->(Ω<!-- O -->)+djcosh⁡<!-- ? -->(Ω<!-- O -->)]j+c0. $$\begin{array}{} \displaystyle \Theta(\Omega)=\frac{\sum_{j=1}^{m}\big[ b_{j}\sinh(\Omega)+ a_{j}\cosh(\Omega)\big]^{j}+a_{0}}{\sum_{j=1}^{m}\big[ c_{j}\sinh(\Omega)+ d_{j}\cosh(\Omega)\big]^{j}+c_{0}}. \end{array}$$(2.9)

In this section, we give the applications of the new extended rational sinh-Gordon equation expansion method.

Consider the complex wave transformation

Θ<!-- T -->(x,y,t)=Θ<!-- T -->(Δ<!-- ? -->)eiθ<!-- ? -->,Θ<!-- T -->∗<!-- * -->(x,y,t)=Θ<!-- T -->(Δ<!-- ? -->)e−<!-- - -->iθ<!-- ? -->Δ<!-- ? -->=ω<!-- ? -->(−<!-- - -->σ<!-- s -->1x−<!-- - -->σ<!-- s -->2y−<!-- - -->μ<!-- ? -->t),θ<!-- ? -->=−<!-- - -->γ<!-- ? -->1x−<!-- - -->γ<!-- ? -->2y+κ<!-- ? -->t+ϕ<!-- ? -->. $$\begin{array}{r} \displaystyle \Theta(x,y,t)=\Theta(\Delta)e^{i\theta},\; \Theta^{*}(x,y,t)=\Theta(\Delta)e^{-i\theta}\; \Delta=\omega(-\sigma_{1}x-\sigma_{2}y-\mu t),\\\theta=-\gamma_{1}x-\gamma_{2}y+\kappa t+\phi. \end{array}$$(3.1)

In Eq. (3.1), γ₁ and γ₂ 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 σ₁ and σ₂ stand for the inverse width of the soliton along x– and y–directions, respectively [40].

Substituting Eq. (3.1) into (1.1), gives

ω<!-- ? -->2α<!-- a -->σ<!-- s -->1σ<!-- s -->2Θ<!-- T -->″−<!-- - -->(κ<!-- ? -->+α<!-- a -->γ<!-- ? -->1γ<!-- ? -->2)Θ<!-- T -->−<!-- - -->2β<!-- ? -->γ<!-- ? -->1Θ<!-- T -->3=0 $$\begin{array}{} \displaystyle \omega^{2}\alpha\sigma_{1}\sigma_{2}\Theta^{''}-(\kappa+\alpha\gamma_{1}\gamma_{2})\Theta -2\beta\gamma_{1}\Theta^{3}=0 \end{array}$$(3.2)

from the real part, and

μ<!-- ? -->=α<!-- a -->(γ<!-- ? -->1σ<!-- s -->2+γ<!-- ? -->2σ<!-- s -->1) $$\begin{array}{} \displaystyle \mu=\alpha(\gamma_{1}\sigma_{2}+\gamma_{2}\sigma_{1}) \end{array}$$(3.3)

from the real part.

Balancing the terms Θ^″ and Θ³, yields m = 1.

With m = 1, Eq. (2.9) turns to

Θ<!-- T -->(Ω<!-- O -->)=b1sinh⁡<!-- ? -->(Ω<!-- O -->)+a1cosh⁡<!-- ? -->(Ω<!-- O -->)+a0c1sinh⁡<!-- ? -->(Ω<!-- O -->)+d1cosh⁡<!-- ? -->(Ω<!-- O -->)+c0. $$\begin{array}{} \displaystyle \Theta(\Omega)=\frac{ b_{1}\sinh(\Omega)+ a_{1}\cosh(\Omega)+a_{0}}{ c_{1}\sinh(\Omega)+ d_{1}\cosh(\Omega)+c_{0}}. \end{array}$$(3.4)

Substituting Eq. (3.4) along with Ω′ = sinh(Ω) and/or Ω′ = cosh(Ω) into Eq. (3.2), yields a polynomial in powers of hyperbolic functions. Collecting the coefficients of the hyperbolic functions of the same power and equating each summation to zero, yields a system of algebraic equations. Solving the system of algebraic equations, produces the values of a₁, b₁, c₁, d₁, and the other parameters involved. Substituting the values of the parameters into Eqs. (2.10)-(2.13) with fixed value of m = 1, gives some new results to Eq. (1.1).

Set-1

When

a0=d1−<!-- - -->(κ<!-- ? -->+α<!-- a -->γ<!-- ? -->1γ<!-- ? -->2)2β<!-- ? -->γ<!-- ? -->1,c0=a1−<!-- - -->2β<!-- ? -->γ<!-- ? -->1(κ<!-- ? -->+α<!-- a -->γ<!-- ? -->1γ<!-- ? -->2),ω<!-- ? -->=−<!-- - -->−<!-- - -->2(κ<!-- ? -->+α<!-- a -->γ<!-- ? -->1γ<!-- ? -->2)α<!-- a -->σ<!-- s -->1σ<!-- s -->2, $$\begin{array}{} \displaystyle a_{0}=d_{1}\sqrt{-\frac{(\kappa+\alpha\gamma_{1}\gamma_{2})}{2\beta\gamma_{1}}},\; c_{0}=a_{1}\sqrt{-\frac{2\beta\gamma_{1}}{(\kappa+\alpha\gamma_{1}\gamma_{2})}},\; \omega=-\sqrt{-\frac{2(\kappa+\alpha\gamma_{1}\gamma_{2})}{\alpha\sigma_{1}\sigma_{2}}}, \end{array}$$

b1=−<!-- - -->i(c12−<!-- - -->d12)(κ<!-- ? -->+α<!-- a -->γ<!-- ? -->1γ<!-- ? -->2)−<!-- - -->2β<!-- ? -->a12γ<!-- ? -->12β<!-- ? -->γ<!-- ? -->1, $\begin{array}{} b_{1}=-i\sqrt{\frac{(c_{1}^{2}-d_{1}^{2})(\kappa+\alpha\gamma_{1}\gamma_{2})-2\beta a_{1}^{2}\gamma_{1}}{2\beta\gamma_{1}}}, \end{array}$ we have the following mixed dark-bright soliton:

Θ<!-- T -->1.1(x,y,t)=(d1−<!-- - -->(κ<!-- ? -->+α<!-- a -->γ<!-- ? -->1γ<!-- ? -->2)2β<!-- ? -->γ<!-- ? -->1+(c12−<!-- - -->d12)(κ<!-- ? -->+α<!-- a -->γ<!-- ? -->1γ<!-- ? -->2)−<!-- - -->2β<!-- ? -->a12γ<!-- ? -->12β<!-- ? -->γ<!-- ? -->1sech[Δ<!-- ? -->]+a1tanh⁡<!-- ? -->[Δ<!-- ? -->])ic1sech[Δ<!-- ? -->]+d1tanh⁡<!-- ? -->[Δ<!-- ? -->]+a1−<!-- - -->2β<!-- ? -->γ<!-- ? -->1(κ<!-- ? -->+α<!-- a -->γ<!-- ? -->1γ<!-- ? -->2)eiθ<!-- ? -->, $$\begin{array}{} \displaystyle \Theta_{1.1}(x,y,t)=\frac{\Bigg(d_{1}\sqrt{-\frac{(\kappa+\alpha\gamma_{1}\gamma_{2})}{2\beta\gamma_{1}}}+ \sqrt{\frac{(c_{1}^{2}-d_{1}^{2})(\kappa+\alpha\gamma_{1}\gamma_{2})-2\beta a_{1}^{2}\gamma_{1}}{2\beta\gamma_{1}}}\mbox{sech}[\Delta] +a_{1}\tanh[\Delta]\Bigg)}{ic_{1}\mbox{sech}[\Delta]+d_{1}\tanh[\Delta] +a_{1}\sqrt{-\frac{2\beta\gamma_{1}}{(\kappa+\alpha\gamma_{1}\gamma_{2})}}}e^{i\theta}, \end{array}$$(3.5)

and the mixed singular soliton

Θ<!-- T -->1.2(x,y,t)=(a1coth⁡<!-- ? -->[Δ<!-- ? -->]+d1−<!-- - -->(κ<!-- ? -->+α<!-- a -->γ<!-- ? -->1γ<!-- ? -->2)2β<!-- ? -->γ<!-- ? -->1+(c12−<!-- - -->d12)(κ<!-- ? -->+α<!-- a -->γ<!-- ? -->1γ<!-- ? -->2)−<!-- - -->2β<!-- ? -->a12γ<!-- ? -->12β<!-- ? -->γ<!-- ? -->1icsch[Δ<!-- ? -->])c1csch[Δ<!-- ? -->]+d1coth⁡<!-- ? -->[Δ<!-- ? -->]+a1−<!-- - -->2β<!-- ? -->γ<!-- ? -->1(κ<!-- ? -->+α<!-- a -->γ<!-- ? -->1γ<!-- ? -->2)eiθ<!-- ? -->, $$\begin{array}{} \displaystyle \Theta_{1.2}(x,y,t)=\frac{\Bigg(a_{1}\coth[\Delta]+d_{1}\sqrt{-\frac{(\kappa+\alpha\gamma_{1}\gamma_{2})}{2\beta\gamma_{1}}}+ \sqrt{\frac{(c_{1}^{2}-d_{1}^{2})(\kappa+\alpha\gamma_{1}\gamma_{2})-2\beta a_{1}^{2}\gamma_{1}}{2\beta\gamma_{1}}}i\; \mbox{csch}[\Delta]\Bigg)}{c_{1}\mbox{csch}[\Delta]+d_{1}\coth[\Delta] +a_{1}\sqrt{-\frac{2\beta\gamma_{1}}{(\kappa+\alpha\gamma_{1}\gamma_{2})}}}e^{i\theta}, \end{array}$$(3.6)

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 Kunduukherjeeaskar 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]:

Θ<!-- T -->(Δ<!-- ? -->)=∑<!-- ? -->j=1mtanhj−<!-- - -->1(Δ<!-- ? -->)[bjsech(Δ<!-- ? -->)+ajtanh(Δ<!-- ? -->)]+a0∑<!-- ? -->j=1mtanhj−<!-- - -->1(Δ<!-- ? -->)[cjsech(Δ<!-- ? -->)+djtanh(Δ<!-- ? -->)]+c0. $$\begin{array}{} \displaystyle \Theta(\Delta)=\frac{\sum_{j=1}^{m}tanh^{j-1}(\Delta)\big[ b_{j}\;\;sech(\Delta)+ a_{j}tanh(\Delta)\big]+a_{0}}{\sum_{j=1}^{m}tanh^{j-1}(\Delta)\big[ c_{j}\;\;sech(\Delta)+ d_{j}tanh(\Delta)\big]+c_{0}}. \end{array}$$(4.1)

In this study, we come up with the following sets of trial solutions that were generated from the sinh-Gordon equation [42]:

Θ<!-- T -->(Δ<!-- ? -->)=∑<!-- ? -->j=1m[±<!-- ? -->bjisech(Δ<!-- ? -->)±<!-- ? -->ajtanh⁡<!-- ? -->(Δ<!-- ? -->)]j+a0∑<!-- ? -->j=1m[±<!-- ? -->cjisech(Δ<!-- ? -->)±<!-- ? -->djtanh⁡<!-- ? -->(Δ<!-- ? -->)]j+c0, $$\begin{array}{} \displaystyle \Theta(\Delta)=\frac{\sum_{j=1}^{m}\big[\pm b_{j}\;i\mbox{sech}(\Delta)\pm a_{j}\tanh(\Delta)\big]^{j}+a_{0}}{\sum_{j=1}^{m}\big[\pm c_{j}\;i\mbox{sech}(\Delta)\pm d_{j}\tanh(\Delta)\big]^{j}+c_{0}}, \end{array}$$(4.2)

Θ<!-- T -->(Δ<!-- ? -->)=∑<!-- ? -->j=1m[±<!-- ? -->bjcsch(Δ<!-- ? -->)±<!-- ? -->ajcoth⁡<!-- ? -->(Δ<!-- ? -->)]j+a0∑<!-- ? -->j=1m[±<!-- ? -->cjcsch(Δ<!-- ? -->)±<!-- ? -->djcoth⁡<!-- ? -->(Δ<!-- ? -->)]j+c0, $$\begin{array}{} \displaystyle \Theta(\Delta)=\frac{\sum_{j=1}^{m}\big[\pm b_{j}\;\mbox{csch}(\Delta)\pm a_{j}\coth(\Delta)\big]^{j}+a_{0}}{\sum_{j=1}^{m}\big[\pm c_{j}\;\mbox{csch}(\Delta)\pm d_{j}\coth(\Delta)\big]^{j}+c_{0}}, \end{array}$$(4.3)

Θ<!-- T -->(Δ<!-- ? -->)=∑<!-- ? -->j=1m[bjtan⁡<!-- ? -->(Δ<!-- ? -->)±<!-- ? -->ajsec⁡<!-- ? -->(Δ<!-- ? -->)]j+a0∑<!-- ? -->j=1m[cjtan⁡<!-- ? -->(Δ<!-- ? -->)±<!-- ? -->djsec⁡<!-- ? -->(Δ<!-- ? -->)]j+c0 $$\begin{array}{} \displaystyle \Theta(\Delta)=\frac{\sum_{j=1}^{m}\big[b_{j}\;\tan(\Delta)\pm a_{j}\sec(\Delta)\big]^{j}+a_{0}}{\sum_{j=1}^{m}\big[c_{j}\;\tan(\Delta)\pm d_{j}\sec(\Delta)\big]^{j}+c_{0}} \end{array}$$(4.4)

and

Θ<!-- T -->(Δ<!-- ? -->)=∑<!-- ? -->j=1m[−<!-- - -->bjcot⁡<!-- ? -->(Δ<!-- ? -->)±<!-- ? -->ajtan⁡<!-- ? -->(Δ<!-- ? -->)]j+a0∑<!-- ? -->j=1m[−<!-- - -->cjcot⁡<!-- ? -->(Δ<!-- ? -->)±<!-- ? -->djtan⁡<!-- ? -->(Δ<!-- ? -->)]j+c0. $$\begin{array}{} \displaystyle \Theta(\Delta)=\frac{\sum_{j=1}^{m}\big[-b_{j}\;\cot(\Delta)\pm a_{j}\tan(\Delta)\big]^{j}+a_{0}}{\sum_{j=1}^{m}\big[-c_{j}\;\cot(\Delta)\pm d_{j}\tan(\Delta)\big]^{j}+c_{0}}. \end{array}$$(4.5)

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.

Figure 1

Figure 2

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 Kunduukherjeeaskar 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.