{"id":190760,"date":"2025-06-17T04:33:22","date_gmt":"2025-06-17T04:33:22","guid":{"rendered":"https:\/\/www.europesays.com\/uk\/190760\/"},"modified":"2025-06-17T04:33:22","modified_gmt":"2025-06-17T04:33:22","slug":"optical-solitons-bifurcation-and-chaos-in-the-nonlinear-conformable-schrodinger-equation-with-group-velocity-dispersion-coefficients-and-second-order-spatiotemporal-terms","status":"publish","type":"post","link":"https:\/\/www.europesays.com\/uk\/190760\/","title":{"rendered":"Optical solitons, bifurcation, and chaos in the nonlinear conformable Schr\u00f6dinger equation with group velocity dispersion coefficients and second-order spatiotemporal terms"},"content":{"rendered":"

Here, we utilize the new direct mapping method to generate several accurate answers in closed-form for the current time-fractional nonlinear Schr\u00f6dinger problem. The inquiry begins by employing the following wave transformations:<\/p>\n

$$\\begin{aligned} \\begin{aligned}&\\rho (x,t)=W(\\sigma ){{e}^{i\\varphi (x,t)}}, \\\\&\\sigma =x-s\\frac{{{t}^{\\alpha }}}{\\alpha }, \\quad \\varphi (x,t)=-\\kappa x+w\\frac{{{t}^{\\alpha }}}{\\alpha }. \\end{aligned} \\end{aligned}$$<\/p>\n

\n (9)\n <\/p>\n

The symbol \\(W(\\sigma )\\) represents the phase component, w represents the wave number, \\(\\kappa\\) represents the frequency of solitons, and s represents the speed of the moving wave. By substituting the above transformations into the nonlinear Schr\u00f6dinger equation (1<\/a>), resulting in the following real part and imaginary part, respectively:<\/p>\n

$$\\begin{aligned} ({{\\eta }_{2}}{{s}^{2}}+{{\\eta }_{3}}){W}”(\\sigma )+(\\kappa -{{\\eta }_{1}}w-{{\\eta }_{2}}{{w}^{2}}-{{\\eta }_{3}}{{\\kappa }^{2}})W(\\sigma )+{{W}^{3}}(\\sigma )=0, \\end{aligned}$$<\/p>\n

\n (10)\n <\/p>\n

and<\/p>\n

$$\\begin{aligned} (1-s{{\\eta }_{1}}-2(sw{{\\eta }_{2}}+{{\\eta }_{3}}\\kappa )){W}'(\\sigma )=0. \\end{aligned}$$<\/p>\n

\n (11)\n <\/p>\n

From the above equation, we obtain the following condition:<\/p>\n

$$\\begin{aligned} s=\\frac{1-2{{\\eta }_{3}}\\kappa }{{{\\eta }_{1}}+2w{{\\eta }_{2}}}. \\end{aligned}$$<\/p>\n

\n (12)\n <\/p>\n

By substituting the value of s into (10<\/a>), we obtain the following equation:<\/p>\n

$$\\begin{aligned} ({{\\eta }_{2}}{{A}_{2}}+{{\\eta }_{3}}{{A}_{1}}){W}”(\\sigma )+{{A}_{1}}{{A}_{3}}W(\\sigma )+{{A}_{1}}{{W}^{3}}(\\sigma )=0, \\end{aligned}$$<\/p>\n

\n (13)\n <\/p>\n

where \\({{A}_{1}}={{({{\\eta }_{1}}+2\\kappa {{\\eta }_{2}})}^{2}}\\), \\({{A}_{2}}={{(1-2{{\\eta }_{3}}\\kappa )}^{2}}\\) , and \\({{A}_{3}}=(\\kappa -{{\\eta }_{1}}w-{{\\eta }_{2}}{{w}^{2}}-{{\\eta }_{3}}{{\\kappa }^{2}})\\).<\/p>\n

In order to calculate the homogeneous balancing constant in (13<\/a>), we focused on the term with the highest order \\(W”(\\sigma )\\), as well as the highest nonlinear term \\(W^{3}(\\sigma )\\). Thus, we establish the equation \\(N + 2 = 3N\\). Therefore, applying the balance principle, the solution for N is determined to be 1. Therefore, the series in (3<\/a>) is reduced to the following form:<\/p>\n

$$\\begin{aligned} W(\\sigma )={{d}_{0}}+{{d}_{1}}\\Lambda (\\sigma ). \\end{aligned}$$<\/p>\n

\n (14)\n <\/p>\n

Substituting (4<\/a>) and (14<\/a>) into (13<\/a>) and then organizing the coefficients of \\({{W}^{i}}(\\sigma )\\), where \\(i=0,1,2,…\\), we obtain the following set of non-linear equations:<\/p>\n

$$\\begin{aligned}&(\\Lambda (\\sigma ))^{0}= \\left( 2\\,\\eta _{2}\\,w+\\eta _{1} \\right) ^{2} \\left( {{ d_{0}}}^{3}+ \\left( -\\eta _{2}\\,{w}^{2}-\\eta _{3}\\,{\\kappa }^{2}-\\eta _{1}\\,w+\\kappa \\right) { d_{0}}\\right) =0\\\\&(\\Lambda (\\sigma ))^{1}= { d_{1}} \\left( 2\\,\\eta _{2}\\,w+\\eta _{1} \\right) ^{2}\\left( -\\eta _{2}\\,{w}^{2}-\\eta _{3}\\,{\\kappa }^{2}+3\\,{{ d_{0}}}^{2}-\\eta _{1}\\,w+\\kappa \\right) +{ d_{1}}\\,{ r_{1}}\\,{u}^{2} ( 4\\,{\\eta _{2}}^{2}\\eta _{3}\\, {w}^{2}+4\\,\\eta _{2}\\,{\\eta _{3}}^{2}{\\kappa }^{2}+4\\,\\eta _{1}\\,\\eta _{2}\\,\\eta _{3}\\,w\\\\&+{\\eta _{1}}^{2}\\eta _{3}-4\\,\\eta _{2}\\,\\eta _{3}\\,\\kappa +\\eta _{2} ) =0,\\\\&(\\Lambda (\\sigma ))^{2}=3{ d0}\\,{{ d1}}^{2}\\, \\left( 2\\,\\eta 2\\,w+\\eta 1 \\right) ^{2}=0,\\\\&{{( \\Lambda (\\sigma ) )}^{3}}={{ d_{1}}}^{3} \\left( 2\\,\\eta _{2}\\,w+\\eta _{1} \\right) ^{2}+{\\frac{2\\,{ d_{1}}\\,{u}^{2} { r_{2}}\\,}{{p}^{2}}} \\left( 4\\,{\\eta _{2}}^{2}\\eta _{3}\\,{w}^{2}+4\\,\\eta _{2}\\,{\\eta _{3}}^{2}{\\kappa }^{2}+4\\,\\eta _{1}\\,\\eta _{2}\\,\\eta _{3}\\,w+{\\eta _{1}}^{2}\\eta _{3}-4\\,\\eta _{2}\\,\\eta _{3}\\,\\kappa +\\eta _{2} \\right) =0. \\end{aligned}$$<\/p>\n

The subsequent outcomes are derived from solving this system:<\/p>\n

Fig. 1<\/b>\"figure<\/a><\/p>\n

The bell-shape plots of \\({{\\left| {{\\rho }_{1}}(x,t) \\right| }^{2}}\\), where \\(\\alpha = 1,w= -1,\\eta _{2}=1,{\\eta }_{1}=0.5,\\kappa = 0.3,\\) and \\(u= -0.36\\).<\/p>\n

Fig. 2<\/b>\"figure<\/a><\/p>\n

The mixed dark-bright plots of \\(\\operatorname {Re}({{\\rho }_{1}}(x,t))\\), where \\(\\alpha = 1,w= -1,\\eta _{2}=1,{\\eta }_{1}=0.5,\\kappa = 0.3,\\) and \\(u= -0.36\\).<\/p>\n

Fig. 3<\/b>\"figure<\/a><\/p>\n

The dark-bright plots of \\(\\operatorname {Im}({{\\rho }_{5}}(x,t))\\), where \\(\\alpha = 1,w= 1.1,\\eta _{3}=1,{\\eta }_{1}=0.9,\\kappa =0.1,\\) and \\(u= -0.1\\).<\/p>\n

Fig. 4<\/b>\"figure<\/a><\/p>\n

The comparison of bright plots of \\(|{{\\rho }_{9}}(x,t)|^2\\) , where \\(w=2,\\eta _{3}=1,{\\eta }_{2}=1,\\kappa =-0.9,w=2\\) and \\(u= -0.1255\\).<\/p>\n

Result 1.<\/p>\n

$$\\begin{aligned} \\begin{aligned}&\\eta _{3}=-{\\frac{4\\,{w}^{2} \\left( { r_{1}}\\,{u}^{2}-{\\kappa }^{2} \\right) {\\eta _{2}}^{2}+ \\left( 4\\,\\eta _{1}\\,{ r_{1}}\\,{u}^{2}w-4\\,\\eta _{1}\\,{\\kappa }^{2}w-4\\,\\kappa \\,{ r_{1}}\\,{u}^{2} \\right) \\eta _{2}+ \\left( { r_{1}}\\,{u}^{2}-{\\kappa }^{ 2} \\right) {\\eta _{1}}^{2} +\\sqrt{{ K_{1}}}}{8{u}^{2} { r_{1}}\\,\\eta _{2}\\, {\\kappa }^{2}}},d_{0}=0,\\\\&d_{1}={\\frac{\\sqrt{\\eta _{2}\\,{ r_{2}}\\, \\left( -4\\,{w}^{2} \\left( { r_{1}}\\,{u}^{2}+{\\kappa }^{2} \\right) {\\eta _{2}}^{2}+ \\left( -4\\,\\eta _{1}\\,{ r_{1}}\\,{u}^{2}w-4\\,\\eta _{1}\\,{\\kappa }^{2}w+4\\, \\kappa \\,{ r_{1}}\\,{u}^{2} \\right) \\eta _{2}+ \\left( { r_{1}}\\,{u}^{2}-{ \\kappa }^{2} \\right) {\\eta _{1}}^{2} +\\sqrt{{ K_{1}}} \\right) }}{2\\eta _{2}\\,u{ r_{1}}\\,p}}, \\end{aligned} \\end{aligned}$$<\/p>\n

\n (15)\n <\/p>\n

where<\/p>\n

\\(K_{1}=\\left( 16\\, \\left( { r_{1}}\\,{u}^{2}+{\\kappa }^{2} \\right) ^{2} \\left( {w}^{2}{\\eta _{2}}^{2}+\\frac{{\\eta _{1}}^{2}}{4}\\right) +16\\, \\left( { r_{1}}\\,{u}^{2}+{\\kappa }^{2} \\right) \\left( \\eta _{1}\\,{ r_{1}}\\,{u}^{2}w+ \\eta _{1}\\,{\\kappa }^{2}w-2\\,\\kappa \\,{ r_{1}}\\,{u}^{2} \\right) \\eta _{2} \\right) \\left( \\eta _{2}\\,w+\\frac{\\eta _{1}}{2} \\right) ^{2} .\\)<\/p>\n

From (5<\/a>), (9<\/a>), (12<\/a>), (14<\/a>), and 15<\/a>, we can have the following optical soliton solution:<\/p>\n

$$\\begin{aligned} \\begin{aligned}&\\rho _{1}(x,t)=\\frac{\\sqrt{-\\eta _{2}{(-4\\,{w}^{2} \\left( {\\kappa }^{2}+{u}^{2} \\right) {\\eta _{2}}^{2}+ \\left( -4 \\,\\eta _{1}\\,{\\kappa }^{2}w-4\\,\\eta _{1}\\,{u}^{2}w+4\\,\\kappa \\,{u}^{2} \\right) \\eta _{2}+{\\eta _{1}}^{2}{u}^{2}-{\\eta _{1}}^{2}{\\kappa }^{2} )}+\\sqrt{{ K_{2}}}}}{\\eta _{2}u}\\\\&\\,\\operatorname {sech}(({-\\frac{ \\left( {\\kappa }^{2}-{u}^{2} \\right) \\left( 4\\,{w}^{2}{ \\eta _{2}}^{2}+4\\,w\\eta _{1}\\,\\eta _{2}+{\\eta _{1}}^{2} \\right) +8\\,\\eta _{2}\\,\\kappa \\,{u} ^{2}+\\sqrt{{ K_{2}}}}{4\\eta _{2}\\,\\kappa \\,{u} \\left( 2\\,\\eta _{2}\\,w+\\eta _{1} \\right) }})\\frac{t^{\\alpha }}{\\alpha })+x) \\textrm{e}^{i ( -\\kappa \\,x+w\\frac{t^{\\alpha }}{\\alpha } )}. \\end{aligned} \\end{aligned}$$<\/p>\n

\n (16)\n <\/p>\n

From (6<\/a>), (9<\/a>), (12<\/a>), (14<\/a>), and 15<\/a>, we can have the following optical soliton solution:<\/p>\n

$$\\begin{aligned} \\begin{aligned}&\\rho _{2}(x,t)=\\frac{\\sqrt{\\eta _{2}{(-4\\,{w}^{2} \\left( {\\kappa }^{2}+{u}^{2} \\right) {\\eta _{2}}^{2}+ \\left( -4 \\,\\eta _{1}\\,{\\kappa }^{2}w-4\\,\\eta _{1}\\,{u}^{2}w+4\\,\\kappa \\,{u}^{2} \\right) \\eta _{2}+{\\eta _{1}}^{2}{u}^{2}-{\\eta _{1}}^{2}{\\kappa }^{2} )}+\\sqrt{{ K_{2}}}}}{\\eta _{2}u}\\\\&\\,\\operatorname {csch}(({-\\frac{ \\left( {\\kappa }^{2}-{u}^{2} \\right) \\left( 4\\,{w}^{2}{ \\eta _{2}}^{2}+4\\,w\\eta _{1}\\,\\eta _{2}+{\\eta _{1}}^{2} \\right) +8\\,\\eta _{2}\\,\\kappa \\,{u} ^{2}+\\sqrt{{ K_{2}}}}{4\\eta _{2}\\,\\kappa \\,{u} \\left( 2\\,\\eta _{2}\\,w+\\eta _{1} \\right) }})\\frac{t^{\\alpha }}{\\alpha })+x) \\textrm{e}^{i ( -\\kappa \\,x+w\\frac{t^{\\alpha }}{\\alpha } )}, \\end{aligned} \\end{aligned}$$<\/p>\n

\n (17)\n <\/p>\n

where<\/p>\n

\\(K_{2}=\\left( 16\\,\\eta _{2}\\, \\left( {\\kappa }^{2}+{u}^{2} \\right) \\left( \\left( \\eta _{2}\\,w+\\eta _{1} \\right) \\left( w{\\kappa }^{2}+{u}^{2}w \\right) -2\\,\\kappa \\,{u}^{2} \\right) +4\\,{\\eta _{1}}^{2} \\left( u-\\kappa \\right) ^{ 2} \\left( u+\\kappa \\right) ^{2} \\right) \\left( \\eta _{2}\\,w+\\frac{\\eta _{1}}{2} \\right) ^{2}.\\)\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0<\/p>\n

From (7<\/a>), (9<\/a>), (12<\/a>), (14<\/a>), and (15<\/a>), we can have the following optical soliton solution:<\/p>\n

$$\\begin{aligned} \\begin{aligned}&\\rho _{3}(x,t)=-\\frac{\\sqrt{\\eta _{2}{(-4\\,{w}^{2} \\left( {\\kappa }^{2}+{u}^{2} \\right) {\\eta _{2}}^{2}+ \\left( -4 \\,\\eta _{1}\\,{\\kappa }^{2}w-4\\,\\eta _{1}\\,{u}^{2}w+4\\,\\kappa \\,{u}^{2} \\right) \\eta _{2}-{\\eta _{1}}^{2}{u}^{2}-{\\eta _{1}}^{2}{\\kappa }^{2} )}+\\sqrt{{ K_{3}}}}}{\\eta _{2}u}\\\\&\\,\\operatorname {sec}(({\\frac{ \\left( {\\kappa }^{2}+{u}^{2} \\right) \\left( -4\\,{w}^{2} {\\eta _{2}}^{2}-4\\,w\\eta _{1}\\,\\eta _{2}-{\\eta _{1}}^{2} \\right) +4\\,\\eta _{2}\\,\\kappa \\,{u }^{2}+\\sqrt{{ K_{3}}}}{4\\eta _{2}\\,\\kappa \\,{u} \\left( 2\\,\\eta _{2}\\,w+\\eta _{1} \\right) }} )\\frac{t^{\\alpha }}{\\alpha })+x) \\textrm{e}^{i ( -\\kappa \\,x+w\\frac{t^{\\alpha }}{\\alpha } )}. \\end{aligned} \\end{aligned}$$<\/p>\n

\n (18)\n <\/p>\n

From (8<\/a>), (9<\/a>), (12<\/a>), (14<\/a>), and (15<\/a>), we can have the following optical soliton solution:<\/p>\n

$$\\begin{aligned} \\begin{aligned}&\\rho _{4}(x,t)=-\\frac{\\sqrt{\\eta _{2}{(-4\\,{w}^{2} \\left( {\\kappa }^{2}+{u}^{2} \\right) {\\eta _{2}}^{2}+ \\left( -4 \\,\\eta _{1}\\,{\\kappa }^{2}w-4\\,\\eta _{1}\\,{u}^{2}w+4\\,\\kappa \\,{u}^{2} \\right) \\eta _{2}-{\\eta _{1}}^{2}{u}^{2}-{\\eta _{1}}^{2}{\\kappa }^{2} )}+\\sqrt{{ K_{3}}}}}{\\eta _{2}u}\\\\&\\,\\operatorname {csc}(({\\frac{ \\left( {\\kappa }^{2}+{u}^{2} \\right) \\left( -4\\,{w}^{2} {\\eta _{2}}^{2}-4\\,w\\eta _{1}\\,\\eta _{2}-{\\eta _{1}}^{2} \\right) +4\\,\\eta _{2}\\,\\kappa \\,{u }^{2}+\\sqrt{{ K_{3}}}}{4\\eta _{2}\\,\\kappa \\,{u} \\left( 2\\,\\eta _{2}\\,w+\\eta _{1} \\right) }} )\\frac{t^{\\alpha }}{\\alpha })+x) \\textrm{e}^{i ( -\\kappa \\,x+w\\frac{t^{\\alpha }}{\\alpha } )}, \\end{aligned} \\end{aligned}$$<\/p>\n

\n (19)\n <\/p>\n

where<\/p>\n

\\(K_{3}=16\\, \\left( \\eta _{2}\\,w+\\frac{\\eta _{1}}{2} \\right) ^{2} \\left( \\eta _{2}\\, \\left( u- \\kappa \\right) \\left( \\left( {w}^{2}\\eta _{2}+\\eta _{1}\\,w-2\\,\\kappa \\right) {u}^{2}-w \\left( \\eta _{2}\\,w+\\eta _{1} \\right) {\\kappa }^{2} \\right) \\left( u+\\kappa \\right) +\\frac{{\\eta 1}^{2} }{4}\\left( {\\kappa }^{2}+{u}^{2} \\right) ^{2} \\right) .\\)<\/p>\n

Result 2.<\/p>\n

$$\\begin{aligned} d_{0}=0,\\eta _{2}=0,\\kappa =\\kappa ,w={\\frac{\\eta _{3}\\,{ r_{1}}\\,{u}^{2}-\\eta _{3}\\,{\\kappa }^{2}+\\kappa }{\\eta _{1}}}, d_{1}={\\frac{u\\sqrt{-2\\,\\eta _{3}\\,{ r_{2}}}}{p}}. \\end{aligned}$$<\/p>\n

\n (20)\n <\/p>\n

Fig. 5<\/b>\"figure<\/a><\/p>\n

The mixed dark-bright plots of \\(\\operatorname {Im}({{\\rho }_{9}}(x,t))\\), where \\(w=2,\\eta _{3}=1,{\\eta }_{2}=1,\\kappa =-0.9,w=2,u= – 0.1255\\) and \\(t=2\\).<\/p>\n

Fig. 6<\/b>\"figure<\/a><\/p>\n

The 2D plots of \\({{\\left| {{\\rho }_{9}}(x,t) \\right| }^{2}}\\) and \\(\\operatorname {Im}({{\\rho }_{9}}(x,t))\\), where \\(w=2,\\eta _{3}=1,{\\eta }_{2}=1,\\kappa =-0.9,w=2,u= -0.1255\\) and \\(t=2\\).<\/p>\n

From (5<\/a>), (9<\/a>), (12<\/a>), (14<\/a>), and (20<\/a>) , we can have the following optical soliton solution:<\/p>\n

$$\\begin{aligned} \\begin{aligned}&\\rho _{5}(x,t)=\\sqrt{2\\eta _{3}} u \\textrm{sech} \\left( u \\left( x +{\\frac{ \\left( 2\\, \\eta _{3}\\,\\kappa -1 \\right) {t}^{\\alpha }}{\\alpha \\,\\eta _{1}}} \\right) \\right) {\\textrm{e}^{i \\left( -\\kappa \\,x+{\\frac{ \\left( -\\eta _{3}\\,{\\kappa }^ {2}+\\eta _{3}\\,{u}^{2}+\\kappa \\right) {t}^{\\alpha }}{\\alpha \\,\\eta _{1}}} \\right) }}. \\end{aligned} \\end{aligned}$$<\/p>\n

\n (21)\n <\/p>\n

From (6<\/a>), (9<\/a>), (12<\/a>), (14<\/a>), and (20<\/a>) , we can have the following optical soliton solution:<\/p>\n

$$\\begin{aligned} \\begin{aligned}&\\rho _{6}(x,t)=\\sqrt{-2\\eta 3} u \\textrm{csch} \\left( u \\left( x +{\\frac{ \\left( 2\\, \\eta _{3}\\,\\kappa -1 \\right) {t}^{\\alpha }}{\\alpha \\,\\eta _{1}}} \\right) \\right) {\\textrm{e}^{i \\left( -\\kappa \\,x+{\\frac{ \\left( -\\eta _{3}\\,{\\kappa }^ {2}+\\eta _{3}\\,{u}^{2}+\\kappa \\right) {t}^{\\alpha }}{\\alpha \\,\\eta _{1}}} \\right) }}. \\end{aligned} \\end{aligned}$$<\/p>\n

\n (22)\n <\/p>\n

From (7<\/a>), (9<\/a>), (12<\/a>), (14<\/a>), and (20<\/a>) , we can have the following optical soliton solution:<\/p>\n

$$\\begin{aligned} \\begin{aligned}&\\rho _{7}(x,t)=\\sqrt{-2\\eta _{3}} u \\textrm{sec} \\left( u \\left( x +{\\frac{ \\left( 2\\, \\eta _{3}\\,\\kappa -1 \\right) {t}^{\\alpha }}{\\alpha \\,\\eta _{1}}} \\right) \\right) {\\textrm{e}^{i \\left( -\\kappa \\,x+{\\frac{ \\left( -\\eta _{3}\\,{\\kappa }^ {2}-\\eta _{3}\\,{u}^{2}+\\kappa \\right) {t}^{\\alpha }}{\\alpha \\,\\eta _{1}}} \\right) }}. \\end{aligned} \\end{aligned}$$<\/p>\n

\n (23)\n <\/p>\n

From (8<\/a>), (9<\/a>), (12<\/a>), (14<\/a>), and (20<\/a>) , we can have the following optical soliton solution:<\/p>\n

$$\\begin{aligned} \\begin{aligned}&\\rho _{8}(x,t)=\\sqrt{-2\\eta 3} u \\textrm{csc} \\left( u \\left( x +{\\frac{ \\left( 2\\, \\eta _{3}\\,\\kappa -1 \\right) {t}^{\\alpha }}{\\alpha \\,\\eta _{1}}} \\right) \\right) {\\textrm{e}^{i \\left( -\\kappa \\,x+{\\frac{ \\left( -\\eta _{3}\\,{\\kappa }^ {2}-\\eta _{3}\\,{u}^{2}+\\kappa \\right) {t}^{\\alpha }}{\\alpha \\,\\eta _{1}}} \\right) }}. \\end{aligned} \\end{aligned}$$<\/p>\n

\n (24)\n <\/p>\n

Result 3<\/p>\n

$$\\begin{aligned} d_{0}=0,\\eta _{2}=0, d_{1}={\\frac{\\sqrt{-2\\,\\eta _{3}\\,{ r_{2}}}u}{p}}, \\eta _{1}={\\frac{ \\left( { r_{1}}\\,{u}^{2}-{\\kappa }^{2} \\right) \\eta _{3}+\\kappa }{w }}. \\end{aligned}$$<\/p>\n

\n (25)\n <\/p>\n

From (5<\/a>), (9<\/a>), (12<\/a>), (14<\/a>), and (25<\/a>) , we can have the following optical soliton solution:<\/p>\n

$$\\begin{aligned} \\begin{aligned}&\\rho _{9}(x,t)=\\sqrt{2\\eta _{3}}u\\textrm{sech} \\left( u \\left( x+{\\frac{ \\left( 2\\, \\eta _{3}\\,\\kappa -1 \\right) w{t}^{\\alpha }}{\\alpha \\, \\left( \\left( -{ \\kappa }^{2}+{u}^{2} \\right) \\eta _{3}+\\kappa \\right) }} \\right) \\right) \\textrm{e}^{i ( -\\kappa \\,x+{\\frac{w{t}^{\\alpha }}{\\alpha }} ) }. \\end{aligned} \\end{aligned}$$<\/p>\n

\n (26)\n <\/p>\n

From (6<\/a>), (9<\/a>), (12<\/a>), (14<\/a>), and (25<\/a>), we can have the following optical soliton solution:<\/p>\n

$$\\begin{aligned} \\begin{aligned}&\\rho _{10}(x,t)=\\sqrt{-2\\eta _{3}}u\\textrm{csch} \\left( u \\left( x+{\\frac{ \\left( 2\\, \\eta _{3}\\,\\kappa -1 \\right) w{t}^{\\alpha }}{\\alpha \\, \\left( \\left( -{ \\kappa }^{2}+{u}^{2} \\right) \\eta _{3}+\\kappa \\right) }} \\right) \\right) \\textrm{e}^{i ( -\\kappa \\,x+{\\frac{w{t}^{\\alpha }}{\\alpha }} ) }.\\\\ \\end{aligned} \\end{aligned}$$<\/p>\n

\n (27)\n <\/p>\n

From (7<\/a>), (9<\/a>), (12<\/a>), (14<\/a>), and (25<\/a>), we can have the following optical soliton solution:<\/p>\n

$$\\begin{aligned} \\begin{aligned}&\\rho _{11}(x,t)=\\sqrt{-2\\eta _{3}}u\\textrm{sec} \\left( u \\left( x+{\\frac{ \\left( 2\\, \\eta _{3}\\,\\kappa -1 \\right) w{t}^{\\alpha }}{\\alpha \\, \\left( \\left( -{ \\kappa }^{2}-{u}^{2} \\right) \\eta _{3}+\\kappa \\right) }} \\right) \\right) \\textrm{e}^{i ( -\\kappa \\,x+{\\frac{w{t}^{\\alpha }}{\\alpha }} ) }.\\\\ \\end{aligned} \\end{aligned}$$<\/p>\n

\n (28)\n <\/p>\n

From (8<\/a>), (9<\/a>), (12<\/a>), (14<\/a>), and (25<\/a>), we can have the following optical soliton solution:<\/p>\n

$$\\begin{aligned} \\begin{aligned}&\\rho _{12}(x,t)=\\sqrt{-2\\eta _{3}}u\\textrm{csc} \\left( u \\left( x+{\\frac{ \\left( 2\\, \\eta _{3}\\,\\kappa -1 \\right) w{t}^{\\alpha }}{\\alpha \\, \\left( \\left( -{ \\kappa }^{2}-{u}^{2} \\right) \\eta _{3}+\\kappa \\right) }} \\right) \\right) \\textrm{e}^{i ( -\\kappa \\,x+{\\frac{w{t}^{\\alpha }}{\\alpha }} ) }.\\\\ \\end{aligned} \\end{aligned}$$<\/p>\n

\n (29)\n <\/p>\n

Fig. 7<\/b>\"figure<\/a><\/p>\n

Phase portraits of the system\u2019s bifurcations are depicted under different conditions for \\(\\mu _{1}\\) and \\(\\mu _{2}\\), using various parameter values for Case 5.1 and Case 5.2.<\/p>\n

Fig. 8<\/b>\"figure<\/a><\/p>\n

Phase portraits of the system\u2019s bifurcations are depicted under different conditions for \\(\\mu _{1}\\) and \\(\\mu _{2}\\), using various parameter values for Case 5.3 and Case 5.4.<\/p>\n","protected":false},"excerpt":{"rendered":"Here, we utilize the new direct mapping method to generate several accurate answers in closed-form for the current…\n","protected":false},"author":2,"featured_media":190761,"comment_status":"","ping_status":"","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[3845],"tags":[78124,78127,78130,3965,78125,3966,78128,78126,4171,78129,74,70,55920,16,15],"class_list":{"0":"post-190760","1":"post","2":"type-post","3":"status-publish","4":"format-standard","5":"has-post-thumbnail","7":"category-physics","8":"tag-applied-mathematics","9":"tag-bifurcation","10":"tag-conformable-derivative","11":"tag-humanities-and-social-sciences","12":"tag-mathematical-model","13":"tag-multidisciplinary","14":"tag-new-direct-mapping-method","15":"tag-nonlinear-conformable-schrodinger-equation","16":"tag-nonlinear-optics","17":"tag-optical-soliton-solutions","18":"tag-physics","19":"tag-science","20":"tag-solitons","21":"tag-uk","22":"tag-united-kingdom"},"share_on_mastodon":{"url":"https:\/\/pubeurope.com\/@uk\/114696841550208079","error":""},"_links":{"self":[{"href":"https:\/\/www.europesays.com\/uk\/wp-json\/wp\/v2\/posts\/190760","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.europesays.com\/uk\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.europesays.com\/uk\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.europesays.com\/uk\/wp-json\/wp\/v2\/users\/2"}],"replies":[{"embeddable":true,"href":"https:\/\/www.europesays.com\/uk\/wp-json\/wp\/v2\/comments?post=190760"}],"version-history":[{"count":0,"href":"https:\/\/www.europesays.com\/uk\/wp-json\/wp\/v2\/posts\/190760\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.europesays.com\/uk\/wp-json\/wp\/v2\/media\/190761"}],"wp:attachment":[{"href":"https:\/\/www.europesays.com\/uk\/wp-json\/wp\/v2\/media?parent=190760"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.europesays.com\/uk\/wp-json\/wp\/v2\/categories?post=190760"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.europesays.com\/uk\/wp-json\/wp\/v2\/tags?post=190760"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}