On the traveling wave solutions to the complex nonlinear second-order modified unstable (1+1)-dimensional Schrödinger equation

It is important to keep in mind that semi-analytic solutions to non-linear partial differential equations fulfill a crucial role in enlightening an extensive selection of unusual and complicated features that are expressed in a range of applied scientific domains. Investigators from all around the globe are attempting to employ these procedures, as more partial differential equations with applications in physics, engineering, and biology have recently been solved using these methods. An additional contention posits that by augmenting the models with computations, these methods assure readers that they generate apt and accurate affirmations. The numerous evolution methods are applied for instance the modified tanh – function and the extended rational sinh – cosh function methods [1], the modified exponential function method [2], in innumerable scientific and mechanical fields, including physical engineering [3], health care biology [4], imagery and digital signal processing [5], and so on. The nonlinear partial differential equations (NLPDEs) are essential and indispensable, excluding the above quotations, Bernoulli sub-equation function methods, the modified exponential rational function method, and two variable (G^′/G, 1/G) — expansion technique [6,7,8,9] have been operated.

The complex dynamical model studied in this research is the modified unstable nonlinear (1+1)-dimensional Schrödinger model. The investigated equation has the subsequent form: (1) −γ∂2ux,t∂x ∂t+∂2ux,t∂x ∂x+i∂ux,t∂t+2λux,t|ux,t|2=0, i=−1, \matrix{{- \gamma {{{\partial^2}u\left({x,t} \right)} \over {\partial x\;\partial t}} + {{{\partial^2}u\left({x,t} \right)} \over {\partial x\;\partial x}} + i{{\partial u\left({x,t} \right)} \over {\partial t}} + 2\lambda u\left({x,t} \right)|u\left({x,t} \right){|^2} = 0,\;\;i = \sqrt {- 1},} \hfill \cr} where γ, λ are non zero parameters to be determined later. As the studied model regulates some instabilities of modified wave-trains, it is a general equation of the family of nonlinear integrable systems in nonlinear Schrödinger system of equations. This equation also describes the temporal evolution of perturbations in mediums that are nominally stable or unstable. The term γ∂2u∂x ∂t \gamma {{{\partial^2}u} \over {\partial x\;\partial t}} dominates the ill-posed-ness of this dynamic model. In order to provide a comprehensive response on the usefulness of the research, we may assert the following: The Schrödinger equation is a fundamental principle in the field of quantum mechanics. Quantum dynamics is the study of how the quantum state of a physical system evolves over time, offering a conceptual structure for comprehending the actions of particles at the atomic and subatomic scales. The solutions to the Schrödinger equation enable physicists to forecast the characteristics and actions of particles, including their energy levels, probability of locating them in certain places, and their interactions with other particles and fields.

Many experts have conducted investigations into the various interpretations of the dynamical model (1) employing more methodological approaches, which include modified extended mapping technique [10], the (G^′/G)-expansion technique [11], the rational-expansion approach on Jacobi elliptic function [12], the modified version of the Kudryashov method and the expansion mod of the sine-Gordon approach [13], and so on.

Here, we try to give some specific semi-analytic methods that play a major role in solving mathematical, physical, and engineering models, such as: Laplace transformation method has been applied to the features of the Caputo-fractional derivatives [14], the improved version of Bernoulli combined with the methods related to the hyperbolic trigonometric functions [15,16,17], the new extended direct algebraic method employed [18], the sine-Gordon expansion method and its rational version have been applied [19], the development version of the ultra-spherical wavelet method utilized [20], employed a Lie symmetry analysis [21], the Hirota's bilinear method is considered [22], describing auxiliary equation approach [23], the bilinear method [24], the Hirota bilinear method and Bell polynomials [25, 26], if you are seeking a unique form of solution that logically is localized in every possible direction, semi-analytic solutions and their interactions are an amazing place to start [27,28,29,30]. The utilized method is trustworthy, dependable, and more practical for solving different mathematical models. To provide evidence of the prior declaration, one can inspect how many researchers applied this method and other different performances, herein are some sources [31,32,33,34,35,36,37,38], diversified arrangements for multi-dimensional space Schrödinger equation became an attractive model to the researcher for investigation [39,40,41,42].

The subsequent structure of the keeping article is as outlined below: The literature about the dynamical model is discussed in Section 1. The construction of the suggested approach is briefly presented in Section 2. The recommended technique is implemented on the provided model in Section 3, and the resulting solutions are visually shown. Discussing the outcomes and physical meaning of the illustrated figures are demonstrated in Section 4. Ultimately, the concluding views are expressed in the last part Section 5.

The improved Bernoulli sub-equation function method (IBSEFM) is presented briefly in the subsequent steps:

Step 1. We consider the general form of NLPDEs given as (2) PE,Ex,Et,Ext,Exx,⋯=0, {\cal P}\left({{\cal E},{{\cal E}_x},{{\cal E}_t},{{\cal E}_{xt}},{{\cal E}_{xx}}, \cdots} \right) = 0, wherein ℰ = ℰ (x,t). Setting (3) Ex,t=UL, L=δ1x−δ2t, {\cal E}\left({x,t} \right) = {\cal U}\left({\cal L} \right),\;{\cal L} = {\delta_1}x - {\delta_2}t, here the constants δ_i for i = 1, 2 are arbitrary parameters. Upon substituting equation (3) into equation (2), the resulting expression is as follows: (4) NU,U',U'',⋯=0, {\cal N}\left({{\cal U},{\cal U}{\rm{'}},{\cal U}{\rm{''}}, \cdots} \right) = 0, where U=UL, U'=dUdL, U''=d2UdL2,⋯. {\cal U} = {\cal U}\left({\cal L} \right),\;{\cal U}{\rm{'}} = {{d{\cal U}} \over {d{\cal L}}},\;{\cal U}{\rm{''}} = {{{d^2}{\cal U}} \over {d{{\cal L}^2}}}, \cdots.

Step 2. Let us consider that the solution of equation (4) might be expressed in the following form: (5) UL=ω0+ω1T+ω2T2+⋯+ωnTnν0+ν1T+ν2T2+⋯+νmTm⋅ \matrix{{{\cal U}\left({\cal L} \right) = {{{\omega_0} + {\omega_1}{\cal T} + {\omega_2}{{\cal T}^2} + \cdots + {\omega_n}{{\cal T}^n}} \over {{\nu_0} + {\nu_1}{\cal T} + {\nu_2}{{\cal T}^2} + \cdots + {\nu_m}{{\cal T}^m}}} \cdot} \hfill \cr} Here, must be both ω_m, ν_n are different from zero, and they should be determined afterwards. Also, 𝒯 (ℒ) satisfies Bernoulli ODE: (6) T'=αT+βTM, α≠0, β≠0, M∈ℝ−0,1. {\cal T}{\rm{'}} = \alpha {\cal T} + \beta {{\cal T}^{\cal M}},\;\alpha \ne 0,\;\beta \ne 0,\;{\cal M} \in {\mathbb R} - \left\{ {0,1} \right\}.

By combining equation (5) with equation (6) and inserting in equation (4), the following result is obtained: (7) ΦTL=εsT(L)s+⋯+ε1TL+ε0=0. {\rm{\Phi}}\left({{\cal T}\left({\cal L} \right)} \right) = {\varepsilon_s}{\cal T}{({\cal L})^s} + \cdots + {\varepsilon_1}{\cal T}\left({\cal L} \right) + {\varepsilon_0} = 0. Utilizing the concepts of balance, one can come up with a formula for n, m, and M by comparing the highest-order derivatives with the terms of the highest degree in equation (4).

Step 3. When the coefficients of Φ(𝒯 (ℒ)) were set to zero, the following algebraic system of equations was produced: (8) εi=0, i=0,⋯,s. {\varepsilon_i} = 0,\;\;i = 0, \cdots,s.

The values of α, β, ν₀,…, ν_n, ω₀,…, ω_m, and so solutions of (4) are discovered if one knows how to solve equation (6). These values are obtained by solving the system (8).

Step 4. The commanded solutions for equation (6) have the following manners: (9) TK=−βα+CeαM−1K11−M, α≠β,TK=C−1+C+1tanhα1−MK21−tanhα1−MK211−M, α=β, C∈ℝ. \matrix{{{\cal T}\left( {\cal K} \right)} \hfill & {= {{\left[ {- {\beta \over \alpha} + {{\cal C} \over {{e^{\alpha \left( {{\cal M} - 1} \right){\cal K}}}}}} \right]}^{{1 \over {1 - {\cal M}}}}},\;\;\alpha \ne \beta ,} \hfill \cr {{\cal T}\left( {\cal K} \right)} \hfill & {= {{\left[ {{{\left( {{\cal C} - 1} \right) + \left( {{\cal C} + 1} \right){\rm{tanh}}\left( {{{\alpha \left( {1 - {\cal M}} \right){\cal K}} \over 2}} \right)} \over {1 - {\rm{tanh}}\left( {{{\alpha \left( {1 - {\cal M}} \right){\cal K}} \over 2}} \right)}}} \right]}^{{1 \over {1 - {\cal M}}}}},\;\;\alpha = \beta ,\;\;{\cal C} \in {\mathbb R}.} \hfill \cr}

Consider equation (1) has a solution in the form of (10) Eη=eiθVη, θ=β1x−β2t, η=δ1x−δ2t. {\cal E}\left(\eta \right) = {e^{i\theta}}V\left(\eta \right),\;\theta = {\beta_1}x - {\beta_2}t,\;\eta = {\delta_1}x - {\delta_2}t.

The following produces are straight away obtained by plugging wave transformation formula (10) into equation (1): (11) eiβ1x−β2t2λV3+δ1γδ2+δ1V''+β2V+iγδ1V'−β12V+eiβ1x−β2tβ1−β2γV+iγδ2+2δ1V'−iδ2V'=0. \matrix{{} \hfill & {{e^{i\left({{\beta_1}x - {\beta_2}t} \right)}}\left({2\lambda {V^3} + {\delta_1}\left({\gamma {\delta_2} + {\delta_1}} \right)V'' + {\beta_2}\left({V + i\gamma {\delta_1}V'} \right) - \beta_1^2V} \right)} \hfill \cr {} \hfill & {+ {e^{i\left({{\beta_1}x - {\beta_2}t} \right)}}\left({{\beta_1}\left({- {\beta_2}\gamma V + i\left({\gamma {\delta_2} + 2{\delta_1}} \right)V'} \right) - i{\delta_2}V'} \right) = 0.} \hfill \cr}

After reestablishing equation (11), it may be interpreted by spreading the real and complex components into two formulations: (12) 2λV3+γδ1δ2V''+δ12V''−β1β2γV−β12V+β2V=0, 2\lambda {V^3} + \gamma {\delta_1}{\delta_2}V'' + \delta_1^2V'' - {\beta_1}{\beta_2}\gamma V - \beta_1^2V + {\beta_2}V = 0, and (13) δ1β2γ+2β1+δ2β1γ−1V'=0, \left({{\delta_1}\left({{\beta_2}\gamma + 2{\beta_1}} \right) + {\delta_2}\left({{\beta_1}\gamma - 1} \right)} \right)V' = 0, from equation (13) one achieves: (14) δ1β2γ+2β1+δ2β1γ−1=0, {\delta_1}\left({{\beta_2}\gamma + 2{\beta_1}} \right) + {\delta_2}\left({{\beta_1}\gamma - 1} \right) = 0, the equation (14) is a selected equation to be added to the obtained systems of algebraic equations in all cases.

Considering the balancing principles connecting the highest order and a greater degree of the nonlinear component in equation (12) we can obtain an essential connection between n, m and M which is as follows: (15) n=M+m−1⋅ n = M + m - 1 \cdot For the recommendation of the values of positive integers in equation (15), we have the following cases:

Case 1. In equation (15), if m = 1 and M = 3, then n = 3, so equation (5) and equation (6) are taking the following forms: (16) Vη=ω0+ω1T+ω2T2+ω3T3ν0+ν1T, \matrix{{V\left(\eta \right) = {{{\omega_0} + {\omega_1}{\cal T} + {\omega_2}{{\cal T}^2} + {\omega_3}{{\cal T}^3}} \over {{\nu_0} + {\nu_1}{\cal T}}},} \hfill \cr} and (17) T'=αT+βT3, α≠0, β≠0, {\cal T}{\rm{'}} = \alpha {\cal T} + \beta {{\cal T}^3},\;\;\alpha \ne 0,\;\;\beta \ne 0, hence, (18) V'=αT+βT3ω1+2ω2T+3ω3T2ν0+ν1T−ν1ω0+ω1T+ω2T2+ω3T3ν0+ν1T2=ΨTΩT, \matrix{{V'} \hfill & {= \left({\alpha {\cal T} + \beta {{\cal T}^3}} \right)\left({{{{\omega_1} + 2{\omega_2}{\cal T} + 3{\omega_3}{{\cal T}^2}} \over {{\nu_0} + {\nu_1}{\cal T}}} - {{{\nu_1}\left({{\omega_0} + {\omega_1}{\cal T} + {\omega_2}{{\cal T}^2} + {\omega_3}{{\cal T}^3}} \right)} \over {{{\left({{\nu_0} + {\nu_1}{\cal T}} \right)}^2}}}} \right)} \hfill \cr {} \hfill & {= {{{\rm{\Psi}}\left({\cal T} \right)} \over {{\rm{\Omega}}\left({\cal T} \right)}},} \hfill \cr} notice that there should be at least ω₃ ≠ 0, ν₁ ≠ 0, one can reach this: (19) V''=ΩTΨ′T−ΨTΩ′T(ΩT)2. V'' = {{{\rm{\Omega}}\left({\cal T} \right){\rm{\Psi '}}\left({\cal T} \right) - {\rm{\Psi}}\left({\cal T} \right){\rm{\Omega '}}\left({\cal T} \right)} \over {{{({\rm{\Omega}}\left({\cal T} \right))}^2}}}. By lodging equations (16)–(19) into equation (12), we acquire the following algebraic system of equations by correlating to zero all the coefficients of the same powers of 𝒯ⁱ for i = 0, 1, 2, …. (20) Ci=0, where Ci is the coefficientof Ti, for anyi=0,⋯,10. {{\cal C}_i} = 0,\;\;{\rm{where}}\;\;{{\cal C}_i}\;\;{\rm{isthecoefficientof}}\;{{\cal T}^i},\;\;{\rm{forany}}i = 0, \cdots,10.

The following sub-cases are constructed from the solving of the system (20), where α ≠ β.

Case 1.1. The following is a set of the obtained parameters: (21) δ2=−ω32λ4ν12γβ2δ1−δ1γ,β2=2ω32λ−4ν12β2δ12ω32γ2δ12λω32α2γ2λ−2ν12β2ω32ν12β2δ12λ−8γ4γ3,ω0=0,ω1=ω3α2β,ω2=0,ν0=0,β1=2ω32γ2δ12λω32α2γ2λ−2ν12β2ω32λ+γγ2. \matrix{{{\delta_2} = - {{\omega_3^2\lambda} \over {4\nu_1^2\gamma {\beta^2}{\delta_1}}} - {{{\delta_1}} \over \gamma},{\beta_2} = {{{{\sqrt 2 \left({\omega_3^2\lambda - 4\nu_1^2{\beta^2}\delta_1^2} \right)\sqrt {\omega_3^2{\gamma^2}\delta_1^2\lambda \left({\omega_3^2{\alpha^2}{\gamma^2}\lambda - 2\nu_1^2{\beta^2}} \right)}} \over {\omega_3^2\nu_1^2{\beta^2}\delta_1^2\lambda}} - 8\gamma} \over {4{\gamma^3}}},} \hfill \cr {\matrix{{} \hfill & {} \hfill \cr {} \hfill & {{\omega_0} = 0,{\omega_1} = {{{\omega_3}\alpha} \over {2\beta}},{\omega_2} = 0,{\nu_0} = 0,{\beta_1} = {{{{\sqrt 2 \sqrt {\omega_3^2{\gamma^2}\delta_1^2\lambda \left({\omega_3^2{\alpha^2}{\gamma^2}\lambda - 2\nu_1^2{\beta^2}} \right)}} \over {\omega_3^2\lambda}} + \gamma} \over {{\gamma^2}}}.} \hfill \cr}} \hfill \cr} With the cited values of the parameters in equation (21), the following solution is attained where 𝒞 = 1: (22) u1,1x,t=ω32ν1β2αβαeα−ω32λtν12β2−4δ12t+γx2γδ1−β+αei2B4ν12β2δ12t+γx−ω32λtω32ν12β2δ12λ+4γ2t+γx4γ3, {u_{1,1}}\left({x,t} \right) = {{{\omega_3}} \over {2{\nu_1}\beta}}\left({{{2\alpha \beta} \over {\alpha {e^{\left({{{\alpha \left({- {{\omega_3^2\lambda t} \over {\nu_1^2{\beta^2}}} - 4\delta_1^2\left({t + \gamma x} \right)} \right)} \over {2\gamma {\delta_1}}}} \right) - \beta}}}} + \alpha} \right){e^{{{i\left({{{\sqrt 2 {\cal B}\left({4\nu_1^2{\beta^2}\delta_1^2\left({t + \gamma x} \right) - \omega_3^2\lambda t} \right)} \over {\omega_3^2\nu_1^2{\beta^2}\delta_1^2\lambda}} + 4\gamma \left({2t + \gamma x} \right)} \right)} \over {4{\gamma^3}}}}}, where B=ω32γ2δ12λω32α2γ2λ−2ν12β2. {\cal B} = \sqrt {\omega_3^2{\gamma^2}\delta_1^2\lambda \left({\omega_3^2{\alpha^2}{\gamma^2}\lambda - 2\nu_1^2{\beta^2}} \right)}. Profile of the obtained solution in equation (22) where γ=12 \gamma = {1 \over 2} , β=14 \beta = {1 \over 4} , α=−32 \alpha = - {3 \over 2} , λ=−23 \lambda = - {2 \over 3} , ν1=13 {\nu_1} = {1 \over 3} , ω3=53 {\omega_3} = {5 \over 3} , δ1=23 {\delta_1} = {2 \over 3} , and −10 ≤ x ≤ 10, −10 ≤ t ≤ 10 are graphed below

Case 1.2. Then following coefficients are obtained: (23) β2=−2δ1δ12α2γ2δ1γδ2+δ1+1+γδ2+δ1−γδ2δ12α2γ2δ1γδ2+δ1+1γ2δ1γδ2+δ1,β=ω3λ2−ν12δ1γδ2+δ1,ω0=αν0−ν12δ1γδ2+δ1ν1λ,ω1=α−ν12δ1γδ2+δ1λ,β1=δ12α2γ2δ1γδ2+δ1+1γδ2+δ1+1γ,ω2=ω3ν0ν1, \matrix{{{\beta_2} = {{- 2{\delta_1}\left({\sqrt {{\delta_1}\left({2{\alpha^2}{\gamma^2}{\delta_1}\left({\gamma {\delta_2} + {\delta_1}} \right) + 1} \right)} + \sqrt {\gamma {\delta_2} + {\delta_1}}} \right) - \gamma {\delta_2}\sqrt {{\delta_1}\left({2{\alpha^2}{\gamma^2}{\delta_1}\left({\gamma {\delta_2} + {\delta_1}} \right) + 1} \right)}} \over {{\gamma^2}{\delta_1}\sqrt {\gamma {\delta_2} + {\delta_1}}}},} \hfill \cr {\beta = {{{\omega_3}\sqrt \lambda} \over {2\sqrt {- \nu_1^2{\delta_1}\left({\gamma {\delta_2} + {\delta_1}} \right)}}},{\omega_0} = {{\alpha {\nu_0}\sqrt {- \nu_1^2{\delta_1}\left({\gamma {\delta_2} + {\delta_1}} \right)}} \over {{\nu_1}\sqrt \lambda}},{\omega_1} = {{\alpha \sqrt {- \nu_1^2{\delta_1}\left({\gamma {\delta_2} + {\delta_1}} \right)}} \over {\sqrt \lambda}},} \hfill \cr {{\beta_1} = {{{{\sqrt {{\delta_1}\left({2{\alpha^2}{\gamma^2}{\delta_1}\left({\gamma {\delta_2} + {\delta_1}} \right) + 1} \right)}} \over {\sqrt {\gamma {\delta_2} + {\delta_1}}}} + 1} \over \gamma},{\omega_2} = {{{\omega_3}{\nu_0}} \over {{\nu_1}}},} \hfill \cr} with the referenced values in equation (23), the following solution has been acquired: (24) u1,2x,t=−α2Cαν12δ1γδ2+δ1−ω3λ−ν12δ1γδ2+δ1e2αδ1x−δ2teiB2Cαν1λ−ν12δ1γδ2+δ1−ω3ν1λe2αδ1x−δ2t {u_{1,2}}\left({x,t} \right) = - {{\alpha \left({2{\cal C}\alpha \nu_1^2{\delta_1}\left({\gamma {\delta_2} + {\delta_1}} \right) - {\omega_3}\sqrt \lambda \sqrt {- \nu_1^2{\delta_1}\left({\gamma {\delta_2} + {\delta_1}} \right)} {e^{2\alpha \left({{\delta_1}x - {\delta_2}t} \right)}}} \right){e^{i{\cal B}}}} \over {2{\cal C}\alpha {\nu_1}\sqrt \lambda \sqrt {- \nu_1^2{\delta_1}\left({\gamma {\delta_2} + {\delta_1}} \right)} - {\omega_3}{\nu_1}\lambda {e^{2\alpha \left({{\delta_1}x - {\delta_2}t} \right)}}}} where B=γδ2tδ12α2γ2δ1γδ2+δ1+1+δ1δ12α2γ2δ1γδ2+δ1+1+γδ2+δ12t+γxγ2δ1γδ2+δ1. {\cal B} = {{\left({\gamma {\delta_2}t\sqrt {{\delta_1}\left({2{\alpha^2}{\gamma^2}{\delta_1}\left({\gamma {\delta_2} + {\delta_1}} \right) + 1} \right)} + {\delta_1}\left({\sqrt {{\delta_1}\left({2{\alpha^2}{\gamma^2}{\delta_1}\left({\gamma {\delta_2} + {\delta_1}} \right) + 1} \right)} + \sqrt {\gamma {\delta_2} + {\delta_1}}} \right)\left({2t + \gamma x} \right)} \right)} \over {{\gamma^2}{\delta_1}\sqrt {\gamma {\delta_2} + {\delta_1}}}}. Profile of the obtained solution in equation (24) where α=45 \alpha = {4 \over 5} , δ1=23 {\delta_1} = {2 \over 3} , δ2=34 {\delta_2} = {3 \over 4} , C=43 {\cal C} = {4 \over 3} , λ=−13 \lambda = - {1 \over 3} , ν1=35 {\nu_1} = {3 \over 5} , ω3=32 {\omega_3} = {3 \over 2} , γ=12 \gamma = {1 \over 2} and −10 ≤ x ≤ 10, −10 ≤ t ≤ 10 are graphed in the following:

Case 1.3. We will get the following coefficients: (25) ω0=iν0−β2β2β1γ−1+β122βλ,ω2=i2ω0−β2β2β1γ−1+β12αλ,ν1=−iω3αλ2−β2β2β1γ−1+β12,δ1=−iβ1γ−1β2β1γ−1+β122−α2γβ2γ+β1+1,ω1=ω3α2β,δ2=iβ2γ+2β1β2β1γ−1+β122β1γ−2−α2γβ2γ+β1+1, \matrix{{{\omega_0} = {{i{\nu_0}\sqrt {- {\beta^2}\left({{\beta_2}\left({{\beta_1}\gamma - 1} \right) + \beta_1^2} \right)}} \over {\sqrt 2 \beta \sqrt \lambda}},{\omega_2} = {{i\sqrt 2 {\omega_0}\sqrt {- {\beta^2}\left({{\beta_2}\left({{\beta_1}\gamma - 1} \right) + \beta_1^2} \right)}} \over {\alpha \sqrt \lambda}},} \hfill \cr {{\nu_1} = - {{i{\omega_3}\alpha \sqrt \lambda} \over {\sqrt 2 \sqrt {- {\beta^2}\left({{\beta_2}\left({{\beta_1}\gamma - 1} \right) + \beta_1^2} \right)}}},{\delta_1} = - {{i\sqrt {{\beta_1}\gamma - 1} \sqrt {{\beta_2}\left({{\beta_1}\gamma - 1} \right) + \beta_1^2}} \over {\sqrt 2 \sqrt {- {\alpha^2}\left({\gamma \left({{\beta_2}\gamma + {\beta_1}} \right) + 1} \right)}}},} \hfill \cr {{\omega_1} = {{{\omega_3}\alpha} \over {2\beta}},{\delta_2} = {{i\left({{\beta_2}\gamma + 2{\beta_1}} \right)\sqrt {{\beta_2}\left({{\beta_1}\gamma - 1} \right) + \beta_1^2}} \over {\sqrt {2{\beta_1}\gamma - 2} \sqrt {- {\alpha^2}\left({\gamma \left({{\beta_2}\gamma + {\beta_1}} \right) + 1} \right)}}},} \hfill \cr} the considered parameters in equation (25) are generate the following solution (26) u1,3x,t=i−β2β2β1γ−1+β121β−2β−Cα ei2αβ2β1γ−1+β12β2γt+β12t+γx−xβ1γ−1−α2γβ2γ+β1+1eiβ1x−β2t2λ. {u_{1,3}}\left({x,t} \right) = {{i\sqrt {- {\beta^2}\left({{\beta_2}\left({{\beta_1}\gamma - 1} \right) + \beta_1^2} \right)} \left({{1 \over \beta} - {2 \over {\beta - {\cal C}\alpha \;{e^{{{i\sqrt 2 \alpha \sqrt {{\beta_2}\left({{\beta_1}\gamma - 1} \right) + \beta_1^2} \left({{\beta_2}\gamma t + {\beta_1}\left({2t + \gamma x} \right) - x} \right)} \over {\sqrt {{\beta_1}\gamma - 1} \sqrt {- {\alpha^2}\left({\gamma \left({{\beta_2}\gamma + {\beta_1}} \right) + 1} \right)}}}}}}}} \right){e^{i\left({{\beta_1}x - {\beta_2}t} \right)}}} \over {\sqrt 2 \sqrt \lambda}}. Profile of the obtained solution in equation (26) where C=14 {\cal C} = {1 \over 4} , β=23 \beta = {2 \over 3} , β1=25 {\beta_1} = {2 \over 5} , β2=43 {\beta_2} = {4 \over 3} , γ=34 \gamma = {3 \over 4} , λ=25 \lambda = {2 \over 5} , α=56 \alpha = {5 \over 6} , and −10 ≤ x ≤ 10, −10 ≤ t ≤ 10 are plotted in the following:

Case 2. In equation (15), if m = 1 and M = 4, then n = 4, so equations (5) and (6) are appearing in the following forms: (27) Vη=ω0+ω1T+ω2T2+ω3T3+ω4T4ν0+ν1T, V\left(\eta \right) = {{{\omega_0} + {\omega_1}{\cal T} + {\omega_2}{{\cal T}^2} + {\omega_3}{{\cal T}^3} + {\omega_4}{{\cal T}^4}} \over {{\nu_0} + {\nu_1}{\cal T}}}, and (28) T'=αT+βT4, α≠0, β≠0, {\cal T}{\rm{'}} = \alpha {\cal T} + \beta {{\cal T}^4},\;\alpha \ne 0,\;\beta \ne 0, thereafter, (29) V'=αT+βT4ω1+2ω2T+3ω3T2+4ω4T3ν0+ν1T−ν1ω0+ω1T+ω2T2+ω3T3+ω4T4ν0+ν1T2=ΨTΩT, \matrix{{V'} \hfill & {= \left({\alpha {\cal T} + \beta {{\cal T}^4}} \right)\left({{{{\omega_1} + 2{\omega_2}{\cal T} + 3{\omega_3}{{\cal T}^2} + 4{\omega_4}{{\cal T}^3}} \over {{\nu_0} + {\nu_1}{\cal T}}} - {{{\nu_1}\left({{\omega_0} + {\omega_1}{\cal T} + {\omega_2}{{\cal T}^2} + {\omega_3}{{\cal T}^3} + {\omega_4}{{\cal T}^4}} \right)} \over {{{\left({{\nu_0} + {\nu_1}{\cal T}} \right)}^2}}}} \right)} \hfill \cr {} \hfill & {= {{{\rm{\Psi}}\left({\cal T} \right)} \over {{\rm{\Omega}}\left({\cal T} \right)}},} \hfill \cr} there should be ω₄ ≠ 0, ν₁ ≠ 0, one can find this: (30) V''=ΩTΨ'T−ΨTΩ'T(ΩT)2⋅ V'' = {{{\rm{\Omega}}\left({\cal T} \right){\rm{\Psi}}'\left({\cal T} \right) - {\rm{\Psi}}\left({\cal T} \right){\rm{\Omega}}'\left({\cal T} \right)} \over {{{({\rm{\Omega}}\left({\cal T} \right))}^2}}} \cdot By appointing equations (27)–(30) into equation (12), we acquire an algebraic system of equations involving coefficients of equation (12). Henceforth, we will try to equate the coefficients of the same powers of 𝒯ⁱ for i = 0, 1, 2, …, to zero, aiming to obtain the following algebraic system: (31) Ci=0, where Ci is the coefficientof Ti, for anyi=0,⋯,13. {{\cal C}_i} = 0,\;\;{\rm{where}}\;\;{{\cal C}_i}\;\;{\rm{isthecoefficientof}}\;{{\cal T}^i},\;\;{\rm{forany}}i = 0, \cdots,13.

The following sub-cases have been generated using computer software programs to solve (31), where α ≠ β.

Case 2.1. We can get the following parameters: (32) β1=−2δ1γδ2+δ19α2γ2δ1γδ2+δ1+2+2γδ2+2δ12γγδ2+δ1,λ=−9ν12β2δ1γδ2+δ1ω42,β2=2γδ2+2δ1δ1γδ2+δ19α2γ2δ1γδ2+δ1+2δ1γδ2+δ1−42γ2,ω0=ω4αν02ν1β,ω1=ω4α2β,ω2=0,ω3=ω4ν0ν1, \matrix{{{\beta_1} = {{- \sqrt 2 \sqrt {{\delta_1}\left({\gamma {\delta_2} + {\delta_1}} \right)\left({9{\alpha^2}{\gamma^2}{\delta_1}\left({\gamma {\delta_2} + {\delta_1}} \right) + 2} \right)} + 2\gamma {\delta_2} + 2{\delta_1}} \over {2\gamma \left({\gamma {\delta_2} + {\delta_1}} \right)}},\lambda = - {{9\nu_1^2{\beta^2}{\delta_1}\left({\gamma {\delta_2} + {\delta_1}} \right)} \over {\omega_4^2}},} \hfill \cr {{\beta_2} = {{{{\sqrt 2 \left({\gamma {\delta_2} + 2{\delta_1}} \right)\sqrt {{\delta_1}\left({\gamma {\delta_2} + {\delta_1}} \right)\left({9{\alpha^2}{\gamma^2}{\delta_1}\left({\gamma {\delta_2} + {\delta_1}} \right) + 2} \right)}} \over {{\delta_1}\left({\gamma {\delta_2} + {\delta_1}} \right)}} - 4} \over {2{\gamma^2}}},{\omega_0} = {{{\omega_4}\alpha {\nu_0}} \over {2{\nu_1}\beta}},{\omega_1} = {{{\omega_4}\alpha} \over {2\beta}},{\omega_2} = 0,{\omega_3} = {{{\omega_4}{\nu_0}} \over {{\nu_1}}},} \hfill \cr} the declared values of parameters in equation (32) are generates the following solution: (33) u2,1x,t=ω42βCe−3αδ1x−δ2t−βα+α2ν1β eix−2B+2γδ2+2δ12γγδ2+δ1−t2Bγδ2+2δ1δ1γδ2+δ1−42γ2, {u_{2,1}}\left({x,t} \right) = {{{\omega_4}\left({{{2\beta} \over {{\cal C}{e^{- 3\alpha \left({{\delta_1}x - {\delta_2}t} \right)}} - {\beta \over \alpha}}} + \alpha} \right)} \over {2{\nu_1}\beta}}\;{e^{i\left({{{x\left({- \sqrt 2 {\cal B} + 2\gamma {\delta_2} + 2{\delta_1}} \right)} \over {2\gamma \left({\gamma {\delta_2} + {\delta_1}} \right)}} - {{t\left({{{\sqrt 2 {\cal B}\left({\gamma {\delta_2} + 2{\delta_1}} \right)} \over {{\delta_1}\left({\gamma {\delta_2} + {\delta_1}} \right)}} - 4} \right)} \over {2{\gamma^2}}}} \right)}}, where B=δ1γδ2+δ19α2γ2δ1γδ2+δ1+2. {\cal B} = \sqrt {{\delta_1}\left({\gamma {\delta_2} + {\delta_1}} \right)\left({9{\alpha^2}{\gamma^2}{\delta_1}\left({\gamma {\delta_2} + {\delta_1}} \right) + 2} \right)}.

Profile of the obtained solution in equation (33) where γ=32 \gamma = {3 \over 2} , β=34 \beta = {3 \over 4} , α=12 \alpha = {1 \over 2} , ν1=45 {\nu_1} = {4 \over 5} , ω4=53 {\omega_4} = {5 \over 3} , δ1=12 {\delta_1} = {1 \over 2} , δ2=25 {\delta_2} = {2 \over 5} , C=34 {\cal C} = {3 \over 4} , and −10 ≤ x ≤ 10, −10 ≤ t ≤ 10, are figured in the following:

Case 2.2. One will obtain the subsequent coefficients: (34) β2=−2ω42λ−9ν12β2δ12ω42γ2δ12λω42α2γ2λ−2ν12β2ω42ν12β2δ12λ+12γ6γ3,ω2=0,ω0=ω4αν02ν1β,ω1=ω4α2β,β1=1γ−3ω42γ2δ12λω42α2γ2λ−2ν12β22ω42γ2λ,ω3=ω4ν0ν1,δ2=−ω42λ9ν12γβ2δ1−δ1γ, \matrix{{{\beta_2} = - {{{{\sqrt 2 \left({\omega_4^2\lambda - 9\nu_1^2{\beta^2}\delta_1^2} \right)\sqrt {\omega_4^2{\gamma^2}\delta_1^2\lambda \left({\omega_4^2{\alpha^2}{\gamma^2}\lambda - 2\nu_1^2{\beta^2}} \right)}} \over {\omega_4^2\nu_1^2{\beta^2}\delta_1^2\lambda}} + 12\gamma} \over {6{\gamma^3}}},{\omega_2} = 0,{\omega_0} = {{{\omega_4}\alpha {\nu_0}} \over {2{\nu_1}\beta}},{\omega_1} = {{{\omega_4}\alpha} \over {2\beta}},} \hfill \cr {{\beta_1} = {1 \over \gamma} - {{3\sqrt {\omega_4^2{\gamma^2}\delta_1^2\lambda \left({\omega_4^2{\alpha^2}{\gamma^2}\lambda - 2\nu_1^2{\beta^2}} \right)}} \over {\sqrt 2 \omega_4^2{\gamma^2}\lambda}},{\omega_3} = {{{\omega_4}{\nu_0}} \over {{\nu_1}}},{\delta_2} = - {{\omega_4^2\lambda} \over {9\nu_1^2\gamma {\beta^2}{\delta_1}}} - {{{\delta_1}} \over \gamma},} \hfill \cr} with the specified values in equation (34) one gets the following solution (35) u2,2x,t=α2Cαν12δ1γδ2+δ1−ω3Zλe2αδ1x−δ2teiBγδ2t+δ1B+γδ2+δ12t+γxγ2δ1γδ2+δ12Cαν1Zλ−ω3ν1λe2αδ1x−δ2t, {u_{2,2}}\left({x,t} \right) = {{\alpha \left({2{\cal C}\alpha \nu_1^2{\delta_1}\left({\gamma {\delta_2} + {\delta_1}} \right) - {\omega_3}{\cal Z}\sqrt \lambda {e^{2\alpha \left({{\delta_1}x - {\delta_2}t} \right)}}} \right){e^{{{i\left({{\cal B}\gamma {\delta_2}t + {\delta_1}\left({{\cal B} + \sqrt {\gamma {\delta_2} + {\delta_1}}} \right)\left({2t + \gamma x} \right)} \right)} \over {{\gamma^2}{\delta_1}\sqrt {\gamma {\delta_2} + {\delta_1}}}}}}} \over {2{\cal C}\alpha {\nu_1}{\cal Z}\sqrt \lambda - {\omega_3}{\nu_1}\lambda {e^{2\alpha \left({{\delta_1}x - {\delta_2}t} \right)}}}}, where B=δ12α2γ2δ1γδ2+δ1+1, Z=−ν12δ1γδ2+δ1. {\cal B} = \sqrt {{\delta_1}\left({2{\alpha^2}{\gamma^2}{\delta_1}\left({\gamma {\delta_2} + {\delta_1}} \right) + 1} \right)},\;\;{\cal Z} = \sqrt {- \nu_1^2{\delta_1}\left({\gamma {\delta_2} + {\delta_1}} \right)}.

Case 2.3. One gets the following coefficients: (36) β1=2γδ2+2δ1−2Q2γγδ2+δ1,β2=−4δ12+2δ12Q−2γδ2+2γδ2Q2γ2δ1γδ2+δ1,ω0=0,ω1=ω4α2β,ω2=0,ω3=0,ν0=0,λ=−9ν12β2δ1γδ2+δ1ω42,Q=δ1γδ2+δ19α2γ2δ1γδ2+δ1+2, \matrix{{{\beta_1} = {{2\gamma {\delta_2} + 2{\delta_1} - \sqrt 2 {\cal Q}} \over {2\gamma \left({\gamma {\delta_2} + {\delta_1}} \right)}},{\beta_2} = {{- 4\delta_1^2 + 2{\delta_1}\left({\sqrt 2 {\cal Q} - 2\gamma {\delta_2}} \right) + \sqrt 2 \gamma {\delta_2}{\cal Q}} \over {2{\gamma^2}{\delta_1}\left({\gamma {\delta_2} + {\delta_1}} \right)}},{\omega_0} = 0,{\omega_1} = {{{\omega_4}\alpha} \over {2\beta}},} \hfill \cr {{\omega_2} = 0,{\omega_3} = 0,{\nu_0} = 0,\lambda = - {{9\nu_1^2{\beta^2}{\delta_1}\left({\gamma {\delta_2} + {\delta_1}} \right)} \over {\omega_4^2}},{\cal Q} = \sqrt {{\delta_1}\left({\gamma {\delta_2} + {\delta_1}} \right)\left({9{\alpha^2}{\gamma^2}{\delta_1}\left({\gamma {\delta_2} + {\delta_1}} \right) + 2} \right)},} \hfill \cr} with the known parameters mentioned in equation (36) we will obtain the following solution: (37) u2,3x,t=ω4αCα+βe3αδ1x−δ2te−i2Qγδ2t+δ12Q−2γδ22t+γx−2δ122t+γx2γ2δ1γδ2+δ12ν1βCα−βe3αδ1x−δ2t. {u_{2,3}}\left({x,t} \right) = {{{\omega_4}\alpha \left({{\cal C}\alpha + \beta {e^{3\alpha \left({{\delta_1}x - {\delta_2}t} \right)}}} \right){e^{- {{i\left({\sqrt 2 {\cal Q}\gamma {\delta_2}t + {\delta_1}\left({\sqrt 2 {\cal Q} - 2\gamma {\delta_2}} \right)\left({2t + \gamma x} \right) - 2\delta_1^2\left({2t + \gamma x} \right)} \right)} \over {2{\gamma^2}{\delta_1}\left({\gamma {\delta_2} + {\delta_1}} \right)}}}}} \over {2{\nu_1}\beta \left({{\cal C}\alpha - \beta {e^{3\alpha \left({{\delta_1}x - {\delta_2}t} \right)}}} \right)}}.

Profiles of the obtained solution in equation (37) where C=14 {\cal C} = {1 \over 4} , β=23 \beta = {2 \over 3} , ν1=25 {\nu_1} = {2 \over 5} , δ2=43 {\delta_2} = {4 \over 3} , γ=34 \gamma = {3 \over 4} , δ1=45 {\delta_1} = {4 \over 5} , α=56 \alpha = {5 \over 6} , ω4=32 {\omega_4} = {3 \over 2} , and −10 ≤ x ≤ 10, −10 ≤ t ≤ 10, are outlined in the following:

Case 3. In equation (15) suppose m = 2 and M = 3, and n = 4, thereafter equations (5) and (6) are taking the following styles: (38) Vη=ω0+ω1T+ω2T2+ω3T3+ω4T4ν0+ν1T+ν2T2, V\left(\eta \right) = {{{\omega_0} + {\omega_1}{\cal T} + {\omega_2}{{\cal T}^2} + {\omega_3}{{\cal T}^3} + {\omega_4}{{\cal T}^4}} \over {{\nu_0} + {\nu_1}{\cal T} + {\nu_2}{{\cal T}^2}}}, and (39) T'=αT+βT3, α≠0,β≠0, {\cal T}{\rm{'}} = \alpha {\cal T} + \beta {{\cal T}^3},\;\alpha \ne 0,\beta \ne 0, hence, (40) V'=−ω4T4+ω3T3+ω2T2+ω1T+ω02ν2T+ν1T'ν2T2+ν1T+ν02+4ω4T3+3ω3T2+2ω2T+ω1T'ν2T2+ν1T+ν0=ΨTΩT, \matrix{{V'} \hfill & {= - {{\left({{\omega_4}{{\cal T}^4} + {\omega_3}{{\cal T}^3} + {\omega_2}{{\cal T}^2} + {\omega_1}{\cal T} + {\omega_0}} \right)\left({2{\nu_2}{\cal T} + {\nu_1}} \right){\cal T}{\rm{'}}} \over {{{\left({{\nu_2}{{\cal T}^2} + {\nu_1}{\cal T} + {\nu_0}} \right)}^2}}}} \hfill \cr {} \hfill & {+ {{\left({4{\omega_4}{{\cal T}^3} + 3{\omega_3}{{\cal T}^2} + 2{\omega_2}{\cal T} + {\omega_1}} \right){\cal T}{\rm{'}}} \over {{\nu_2}{{\cal T}^2} + {\nu_1}{\cal T} + {\nu_0}}}} \hfill \cr {} \hfill & {= {{{\rm{\Psi}}\left({\cal T} \right)} \over {{\rm{\Omega}}\left({\cal T} \right)}},} \hfill \cr} there should be ω₄ ≠ 0, ν₂ ≠ 0, one can reach the following: (41) V''=ΩTΨ'T−ΨTΩ'T(ΩT)2⋅ V'' = {{{\rm{\Omega}}\left({\cal T} \right){\rm{\Psi}}'\left({\cal T} \right) - {\rm{\Psi}}\left({\cal T} \right){\rm{\Omega}}'\left({\cal T} \right)} \over {{{({\rm{\Omega}}\left({\cal T} \right))}^2}}} \cdot By restoring equations (38)–(41) into equation (12) and equating to zero all the coefficients of the same powers of 𝒯ⁱ for i = 0, 1, 2, …, we obtain an algebraic system of equations as follows: (42) Ci=0, where Ci is the coefficientof Ti, for anyi=0,⋯,13. {{\cal C}_i} = 0,\;\;{\rm{where}}\;\;{{\cal C}_i}\;\;{\rm{isthecoefficientof}}\;{{\cal T}^i},\;\;{\rm{forany}}i = 0, \cdots,13.

The following sub-cases have been attained using computer software packages to solve (42) when α ≠ β.

Case 3.1. We obtain the following parameters: (43) β1=δ1γδ2+δ12α2γ2δ1γδ2+δ1+1+γδ2+δ1γγδ2+δ1,ω1=ω4αν12ν2β,β2=−γδ2+2δ1δ1γδ2+δ12α2γ2δ1γδ2+δ1+1δ1γδ2+δ1+2γ2,ω0=ω4αν02ν2β,ω3=ω4ν1ν2,ω2=12ω4αβ+2ν0ν2,λ=−4ν22β2δ1γδ2+δ1ω42. \matrix{{{\beta_1} = {{\sqrt {{\delta_1}\left({\gamma {\delta_2} + {\delta_1}} \right)\left({2{\alpha^2}{\gamma^2}{\delta_1}\left({\gamma {\delta_2} + {\delta_1}} \right) + 1} \right)} + \gamma {\delta_2} + {\delta_1}} \over {\gamma \left({\gamma {\delta_2} + {\delta_1}} \right)}},{\omega_1} = {{{\omega_4}\alpha {\nu_1}} \over {2{\nu_2}\beta}},} \hfill \cr {{\beta_2} = - {{{{\left({\gamma {\delta_2} + 2{\delta_1}} \right)\sqrt {{\delta_1}\left({\gamma {\delta_2} + {\delta_1}} \right)\left({2{\alpha^2}{\gamma^2}{\delta_1}\left({\gamma {\delta_2} + {\delta_1}} \right) + 1} \right)}} \over {{\delta_1}\left({\gamma {\delta_2} + {\delta_1}} \right)}} + 2} \over {{\gamma^2}}},{\omega_0} = {{{\omega_4}\alpha {\nu_0}} \over {2{\nu_2}\beta}},{\omega_3} = {{{\omega_4}{\nu_1}} \over {{\nu_2}}},} \hfill \cr {{\omega_2} = {1 \over 2}{\omega_4}\left({{\alpha \over \beta} + {{2{\nu_0}} \over {{\nu_2}}}} \right),\lambda = - {{4\nu_2^2{\beta^2}{\delta_1}\left({\gamma {\delta_2} + {\delta_1}} \right)} \over {\omega_4^2}}.} \hfill \cr} By utilizing the declared parameter values in equation (43), the subsequent solution is constructed: (44) u3,1x,t=ω4ν1A3/2ν2+ω4A2+ω4αβ+2ν0ν22A+ω4αν12Aν2β+ω4αν02ν2βeiBtγ2+xDγγδ2+δ1ν1A+ν2A+ν0, {u_{3,1}}\left({x,t} \right) = {{\left({{{{\omega_4}{\nu_1}} \over {{{\cal A}^{3/2}}{\nu_2}}} + {{{\omega_4}} \over {{{\cal A}^2}}} + {{{\omega_4}\left({{\alpha \over \beta} + {{2{\nu_0}} \over {{\nu_2}}}} \right)} \over {2{\cal A}}} + {{{\omega_4}\alpha {\nu_1}} \over {2\sqrt {\cal A} {\nu_2}\beta}} + {{{\omega_4}\alpha {\nu_0}} \over {2{\nu_2}\beta}}} \right){e^{i\left({{{{\cal B}t} \over {{\gamma^2}}} + {{x{\cal D}} \over {\gamma \left({\gamma {\delta_2} + {\delta_1}} \right)}}} \right)}}} \over {{{{\nu_1}} \over {\sqrt {\cal A}}} + {{{\nu_2}} \over {\cal A}} + {\nu_0}}}, where B=γδ2+2δ1δ1γδ2+δ12α2γ2δ1γδ2+δ1+1δ1γδ2+δ1+2,D=δ1γδ2+δ12α2γ2δ1γδ2+δ1+1+γδ2+δ1, and A=Ce−2αδ1x−δ2t−βα⋅ \matrix{{{\cal B} = {{\left({\gamma {\delta_2} + 2{\delta_1}} \right)\sqrt {{\delta_1}\left({\gamma {\delta_2} + {\delta_1}} \right)\left({2{\alpha^2}{\gamma^2}{\delta_1}\left({\gamma {\delta_2} + {\delta_1}} \right) + 1} \right)}} \over {{\delta_1}\left({\gamma {\delta_2} + {\delta_1}} \right)}} + 2,} \cr {{\cal D} = \sqrt {{\delta_1}\left({\gamma {\delta_2} + {\delta_1}} \right)\left({2{\alpha^2}{\gamma^2}{\delta_1}\left({\gamma {\delta_2} + {\delta_1}} \right) + 1} \right)} + \gamma {\delta_2} + {\delta_1},{\rm{\;}}\;\;\;\;{\rm{and\;}}\;\;\;\;{\cal A} = {\cal C}{e^{- 2\alpha \left({{\delta_1}x - {\delta_2}t} \right)}} - {\beta \over \alpha} \cdot} \cr}

Case 3.2. We can obtain the following coefficients: (45) ω0=ω4αν02ν2β,ω1=ω4αν12ν2β,ω2=12ω4αβ+2ν0ν2,γ=ω42α2λν22β2−2β12β2+22β1,δ1=−ω4β1λω42α2λ−2ν22β12β22ω44α4λ2−4ω42α2ν22β12−2β2β2λ+8ν24β22β4,ω3=ω4ν1ν2,δ2=ω4β2λω42α2λ+2ν22β12+β2β22ω42α2λ−4ν22β12β2ω44α4λ2−2ω42α2ν22β12−2β2β2λ+4ν24β22β4. \matrix{{{\omega_0} = {{{\omega_4}\alpha {\nu_0}} \over {2{\nu_2}\beta}},{\omega_1} = {{{\omega_4}\alpha {\nu_1}} \over {2{\nu_2}\beta}},{\omega_2} = {1 \over 2}{\omega_4}\left({{\alpha \over \beta} + {{2{\nu_0}} \over {{\nu_2}}}} \right),\gamma = {{{{{{\omega_4^2{\alpha^2}\lambda} \over {\nu_2^2{\beta^2}}} - 2\beta_1^2} \over {{\beta_2}}} + 2} \over {2{\beta_1}}},} \hfill \cr {{\delta_1} = - {{{\omega_4}{\beta_1}\sqrt \lambda \sqrt {\omega_4^2{\alpha^2}\lambda - 2\nu_2^2\beta_1^2{\beta^2}}} \over {\sqrt {2\omega_4^4{\alpha^4}{\lambda^2} - 4\omega_4^2{\alpha^2}\nu_2^2\left({\beta_1^2 - 2{\beta_2}} \right){\beta^2}\lambda + 8\nu_2^4\beta_2^2{\beta^4}}}},{\omega_3} = {{{\omega_4}{\nu_1}} \over {{\nu_2}}},} \hfill \cr {{\delta_2} = {{{\omega_4}{\beta_2}\sqrt \lambda \left({\omega_4^2{\alpha^2}\lambda + 2\nu_2^2\left({\beta_1^2 + {\beta_2}} \right){\beta^2}} \right)} \over {\sqrt {2\omega_4^2{\alpha^2}\lambda - 4\nu_2^2\beta_1^2{\beta^2}} \sqrt {\omega_4^4{\alpha^4}{\lambda^2} - 2\omega_4^2{\alpha^2}\nu_2^2\left({\beta_1^2 - 2{\beta_2}} \right){\beta^2}\lambda + 4\nu_2^4\beta_2^2{\beta^4}}}}.} \hfill \cr} The provided quantities in equation (45) lead to the following solution: (46) u3,2x,t=ω4α2ν21β−2β−Cα e2ω4αλω42α2λβ2t+β1x+2ν22β2β2β12t+β22t+β13−xω42α2λ−2ν22β12β2ω44α4λ2−2ω42α2ν22β12−2β2β2λ+4ν24β22β4eiβ1x−β2t. {u_{3,2}}\left({x,t} \right) = {{{\omega_4}\alpha} \over {2{\nu_2}}}\left({{1 \over \beta} - {2 \over {\beta - {\cal C}\alpha \;\;{e^{{{\sqrt 2 {\omega_4}\alpha \sqrt \lambda \left({\omega_4^2{\alpha^2}\lambda \left({{\beta_2}t + {\beta_1}x} \right) + 2\nu_2^2{\beta^2}\left({{\beta_2}\beta_1^2t + \beta_2^2t + \beta_1^3\left({- x} \right)} \right)} \right)} \over {\sqrt {\omega_4^2{\alpha^2}\lambda - 2\nu_2^2\beta_1^2{\beta^2}} \sqrt {\omega_4^4{\alpha^4}{\lambda^2} - 2\omega_4^2{\alpha^2}\nu_2^2\left({\beta_1^2 - 2{\beta_2}} \right){\beta^2}\lambda + 4\nu_2^4\beta_2^2{\beta^4}}}}}}}}} \right){e^{i\left({{\beta_1}x - {\beta_2}t} \right)}}. Profile of the obtained solution in equation (46) where α=53 \alpha = {5 \over 3} , β=13 \beta = {1 \over 3} , β1=23 {\beta_1} = {2 \over 3} , β2=34 {\beta_2} = {3 \over 4} , λ=−23 \lambda = - {2 \over 3} , ν2=52 {\nu_2} = {5 \over 2} , ω4=72 {\omega_4} = {7 \over 2} , C=35 {\cal C} = {3 \over 5} , and −10 ≤ x ≤ 10, −10 ≤ t ≤ 10, are figured out in the following:

Case 3.3. The following coefficients are gained. (47) β2=−2δ12+2δ1L−γδ2+γδ2Lγ2δ1γδ2+δ1,β1=γδ2+δ1−Lγγδ2+δ1,ω0=ω4α24β2,ω1=0,ω2=ω4αβ,ω3=0,ν0=αν22β,ν1=0,λ=−4ν22β2δ1γδ2+δ1ω42,L=δ1γδ2+δ12α2γ2δ1γδ2+δ1+1, \matrix{{{\beta_2} = {{- 2\delta_1^2 + 2{\delta_1}\left({{\cal L} - \gamma {\delta_2}} \right) + \gamma {\delta_2}{\cal L}} \over {{\gamma^2}{\delta_1}\left({\gamma {\delta_2} + {\delta_1}} \right)}},{\beta_1} = {{\gamma {\delta_2} + {\delta_1} - {\cal L}} \over {\gamma \left({\gamma {\delta_2} + {\delta_1}} \right)}},{\omega_0} = {{{\omega_4}{\alpha^2}} \over {4{\beta^2}}},{\omega_1} = 0,{\omega_2} = {{{\omega_4}\alpha} \over \beta},{\omega_3} = 0,} \hfill \cr {{\nu_0} = {{\alpha {\nu_2}} \over {2\beta}},{\nu_1} = 0,\lambda = - {{4\nu_2^2{\beta^2}{\delta_1}\left({\gamma {\delta_2} + {\delta_1}} \right)} \over {\omega_4^2}},{\cal L} = \sqrt {{\delta_1}\left({\gamma {\delta_2} + {\delta_1}} \right)\left({2{\alpha^2}{\gamma^2}{\delta_1}\left({\gamma {\delta_2} + {\delta_1}} \right) + 1} \right)},} \hfill \cr} the declared values of parameters in equation (47) gives the following solution (48) u3,3x,t=ω42βCe−2αδ1x−δ2t−βα+α2ν2β eix−L+γδ2+δ1γγδ2+δ1−t2δ1L−γδ2+Lγδ2−2δ12γ2δ1γδ2+δ1. {u_{3,3}}\left({x,t} \right) = {{{\omega_4}\left({{{2\beta} \over {{\cal C}{e^{- 2\alpha \left({{\delta_1}x - {\delta_2}t} \right)}} - {\beta \over \alpha}}} + \alpha} \right)} \over {2{\nu_2}\beta}}{\rm{\;}}{e^{i\left({{{x\left({- {\cal L} + \gamma {\delta_2} + {\delta_1}} \right)} \over {\gamma \left({\gamma {\delta_2} + {\delta_1}} \right)}} - {{t\left({2{\delta_1}\left({{\cal L} - \gamma {\delta_2}} \right) + {\cal L}\gamma {\delta_2} - 2\delta_1^2} \right)} \over {{\gamma^2}{\delta_1}\left({\gamma {\delta_2} + {\delta_1}} \right)}}} \right)}}.

Here, we give a summary of the dynamic behaviors of the figures, in general, the three-dimensional plots: demonstrate how the solution becomes over both space and time, providing a complete view of the solution's dynamics in three dimensions. Two-dimensional time evolution graphs concentrate on the material evolution at typical spatial points, emphasizing how the solution modifies over time. Three-dimensional revolving plots offer a dynamic perspective by allowing the visualization of the solution from different angles. While, contour figures provide a top-down view of the solution's surface, emphasizing regions with similar values and gradients. These visualizations are essential for comprehending the complex demeanor of solutions to differential equations, presenting perspicuity in their spatial and material dynamics. For the Multi-periodic wave see Figure 1, periodic time evolution in two dimension Figure 2, breather wave in three dimensions and its revolving plots are in Figures 3 and 4, the piece-wise periodic traveling-waves Figure 5, multi-hole traveling waves in three dimensions and contour surfaces are presented in Figures 6 and 7, the periodic time-evolution waves with one growth up point and one going down point Figures 8 and 11, multi-hole traveling waves, multiple breather waves in three dimensions, and contour surfaces are presented in Figures 9 and 10.

Fig. 1

Three-dimensional figures of (22).

Fig. 2

Two-dimensional time evolution graphs of equation (22) where values of t are presented in the legend.

Fig. 3

Three-dimensional figures of equation (24).

Fig. 4

Three-dimensional revolving plot of equation (24).

Fig. 5

Three-dimensional figures of equation (26).

Fig. 6

Three-dimensional figures of equation (33).

Fig. 7

Contour surface plots of equation (33).

Fig. 8

Two dimensional time-evolution graphs of equation (37) where t values are existed in the legend.

Fig. 9

Three-dimensional figures of equation (46).

Fig. 10

Contour surface plots of equation (46).

Fig. 11

Two-dimensional time-evolution graphs of equation (46) where −20 ≤ x ≤ 20, and t values are mentioned in the legend.

This paper provides the comprehensive and accurate solutions for a nonlinear second-order modified unstable Schrödinger model in a (1+1)-dimensional space. This has been accomplished by using IBSEFM. This is the first implementation of the approach to the specified model. Our study's findings are innovative and significant compared to previous investigations since we introduced many unexplored analytically outcomes. The solutions created are complicated exponential rational functions and their combinations. In addition, the results are presented via both two- and three-dimensional graphical representations, as well as two-dimensional visualizations showing the effects of temporal changes. Contour plots and 3D rotating plots have been shown to enhance the understanding of the dynamic characteristics of the identified solutions. Moreover, each result has been validated by re-entering them into the corresponding equation using computational software tools. The efficacy and reliability of the method for examining the exact solutions of the given equation have been validated by the research findings, which will be used in future mathematical models. The results show that the unrestricted parameters of the gathered solutions have a significant influence on the waveform and its dynamics, potentially representing many complex and unknown aspects seen in different scientific disciplines. That being said, there have been reports of certain renewed, previously unreported solution types.