Mohammed O Al-Amr^,1, Hadi Rezazadeh^,2, Khalid K Ali^,3, Alper Korkmazki^,41Department of Mathematics, College of Computer Sciences and Mathematics, University of Mosul, Mosul 41002, Iraq
²Faculty of Engineering Technology, Amol University of Special Modern Technologies, Amol, Iran
³Mathematics Department, Faculty of Science, Al-Azhar University, Nasr-City, Cairo, Egypt
⁴Nord Staße 9, Weimar, Germany

Received:2020-01-25Revised:2020-03-13Accepted:2020-04-9Online:2020-05-22


Abstract
The nonlocal nonlinear Schrödinger equation (NNLSE) with competing weakly nonlocal nonlinearity and parabolic law nonlinearity is explored in the current work. A powerful integration tool, which is a modified form of the simple equation method, is used to construct the dark and singular 1-soliton solutions. It is shown that the modified simple equation method provides an effective and powerful mathematical gadget for solving various types of NNLSEs.
Keywords： Schrödinger equation;soliton;integrability;modified simple equation method


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Mohammed O Al-Amr, Hadi Rezazadeh, Khalid K Ali, Alper Korkmazki. N1-soliton solution for Schrödinger equation with competing weakly nonlocal and parabolic law nonlinearities. Communications in Theoretical Physics, 2020, 72(6): 065503- doi:10.1088/1572-9494/ab8a12

1. Introduction

The dimensionless form of the Schrödinger equation with nonlocal and cubic–quintic nonlinearities with spatio-temporal dispersion is given by(1)$\begin{eqnarray}\begin{array}{l}{{\rm{i}}{q}}_{t}+{\rho }_{1}{q}_{{xx}}+{\rho }_{2}{q}_{{xt}}\\ \,+\left({b}_{1}{\left|q\right|}^{2}+{b}_{2}\,{\left|q\right|}^{4}\right)q\\ \,+{b}_{3}{\left({\left|q\right|}^{2}\right)}_{{xx}}q=0,\end{array}\end{eqnarray}$where q=q(x, t) is a complex-valued unknown function with slowly changing amplitude, ρ_i (i=1, 2) are the parameters with ρ₁ being a diffraction coefficient, b_j (j=1, 2, 3) are constant coefficients, and subscripts denote partial derivatives [1, 2]. The coefficient b₃ denotes nonlocal weak nonlinearity as b₁ and b₂ are cubic and quintic nonlinear term coefficients, respectively [3]. ρ₂ represents the coefficient of spatio-temporal dispersion [2]. This term is added to the model in order to satisfy the well-posedness against standard group-velocity dispersion [2].

So far various techniques have been proposed to set solutions in different function forms covering exponential, rational, hyperbolic functions and periodic and soliton solutions [4–32]. Many of these solutions are, in fact, based on predicting a solution with some unknown parameters and determining them by substitution into the governing model. Without any doubt, the $\tanh (.)$ method [33] is one of ancestors of all of those predicted solution approaches. Extended or generalized forms of this method were also used to solve many problems arising in various fields in engineering or in natural sciences [34, 35]. In the subsequent years, lots of smart methods have been derived. Most of them are based on some simple ordinary differential equations (ODEs), their structures and solutions. Various forms of Kudryashov’s approach have been used to solve many model equations [36–38]. Some methods are based on Riccati equations [39, 40]. The (G′/G)-expansion approach is also constructed on some ODE of second order [41–43]. Simple equation methods are also derived for some simple ODEs and the relations among derivative terms, dependent variables and their solutions [44, 45, 46, 47].

In the present paper, we derive a modified form of the simple equation approach that has many solutions in different forms. In the first stage, a compatible wave transform is implemented to reduce the governing equation to some ODE. Since the governing equation is in complex form, the resulting ODE is also derived in complex form. The constraints among the coefficients reduce the resulting system derived from imaginary and real parts of the equation to a unique ODE. In the final stage, the predicted solution is substituted into the resulting ODE after determination of its degree by the homogeneous balance routine. Many kinds of solutions are constructed in various forms covering hyperbolic and exponential function forms.

2. Mathematical Analysis

In order to solve the model, the following hypothesis is selected:(2)$\begin{eqnarray}q(x,t)=P(\eta ){{\rm{e}}}^{{\rm{i}}{\rm{\Phi }}\left(x,t\right)},\end{eqnarray}$where P(η) represents the shape of the pulse and(3)$\begin{eqnarray}\eta =x-{vt},\end{eqnarray}$and the phase component is defined as(4)$\begin{eqnarray}{\rm{\Phi }}\left(x,t\right)=-\kappa x+\omega t+{\theta }_{0},\end{eqnarray}$where κ, ω and θ₀ represent the frequency, the wavenumber and the phase constant respectively.

Substituting (2) into equation (1) and decomposing into real and imaginary parts gives(5)$\begin{eqnarray}\begin{array}{l}\left({\rho }_{1}-v{\rho }_{2}\right)P^{\prime\prime} -(\omega +{\rho }_{1}{\kappa }^{2}-\kappa \omega {\rho }_{2})P+{b}_{1}{P}^{3}\\ \,+\,{b}_{2}{P}^{5}+2{b}_{3}\left\{P{\left(P^{\prime} \right)}^{2}+{P}^{2}P^{\prime\prime} \right\}=0,\end{array}\end{eqnarray}$and(6)$\begin{eqnarray}v-{\rho }_{2}\left(\kappa v+\omega \right)+2{\rho }_{1}\kappa =0.\end{eqnarray}$From (6), setting the coefficients of the linearly independent functions to zero gives the speed of the soliton as(7)$\begin{eqnarray}v=\displaystyle \frac{{\rho }_{2}\omega -2{\rho }_{1}\kappa }{1-{\rho }_{2}\kappa }.\end{eqnarray}$There is a constraint(8)$\begin{eqnarray}{\rho }_{2}\kappa \ne 1.\end{eqnarray}$By applying equation (7) in equation (5), we get(9)$\begin{eqnarray}\begin{array}{l}\left({\rho }_{1}\,M-\left({\rho }_{2}\omega -2{\rho }_{1}\kappa \right){\rho }_{2}\right)P^{\prime\prime} \\ \,-\,M(\omega +{\rho }_{1}{\kappa }^{2}-\kappa \omega {\rho }_{2})P+{b}_{1}\,M\,{P}^{3}\\ \,+\,{b}_{2}\,M\,{P}^{5}+2{b}_{3}\,M\left\{P{\left(P^{\prime} \right)}^{2}+{P}^{2}P^{\prime\prime} \right\}=0,\end{array}\end{eqnarray}$where$\begin{eqnarray*}M=1-{\rho }_{2}\kappa .\end{eqnarray*}$Balancing P⁵ with P²P″ in equation (9), then we get N = 1.

2.1. Application of modified simple equation method

In order to handle equation (9) by means of the modified simple equation method, we assume that equation (9) possesses the following formal solution:(10)$\begin{eqnarray}P(\eta )={A}_{0}+{A}_{1}\left(\displaystyle \frac{\psi ^{\prime} }{\psi }\right),\end{eqnarray}$where A₀ and A₁ are constants such that ${A}_{1}\ne 0$ , and P(η) is an unknown function to be identified. One can easily obtain(11)$\begin{eqnarray}P^{\prime} ={A}_{1}\left[\displaystyle \frac{\psi ^{\prime\prime} }{\psi }-{\left(\displaystyle \frac{\psi ^{\prime} }{\psi }\right)}^{2}\right],\end{eqnarray}$(12)$\begin{eqnarray}P^{\prime\prime} ={A}_{1}\left[\displaystyle \frac{\psi \prime\prime\prime }{\psi }-3\displaystyle \frac{\psi ^{\prime} \psi ^{\prime\prime} }{{\psi }^{2}}+2{\left(\displaystyle \frac{\psi ^{\prime} }{\psi }\right)}^{3}\right].\end{eqnarray}$By substituting equations (10)–(12) into equation (9) and equating the coefficients of ${\psi }^{0},{\psi }^{-1},{\psi }^{-2},{\psi }^{-3},{\psi }^{-4}$ and ψ⁻⁵ to zero, we get(13)$\begin{eqnarray}-M\left({k}^{2}{\rho }_{1}-k\omega {\rho }_{2}+\omega \right){A}_{0}+{b}_{1}{{MA}}_{0}^{3}+{b}_{2}{{MA}}_{0}^{5}=0,\end{eqnarray}$(14)$\begin{eqnarray}\begin{array}{l}-\left(\left(\left(-2{A}_{0}^{2}{b}_{3}-{\rho }_{1}\right)M-2k{\rho }_{1}{\rho }_{2}+\omega {\rho }_{2}^{2}\right)\psi \prime\prime\prime \right.\\ \left.+\,M\psi ^{\prime} \left(-5{A}_{0}^{4}{b}_{2}+{k}^{2}{\rho }_{1}-k\omega {\rho }_{2}-3{A}_{0}^{2}{b}_{1}+\omega \right)\right){A}_{1}=0,\end{array}\end{eqnarray}$(15)$\begin{eqnarray}\begin{array}{l}-3{A}_{1}\left(\displaystyle \frac{-4}{3}M\psi \prime\prime\prime \psi ^{\prime} {A}_{0}{A}_{1}{b}_{3}-\displaystyle \frac{2}{3}M{\left(\psi ^{\prime\prime} \right)}^{2}{A}_{0}{A}_{1}{b}_{3}\right.\\ \,+\,\left(\left(2{A}_{0}^{2}{b}_{3}+{\rho }_{1}\right)M+2k{\rho }_{1}{\rho }_{2}-\omega {\rho }_{2}^{2}\right)\psi ^{\prime} \psi ^{\prime\prime} \\ \left.\,-\,\displaystyle \frac{10}{3}{A}_{0}{A}_{1}{\left(\psi ^{\prime} \right)}^{2}M\left({A}_{0}^{2}{b}_{2}+\displaystyle \frac{3}{10}{b}_{1}\right)\right)=0,\end{array}\end{eqnarray}$(16)$\begin{eqnarray}\begin{array}{l}2\left(M\psi \prime\prime\prime \psi ^{\prime} {A}_{1}^{2}{b}_{3}+M{\left(\psi ^{\prime\prime} \right)}^{2}{A}_{1}^{2}{b}_{3}-8M\psi ^{\prime} \psi ^{\prime\prime} {A}_{0}{A}_{1}{b}_{3}\right.\\ \,+\,\left(\left(\left(5{A}_{0}^{2}{b}_{2}+\displaystyle \frac{1}{2}{b}_{1}\right){A}_{1}^{2}+2{A}_{0}^{2}{b}_{3}+{\rho }_{1}\right)M\right.\\ \,+\,\left.\left.2k{\rho }_{1}{\rho }_{2}-\omega {\rho }_{2}^{2}\right){\left(\psi ^{\prime} \right)}^{2}\Space{0ex}{3.4ex}{0ex}\right){A}_{1}\psi ^{\prime} =0,\end{array}\end{eqnarray}$(17)$\begin{eqnarray}5{A}_{1}^{2}M{\left(\psi ^{\prime} \right)}^{3}\left(-2\psi ^{\prime\prime} {A}_{1}{b}_{3}+{A}_{0}\psi ^{\prime} \left({A}_{1}^{2}{b}_{2}+2{b}_{3}\right)\right)=0,\end{eqnarray}$(18)$\begin{eqnarray}M{\left(\psi ^{\prime} \right)}^{5}{A}_{1}^{3}\left({A}_{1}^{2}{b}_{2}+6{b}_{3}\right)=0.\end{eqnarray}$From equations (13) and (18) we obtain(19)$\begin{eqnarray}\begin{array}{rcl}{A}_{0} & = & 0,\mp \displaystyle \frac{\sqrt{-{b}_{1}+\sqrt{\left(4{k}^{2}{\rho }_{1}-4k\omega {\rho }_{2}+4\omega \right){b}_{2}+{b}_{1}^{2}}}}{\sqrt{2{b}_{2}}},\\ & & \mp \displaystyle \frac{\sqrt{-{b}_{1}-\sqrt{\left(4{k}^{2}{\rho }_{1}-4k\omega {\rho }_{2}+4\omega \right){b}_{2}+{b}_{1}^{2}}}}{\sqrt{2{b}_{2}}},\end{array}\end{eqnarray}$and(20)$\begin{eqnarray}{A}_{1}=\mp \displaystyle \frac{\sqrt{-6{b}_{3}}}{\sqrt{{b}_{2}}}\end{eqnarray}$provided that ${b}_{2}\ne 0$.

From equations (14) and (17) we get(21)$\begin{eqnarray}\begin{array}{rcl}\psi ^{\prime} & = & \displaystyle \frac{2{A}_{1}{b}_{3}}{{A}_{0}\left({A}_{1}^{2}{b}_{2}+2{b}_{3}\right)}\\ \psi ^{\prime\prime} & = & \displaystyle \frac{2{{MA}}_{0}^{2}{b}_{3}+2k{\rho }_{1}{\rho }_{2}-\omega {\rho }_{2}^{2}+M{\rho }_{1}}{M\left(-5{A}_{0}^{4}{b}_{2}+{\rho }_{1}{k}^{2}-k\omega {\rho }_{2}-3{A}_{0}^{2}{b}_{1}+\omega \right)}\psi \prime\prime\prime ,\end{array}\end{eqnarray}$so that(22)$\begin{eqnarray}\displaystyle \frac{\psi \prime\prime\prime }{\psi ^{\prime\prime} }=\displaystyle \frac{2M\left(-5{A}_{0}^{4}{b}_{2}+{\rho }_{1}{k}^{2}-k\omega {\rho }_{2}-3{A}_{0}^{2}{b}_{1}+\omega \right){A}_{1}{b}_{3}}{\left(2{{MA}}_{0}^{2}{b}_{3}+2k{\rho }_{1}{\rho }_{2}-\omega {\rho }_{2}^{2}+M{\rho }_{1}\right){A}_{0}\left({A}_{1}^{2}{b}_{2}+2{b}_{3}\right)}.\end{eqnarray}$Integrating equation (22) with respect to η gives(23)$\begin{eqnarray}\begin{array}{l}\psi ^{\prime\prime} (\eta )={c}_{1}\\ \times \exp \left(\displaystyle \frac{2M\left(-5{A}_{0}^{4}{b}_{2}+{\rho }_{1}{k}^{2}-k\omega {\rho }_{2}-3{A}_{0}^{2}{b}_{1}+\omega \right){A}_{1}{b}_{3}}{\left(2{{MA}}_{0}^{2}{b}_{3}+2k{\rho }_{1}{\rho }_{2}-\omega {\rho }_{2}^{2}+M{\rho }_{1}\right){A}_{0}\left({A}_{1}^{2}{b}_{2}+2{b}_{3}\right)}\right).\end{array}\end{eqnarray}$By using equation (21), we obtain(24)$\begin{eqnarray}\begin{array}{l}\psi ^{\prime} (\eta )=\displaystyle \frac{2{A}_{1}{b}_{3}}{{A}_{0}\left({A}_{1}^{2}{b}_{2}+2{b}_{3}\right)}\times {c}_{1}\\ \times \exp \left(\displaystyle \frac{2M\left(-5{A}_{0}^{4}{b}_{2}+{\rho }_{1}{k}^{2}-k\omega {\rho }_{2}-3{A}_{0}^{2}{b}_{1}+\omega \right){A}_{1}{b}_{3}}{\left(2{{MA}}_{0}^{2}{b}_{3}+2k{\rho }_{1}{\rho }_{2}-\omega {\rho }_{2}^{2}+M{\rho }_{1}\right){A}_{0}\left({A}_{1}^{2}{b}_{2}+2{b}_{3}\right)}\right).\end{array}\end{eqnarray}$Integrating equation (24) gives(25)$\begin{eqnarray}\begin{array}{l}\psi (\eta )={c}_{2}+\displaystyle \frac{2{{MA}}_{0}^{2}{b}_{3}+\left(2k{\rho }_{2}+M\right){\rho }_{1}-\omega {\rho }_{2}^{2}}{M\left(-5{A}_{0}^{4}{b}_{2}+{\rho }_{1}{k}^{2}-k\omega {\rho }_{2}-3{A}_{0}^{2}{b}_{1}+\omega \right)}\\ \times {c}_{1}\exp \left(\displaystyle \frac{2M\left(-5{A}_{0}^{4}{b}_{2}+{\rho }_{1}{k}^{2}-k\omega {\rho }_{2}-3{A}_{0}^{2}{b}_{1}+\omega \right){A}_{1}{b}_{3}}{\left(2{{MA}}_{0}^{2}{b}_{3}+2k{\rho }_{1}{\rho }_{2}-\omega {\rho }_{2}^{2}+M{\rho }_{1}\right){A}_{0}\left({A}_{1}^{2}{b}_{2}+2{b}_{3}\right)}\right).\end{array}\,\end{eqnarray}$

Accordingly, we have the following cases:

Case 1. If A₀=0 and ${A}_{1}=\mp \tfrac{\sqrt{-6{b}_{3}}}{\sqrt{{b}_{2}}}$. We obtain a trivial solution. Hence, this case is refused.

Case 2. If ${A}_{0}=\mp \tfrac{\sqrt{-{b}_{1}+\sqrt{\left(4{k}^{2}{\rho }_{1}-4k\omega {\rho }_{2}+4\omega \right){b}_{2}+{b}_{1}^{2}}}}{\sqrt{2{b}_{2}}}$ and ${A}_{1}=\mp \tfrac{\sqrt{-6{b}_{3}}}{\sqrt{{b}_{2}}}$. Substituting the values of ${A}_{0},{A}_{1},\psi (\eta )$ and $\psi ^{\prime} (\eta )$ into equation (10), the exact solution of equation (9) is as follows:

(26)$\begin{eqnarray}\begin{array}{rcl}P(\eta ) & = & \mp \displaystyle \frac{\sqrt{-{b}_{1}+\delta }}{\sqrt{2{b}_{2}}}\pm \displaystyle \frac{3\sqrt{2}{b}_{3}{c}_{1}}{\sqrt{{b}_{2}(-{b}_{1}+\delta )}}\\ & & \times \exp \left(\pm \displaystyle \frac{\delta \,\sqrt{-3\,{b}_{3}\left(-{b}_{1}+\delta \right)}M\eta }{M\left(-{b}_{1}+\delta \right){b}_{3}+\lambda \,{b}_{2}}\right)\\ & & \times \left({c}_{2}+\displaystyle \frac{\left(\lambda \,{b}_{2}+\left(-{b}_{1}+\delta \right){b}_{3}M\right){c}_{1}}{\left({b}_{1}\delta -{\delta }^{2}\right)M}\right.\\ & & {\left.\times \exp \left(\pm \displaystyle \frac{\delta \sqrt{-3{b}_{3}\left(-{b}_{1}+\delta \right)}M\eta }{M\left(-{b}_{1}+\delta \right){b}_{3}+\lambda {b}_{2}}\right)\right)}^{-1},\end{array}\end{eqnarray}$where $\delta =\sqrt{\left(4{k}^{2}{\rho }_{1}-4k\omega {\rho }_{2}+4\omega \right){b}_{2}+{b}_{1}^{2}}$ and $\lambda \,=2\,k{\rho }_{1}{\rho }_{2}-\omega \,{\rho }_{2}^{2}+M{\rho }_{1}$.

The parameters c₁ and c₂ could be randomly chosen. By setting ${c}_{2}=\tfrac{\lambda \,{b}_{2}+\left(-{b}_{1}+\delta \right){b}_{3}M}{\left({b}_{1}\delta -{\delta }^{2}\right)M}{c}_{1}$, we obtain(27)$\begin{eqnarray}\begin{array}{rcl}{P}_{1}(\eta ) & = & \mp \displaystyle \frac{\sqrt{-{b}_{1}+\delta }}{\sqrt{2{b}_{2}}}-\,\displaystyle \frac{3{b}_{3}\delta \,\sqrt{-2\,{b}_{1}+2\,\delta }M}{\sqrt{{b}_{2}}\left(2\left(-{b}_{1}+\delta \right){b}_{3}M+2\,\lambda \,{b}_{2}\right)}\\ & & \times \left(1\pm \tanh \left(\displaystyle \frac{\delta \,\sqrt{-3\,{b}_{3}\left(-{b}_{1}+\delta \right)}M\eta }{2M\left(-{b}_{1}+\delta \right){b}_{3}+2\lambda \,{b}_{2}}+{\eta }_{0}\right)\right),\end{array}\end{eqnarray}$where η₀ is an arbitrary constant. Accordingly, we have the following dark 1-soliton solution:(28)$\begin{eqnarray}\begin{array}{l}q(x,t)=\left\{\mp \displaystyle \frac{\sqrt{-{b}_{1}+\delta }}{\sqrt{2{b}_{2}}}\right.\\ \,-\,\displaystyle \frac{3{b}_{3}\delta \,\sqrt{-2\,{b}_{1}+2\,\delta }M}{\sqrt{{b}_{2}}\left(2\left(-{b}_{1}+\delta \right){b}_{3}M+2\,\lambda \,{b}_{2}\right)}\\ \left.\,\times \left(1\pm \tanh \left(\displaystyle \frac{\delta \,\sqrt{-3\,{b}_{3}\left(-{b}_{1}+\delta \right)}M\eta }{2M\left(-{b}_{1}+\delta \right){b}_{3}+2\lambda \,{b}_{2}}+{\eta }_{0}\right)\right)\right\}\\ \,\times {{\rm{e}}}^{{\rm{i}}\left(-\kappa x+\omega t+{\theta }_{0}\right)}.\end{array}\end{eqnarray}$Similarly, by setting ${c}_{2}=-\tfrac{\lambda \,{b}_{2}+\left(-{b}_{1}+\delta \right){b}_{3}M}{\left({b}_{1}\delta -{\delta }^{2}\right)M}{c}_{1}$, we obtain(29)$\begin{eqnarray}\begin{array}{rcl}{P}_{2}(\eta ) & = & \mp \displaystyle \frac{\sqrt{-{b}_{1}+\delta }}{\sqrt{2{b}_{2}}}-\,\displaystyle \frac{3{b}_{3}\delta \,\sqrt{-2\,{b}_{1}+2\,\delta }M}{\sqrt{{b}_{2}}\left(2\left(-{b}_{1}+\delta \right){b}_{3}M+2\,\lambda \,{b}_{2}\right)}\\ & & \times \left(1\pm \coth \left(\displaystyle \frac{\delta \,\sqrt{-3\,{b}_{3}\left(-{b}_{1}+\delta \right)}M\eta }{2M\left(-{b}_{1}+\delta \right){b}_{3}+2\lambda \,{b}_{2}}+{\eta }_{0}\right)\right),\end{array}\end{eqnarray}$where η₀ is an arbitrary constant. Therefore, we obtain the following singular 1-soliton solution:(30)$\begin{eqnarray}\begin{array}{l}q(x,t)=\left\{\mp \displaystyle \frac{\sqrt{-{b}_{1}+\delta }}{\sqrt{2{b}_{2}}}\right.\\ \,-\,\displaystyle \frac{3{b}_{3}\delta \,\sqrt{-2\,{b}_{1}+2\,\delta }M}{\sqrt{{b}_{2}}\left(2\left(-{b}_{1}+\delta \right){b}_{3}M+2\,\lambda \,{b}_{2}\right)}\\ \left.\,\times \left(1\pm \coth \left(\displaystyle \frac{\delta \,\sqrt{-3\,{b}_{3}\left(-{b}_{1}+\delta \right)}M\eta }{2M\left(-{b}_{1}+\delta \right){b}_{3}+2\lambda \,{b}_{2}}+{\eta }_{0}\right)\right)\right\}\\ \,\times {{\rm{e}}}^{{\rm{i}}\left(-\kappa x+\omega t+{\theta }_{0}\right)}.\end{array}\end{eqnarray}$

Case 3. If ${A}_{0}=\pm \tfrac{\sqrt{-{b}_{1}-\sqrt{\left(4{k}^{2}{\rho }_{1}-4k\omega {\rho }_{2}+4\omega \right){b}_{2}+{b}_{1}^{2}}}}{\sqrt{2{b}_{2}}}$ and ${A}_{1}=\mp \tfrac{\sqrt{-6{b}_{3}}}{\sqrt{{b}_{2}}}$. Substituting the values of ${A}_{0},{A}_{1},\psi (\eta )$ and $\psi ^{\prime} (\eta )$ into equation (10), the exact solution of equation (9) is as follows:

(31)$\begin{eqnarray}\begin{array}{rcl}P(\eta ) & = & \mp \displaystyle \frac{\sqrt{-{b}_{1}-\delta }}{\sqrt{2{b}_{2}}}\pm \displaystyle \frac{3\sqrt{2}{b}_{3}{c}_{1}}{\sqrt{-{b}_{2}({b}_{1}+\delta )}}\\ & & \times \exp \left(\pm \displaystyle \frac{\delta \,\sqrt{3\,{b}_{3}\left({b}_{1}+\delta \right)}M\eta }{-M\left({b}_{1}+\delta \right){b}_{3}+\lambda \,{b}_{2}}\right)\\ & & \times \left({c}_{2}+\displaystyle \frac{\left(\lambda \,{b}_{2}-\left({b}_{1}+\delta \right){b}_{3}M\right){c}_{1}}{-\left({b}_{1}\delta +{\delta }^{2}\right)M}\right.\\ & & {\left.\times \exp \left(\pm \displaystyle \frac{\delta \sqrt{3{b}_{3}\left({b}_{1}+\delta \right)}M\eta }{-M\left({b}_{1}+\delta \right){b}_{3}+\lambda {b}_{2}}\right)\right)}^{-1},\end{array}\end{eqnarray}$where $\delta =\sqrt{\left(4{k}^{2}{\rho }_{1}-4k\omega {\rho }_{2}+4\omega \right){b}_{2}+{b}_{1}^{2}}$ and $\lambda =2\,k{\rho }_{1}{\rho }_{2}\,-\omega \,{\rho }_{2}^{2}+M{\rho }_{1}$.

The parameters c₁ and c₂ could be randomly chosen. By setting ${c}_{2}=\tfrac{\lambda \,{b}_{2}-\left({b}_{1}+\delta \right){b}_{3}M}{-\left({b}_{1}\delta +{\delta }^{2}\right)M}{c}_{1}$, we obtain(32)$\begin{eqnarray}\begin{array}{rcl}{P}_{1}(\eta ) & = & \mp \displaystyle \frac{\sqrt{-{b}_{1}-\delta }}{\sqrt{2{b}_{2}}}+\,\displaystyle \frac{3{b}_{3}\delta \,\sqrt{-2\,{b}_{1}-2\,\delta }M}{\sqrt{{b}_{2}}\left(2\left(-{b}_{1}-\delta \right){b}_{3}M+2\,\lambda \,{b}_{2}\right)}\\ & & \times \left(1\pm \tanh \left(\displaystyle \frac{\delta \,\sqrt{3\,{b}_{3}\left({b}_{1}+\delta \right)}M\eta }{-2M\left({b}_{1}+\delta \right){b}_{3}+2\lambda \,{b}_{2}}\right)\right),\end{array}\end{eqnarray}$where η₀ is an arbitrary constant. Accordingly, we have the following dark 1-soliton solution:(33)$\begin{eqnarray}\begin{array}{rcl}q(x,t) & = & \left\{\mp \displaystyle \frac{\sqrt{-{b}_{1}-\delta }}{\sqrt{2{b}_{2}}}\right.\\ & & +\,\displaystyle \frac{3{b}_{3}\delta \,\sqrt{-2\,{b}_{1}-2\,\delta }M}{\sqrt{{b}_{2}}\left(2\left(-{b}_{1}-\delta \right){b}_{3}M+2\,\lambda \,{b}_{2}\right)}\\ & & \left.\times \left(1\pm \tanh \left(\displaystyle \frac{\delta \,\sqrt{3\,{b}_{3}\left({b}_{1}+\delta \right)}M\eta }{-2M\left({b}_{1}+\delta \right){b}_{3}+2\lambda \,{b}_{2}}\right)\right)\right\}\\ & & \times {{\rm{e}}}^{{\rm{i}}\left(-\kappa x+\omega t+{\theta }_{0}\right)}.\end{array}\end{eqnarray}$Similarly, by setting ${c}_{2}=-\tfrac{\lambda \,{b}_{2}-\left({b}_{1}+\delta \right){b}_{3}M}{-\left({b}_{1}\delta +{\delta }^{2}\right)M}{c}_{1}$, we obtain(34)$\begin{eqnarray}\begin{array}{rcl}{P}_{2}(\eta ) & = & \mp \displaystyle \frac{\sqrt{-{b}_{1}-\delta }}{\sqrt{2{b}_{2}}}+\,\displaystyle \frac{3{b}_{3}\delta \,\sqrt{-2\,{b}_{1}-2\,\delta }M}{\sqrt{{b}_{2}}\left(2\left(-{b}_{1}-\delta \right){b}_{3}M+2\,\lambda \,{b}_{2}\right)}\\ & & \times \left(1\pm \coth \left(\displaystyle \frac{\delta \,\sqrt{3\,{b}_{3}\left({b}_{1}+\delta \right)}M\eta }{-2M\left({b}_{1}+\delta \right){b}_{3}+2\lambda \,{b}_{2}}\right)\right),\end{array}\end{eqnarray}$where η₀ is an arbitrary constant. Therefore, we obtain the following singular 1-soliton solution:(35)$\begin{eqnarray}\begin{array}{rcl}q(x,t) & = & \left\{\mp \displaystyle \frac{\sqrt{-{b}_{1}-\delta }}{\sqrt{2{b}_{2}}}\right.\\ & & +\,\displaystyle \frac{3{b}_{3}\delta \,\sqrt{-2\,{b}_{1}-2\,\delta }M}{\sqrt{{b}_{2}}\left(2\left(-{b}_{1}-\delta \right){b}_{3}M+2\,\lambda \,{b}_{2}\right)}\\ & & \left.\times \left(1\pm \coth \left(\displaystyle \frac{\delta \,\sqrt{3\,{b}_{3}\left({b}_{1}+\delta \right)}M\eta }{-2M\left({b}_{1}+\delta \right){b}_{3}+2\lambda \,{b}_{2}}\right)\right)\right\}\\ & & \times {{\rm{e}}}^{{\rm{i}}\left(-\kappa x+\omega t+{\theta }_{0}\right)}.\end{array}\end{eqnarray}$Note that the wave variable is $\eta =x-\left(\tfrac{{\rho }_{2}\omega -2{\rho }_{1}\kappa }{1-{\rho }_{2}\kappa }\right)t$ in all the above analytical solutions from cases 2 and 3.

3. Some graphical illustrations

In this section, we give some graphical illustrations of the acquired solutions of our equations with the aid of Mathematica. The two-dimensional and three-dimensional plots of certain solutions are presented in figures 1–4.

4. Results and discussion

One of the most powerful integration schemes has been used in this article to construct some new dark and singular 1-soliton solutions of the nonlocal nonlinear Schrödinger equation (NNLSE) with competing weakly nonlocal nonlinearity and parabolic law nonlinearity. By comparing our outcomes with those obtained by some existing methods in the previous literature [1–3], one can observe that this approach has successfully constructed new complex wave solutions that have not been accounted for earlier. All of these solutions are verified by direct substitution via the commercial software Mathematica. The geometrical shapes (numerical simulations) for certain solutions are depicted in figures 1–4 at particular values of the free parameters; they help us to understand the complex physical phenomena of the dynamical model under consideration. Numerical solvers can count on the constructed solutions to validate their solutions as well as to study the stability analysis. The N1-soliton solutions (28), (30), (33) and (35) signify solitary waves that maintain their identity upon interaction. Figures 1 and 2 elaborate both cases of the dark 1-soliton solutions (28) in two- and three-dimensional shapes by taking some suitable choices of the parameters. In figure 3, the graphical shapes of the singular 1-soliton solution (30) are shown in two and three dimensions at specific values of the parameters. The dark 1-soliton solution (33) is depicted in figure 4 in two- and three-dimensional shapes by taking suitable values of the free parameters.

Figure 1.

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Figure 1.Graph of case 2 for (28) at b₂ = 1, δ = 1, κ = 0.2, λ = 0.1, b₁ = 0.1, ν = 0.1, θ₀ = 5, M = 0.2, ω = 0.1, b₃ = 0.2.

Figure 2.

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Figure 2.Graph of case 2 for (30) at b₂ = 1, δ = 1, κ = 0.2, λ = 0.1, b₁ = 0.1, ν = 0.1, θ₀ = 5, M = 0.2, ω = 0.1, b₃ = 0.2.

Figure 3.

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Figure 3.Graph of case 3 for (33) at b₂ = 1, δ = 1, κ = 0.2, λ = 0.1, b₁ = 0.1, ν = 0.1, θ₀ = 5, M = 0.2, ω = 0.1, b₃ = 0.2.

Figure 4.

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Figure 4.Graph of case 3 for (35) at b₂ = 2, δ = 0.1, κ = 0.1, λ = 0.01, b₁ = 0.1, ν = 0.1, θ₀ = 0.5, M = 0.1, ω = 0.1, b₃ = 0.2.

5. Conclusions

In this work, the NNLSE with competing weakly nonlocal nonlinearity and parabolic law nonlinearity is studied using the modified simple equation method. It should be noted that all the obtained solutions in the current work are dark and singular 1-soliton solutions. This strong integration approach is very effective when other schemes fail to find soliton solutions. Thus, this methodology will be studied in future for various other situations in various types of NNLSEs.

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