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July  2021, 14(7): 2245-2260. doi: 10.3934/dcdss.2021058

Dynamical behaviors and oblique resonant nonlinear waves with dual-power law nonlinearity and conformable temporal evolution

1. 

Department of Mathematics, Chittagong University of Engineering and Technology, Chittagong-4349, Bangladesh

2. 

Department of Electrical and Electronic Engineering, International Islamic University Chittagong, Chattogram-4318, Bangladesh

3. 

Department of Computer Science and Engineering, Chittagong University of Engineering and Technology, Chittagong-4349, Bangladesh

4. 

Division of Applied Mathematics, Thu Dau Mot University, Binh Duong Province, Vietnam

5. 

Department of Medical Research, China Medical University Hospital Taichung 40402, Taiwan

6. 

Department of Sciences, École Normale Supérieure, Moulay Ismail University of Meknes, Morocco

* Corresponding author: Zakia Hammouch, email : hammouch_zakia@tdmu.edu.vn

Received  May 2019 Revised  September 2020 Published  July 2021 Early access  May 2021

In this article, the oblique resonant traveling waves and dynamical behaviors of (2+1)-dimensional Nonlinear Schrödinger equation along with dual-power law nonlinearity, and fractal conformable temporal evolution are reported. The considered equation is converted to an ordinary differential equation by taking the traveling variable wave transform and properties of Khalil's conformable derivative into account. The modified Kudryashov method is implemented to divulge the oblique resonant traveling wave of such an equation. It is found that the obliqueness is only affected on width, but not on amplitude and phase patriots of resonant nonlinear propagating wave dynamics. The research outcomes are very helpful for analyzing the obliquely propagating nonlinear resonant wave phenomena and their dynamical behaviors in several nonlinear systems having Madelung fluids and optical bullets.

Citation: Md. Golam Hafez, Sayed Allamah Iqbal, Asaduzzaman, Zakia Hammouch. Dynamical behaviors and oblique resonant nonlinear waves with dual-power law nonlinearity and conformable temporal evolution. Discrete and Continuous Dynamical Systems - S, 2021, 14 (7) : 2245-2260. doi: 10.3934/dcdss.2021058
References:
[1]

K. Ait Touchent et al., Implementation and convergence analysis of homotopy perturbation coupled with sumudu transform to construct solutions of local-fractional PDEs, Fractal and Fractional, 2 (2018), 22.

[2]

F. Belgacem et al., New and Extended Applications of the Natural and Sumudu Transforms: Fractional Diffusion and Stokes Fluid Flow Realms, Advances in Real and Complex Analysis with Applications. Birkhauser, Singapore, 2017.

[3]

A. Biswas, Quasi-stationary optical solitons with dual-power law nonlinearity, Optics Communications, 235 (2004), 183-194. 

[4]

A. Biswas, Soliton solutions of the perturbed resonant nonlinear Schrodinger's equation with full nonlinearity by semi-inverse variational principle, Quantum Phys. Lett, 1 (2012), 79-89. 

[5]

D. Bohm, A suggested interpretation of the quantum theory in terms of hidden variables, Phys. Rev., 85 (1952), 166-179.  doi: 10.1103/PhysRev.85.166.

[6]

M. EslamiM. Mirzazadeh and A. Biswas, Soliton solutions of the resonant nonlinear Schrodinger's equation in optical fibers with time-dependent coefficients by simplest equation approach, Journal of Modern Optics, 60 (2013), 1627-1636. 

[7]

F. Ferdous and M. G. Hafez, Nonlinear time fractional Korteweg-de Vries equations for the interaction of wave phenomena in fluid-filled elastic tubes, The European Physical Journal Plus, 133 (2018), 384. doi: 10.1140/epjp/i2018-12195-6.

[8]

F. Ferdous, Oblique resonant optical solitons with Kerr and parabolic law nonlinearities and fractional temporal evolution by generalized $\exp(-\phi(\xi))$-expansion, Optik, 178 (2019), 439-448. 

[9]

M. M. Ghalib, Analytical results on the unsteady rotational flow of fractional-order non-Newtonian fluids with shear stress on the boundary, Discrete Contin. Dyn. Syst. Ser. S, 13 (2020), 683-693.  doi: 10.3934/dcdss.2020037.

[10]

M. G. Hafez, Nonlinear ion acoustic solitary waves with dynamical behaviours in the relativistic plasmas, Astrophys. Space Sci., 365 (2020), Paper No. 78, 11 pp. doi: 10.1007/s10509-020-03791-9.

[11]

M. G. Hafez, Exact solutions to the (3+1)-dimensional coupled Klein-Gordon-Zakharov equation using $\exp(-\phi(\xi))$-expansion method, Alexandria Engineering Journal, 55 (2016), 1635-1645. 

[12]

M. G. HafezR. Sakthivel and M. R. Talukder, Some new electrostatic potential functions used to analyze the ion-acoustic waves in a Thomas Fermi plasma with degenerate electrons, Chinese Journal of Physics, 35 (2015), 1-13. 

[13]

M. G. HafezM. R. Talukder and M. H. Ali, New analytical solutions for propagation of small but finite amplitude ion-acoustic waves in a dense plasma, Waves in Random and Complex Media, 26 (2016), 68-80.  doi: 10.1080/17455030.2015.1111543.

[14]

Z. Hammouch and T. Mekkaoui, Traveling-wave solutions of the generalized Zakharov equation with time-space fractional derivatives, Journal MESA, 5 (2014), 489-498. 

[15]

Z. Hammouch and T. Mekkaoui, Travelling-wave solutions for some fractional partial differential equation by means of generalized trigonometry functions, International Journal of Applied Mathematical Research, 1 (2012), 206-212. 

[16]

Z. Hammouch, T. Mekkaoui and P. Agarwal, Optical solitons for the Calogero-Bogoyavlenskii-Schiff equation in (2+1) dimensions with time-fractional conformable derivative, The European Physical Journal Plus, 133 (2018), 248.

[17]

K. Hosseini and R. Ansari, New exact solutions of nonlinear conformable time-fractional Boussinesq equations using the modified Kudryashov method, Waves in Random and Complex Media, 27 (2017), 628-636.  doi: 10.1080/17455030.2017.1296983.

[18]

K. Hosseini, New optical solitons of cubic-quartic nonlinear Schrodinger equation, Optik, 157 (2018), 1101-1105. 

[19]

K. Hosseini, Resonant optical solitons with perturbation terms and fractional temporal evolution using improved $\tan(\phi(\eta)/2)$-expansion method and exp function approach, Optik, 158 (2018), 933-939. 

[20]

A. Houwe et al., Nonlinear Schrodingers equations with cubic nonlinearity: M-derivative soliton solutions by $\exp (-\Phi (\xi)) $-Expansion method, https://www.eprints.org/, 2019.

[21]

M. Ilie MousaJ. Biazar and Z. Ayati., Resonant solitons to the nonlinear Schrodinger equation with different forms of nonlinearities, Optik, 164 (2018), 201-209. 

[22]

T. Iizuka and Y. S. Kivshar, Optical gap solitons in nonresonant quadratic media, Phys. Rev. E, 59 (1999), 7148-7151.  doi: 10.1103/PhysRevE.59.7148.

[23]

R. Khalil, A new definition of fractional derivative, J. Comput. Appl. Math., 264 (2014), 65-70.  doi: 10.1016/j.cam.2014.01.002.

[24]

J.-H. Lee and O. K. Pashaev, Solitons of the resonant nonlinear Schrodinger equation with nontrivial boundary conditions: Hirota bilinear method, Theoret. and Math. Phys., 152 (2007), 991-1003.  doi: 10.1007/s11232-007-0083-3.

[25]

J. H. Lee, The resonant nonlinear Schrodinger equation in cold plasma physics : Application of Backlund-Darboux transformations and superposition principles, Journal of Plasma Physics, 73 (2007), 257-272. 

[26]

H. LiJ. W. Haus and P. P. Banerjee, Application of transfer matrix method to second-harmonic generation in nonlinear photonic bandgap structures: Oblique incidence, JOSA B, 32 (2015), 1456-1462.  doi: 10.1364/JOSAB.32.001456.

[27]

B. Lu, The first integral method for some time fractional differential equations, J. Math. Anal. Appl., 395 (2012), 684-693.  doi: 10.1016/j.jmaa.2012.05.066.

[28]

V. V. Nemytskii and V. V. Stepanov, Qualitative Theory of Differential Equations, Princeton University Press, Princeton, N.J. 1960.

[29]

O. K. Pashaev and L. Jyh-Hao, Resonance solitons as black holes in Madelung fluid, Modern Physics Letters A, 17 (2002), 1601-1619. 

[30]

K. Porsezian, Soliton models in resonant and nonresonant optical fibers, Pramana, 57 (2001), 1003-1039. 

[31]

L. Singh, S. Konar and A. K. Sharma, Resonant cross-modulation of two laser beams in a semiconductor slab, Journal of Physics D: Applied Physics, 34 (2001), 2237.

[32]

D. V. Skryabin and W. J. Firth, Dynamics of self-trapped beams with phase dislocation in saturable Kerr and quadratic nonlinear media, Phys. Rev. E, 58 (1998), 3916. doi: 10.1103/PhysRevE.58.3916.

[33]

L. Wenjun and K. Chen, The functional variable method for finding exact solutions of some nonlinear time-fractional differential equations, Pramana, 81 (2013), 377-384. 

[34]

B. Zheng, $(G'/G)$-expansion method for solving fractional partial differential equations in the theory of mathematical physics, Commun. Theor. Phys. (Beijing), 58 (2012), 623-630.  doi: 10.1088/0253-6102/58/5/02.

show all references

References:
[1]

K. Ait Touchent et al., Implementation and convergence analysis of homotopy perturbation coupled with sumudu transform to construct solutions of local-fractional PDEs, Fractal and Fractional, 2 (2018), 22.

[2]

F. Belgacem et al., New and Extended Applications of the Natural and Sumudu Transforms: Fractional Diffusion and Stokes Fluid Flow Realms, Advances in Real and Complex Analysis with Applications. Birkhauser, Singapore, 2017.

[3]

A. Biswas, Quasi-stationary optical solitons with dual-power law nonlinearity, Optics Communications, 235 (2004), 183-194. 

[4]

A. Biswas, Soliton solutions of the perturbed resonant nonlinear Schrodinger's equation with full nonlinearity by semi-inverse variational principle, Quantum Phys. Lett, 1 (2012), 79-89. 

[5]

D. Bohm, A suggested interpretation of the quantum theory in terms of hidden variables, Phys. Rev., 85 (1952), 166-179.  doi: 10.1103/PhysRev.85.166.

[6]

M. EslamiM. Mirzazadeh and A. Biswas, Soliton solutions of the resonant nonlinear Schrodinger's equation in optical fibers with time-dependent coefficients by simplest equation approach, Journal of Modern Optics, 60 (2013), 1627-1636. 

[7]

F. Ferdous and M. G. Hafez, Nonlinear time fractional Korteweg-de Vries equations for the interaction of wave phenomena in fluid-filled elastic tubes, The European Physical Journal Plus, 133 (2018), 384. doi: 10.1140/epjp/i2018-12195-6.

[8]

F. Ferdous, Oblique resonant optical solitons with Kerr and parabolic law nonlinearities and fractional temporal evolution by generalized $\exp(-\phi(\xi))$-expansion, Optik, 178 (2019), 439-448. 

[9]

M. M. Ghalib, Analytical results on the unsteady rotational flow of fractional-order non-Newtonian fluids with shear stress on the boundary, Discrete Contin. Dyn. Syst. Ser. S, 13 (2020), 683-693.  doi: 10.3934/dcdss.2020037.

[10]

M. G. Hafez, Nonlinear ion acoustic solitary waves with dynamical behaviours in the relativistic plasmas, Astrophys. Space Sci., 365 (2020), Paper No. 78, 11 pp. doi: 10.1007/s10509-020-03791-9.

[11]

M. G. Hafez, Exact solutions to the (3+1)-dimensional coupled Klein-Gordon-Zakharov equation using $\exp(-\phi(\xi))$-expansion method, Alexandria Engineering Journal, 55 (2016), 1635-1645. 

[12]

M. G. HafezR. Sakthivel and M. R. Talukder, Some new electrostatic potential functions used to analyze the ion-acoustic waves in a Thomas Fermi plasma with degenerate electrons, Chinese Journal of Physics, 35 (2015), 1-13. 

[13]

M. G. HafezM. R. Talukder and M. H. Ali, New analytical solutions for propagation of small but finite amplitude ion-acoustic waves in a dense plasma, Waves in Random and Complex Media, 26 (2016), 68-80.  doi: 10.1080/17455030.2015.1111543.

[14]

Z. Hammouch and T. Mekkaoui, Traveling-wave solutions of the generalized Zakharov equation with time-space fractional derivatives, Journal MESA, 5 (2014), 489-498. 

[15]

Z. Hammouch and T. Mekkaoui, Travelling-wave solutions for some fractional partial differential equation by means of generalized trigonometry functions, International Journal of Applied Mathematical Research, 1 (2012), 206-212. 

[16]

Z. Hammouch, T. Mekkaoui and P. Agarwal, Optical solitons for the Calogero-Bogoyavlenskii-Schiff equation in (2+1) dimensions with time-fractional conformable derivative, The European Physical Journal Plus, 133 (2018), 248.

[17]

K. Hosseini and R. Ansari, New exact solutions of nonlinear conformable time-fractional Boussinesq equations using the modified Kudryashov method, Waves in Random and Complex Media, 27 (2017), 628-636.  doi: 10.1080/17455030.2017.1296983.

[18]

K. Hosseini, New optical solitons of cubic-quartic nonlinear Schrodinger equation, Optik, 157 (2018), 1101-1105. 

[19]

K. Hosseini, Resonant optical solitons with perturbation terms and fractional temporal evolution using improved $\tan(\phi(\eta)/2)$-expansion method and exp function approach, Optik, 158 (2018), 933-939. 

[20]

A. Houwe et al., Nonlinear Schrodingers equations with cubic nonlinearity: M-derivative soliton solutions by $\exp (-\Phi (\xi)) $-Expansion method, https://www.eprints.org/, 2019.

[21]

M. Ilie MousaJ. Biazar and Z. Ayati., Resonant solitons to the nonlinear Schrodinger equation with different forms of nonlinearities, Optik, 164 (2018), 201-209. 

[22]

T. Iizuka and Y. S. Kivshar, Optical gap solitons in nonresonant quadratic media, Phys. Rev. E, 59 (1999), 7148-7151.  doi: 10.1103/PhysRevE.59.7148.

[23]

R. Khalil, A new definition of fractional derivative, J. Comput. Appl. Math., 264 (2014), 65-70.  doi: 10.1016/j.cam.2014.01.002.

[24]

J.-H. Lee and O. K. Pashaev, Solitons of the resonant nonlinear Schrodinger equation with nontrivial boundary conditions: Hirota bilinear method, Theoret. and Math. Phys., 152 (2007), 991-1003.  doi: 10.1007/s11232-007-0083-3.

[25]

J. H. Lee, The resonant nonlinear Schrodinger equation in cold plasma physics : Application of Backlund-Darboux transformations and superposition principles, Journal of Plasma Physics, 73 (2007), 257-272. 

[26]

H. LiJ. W. Haus and P. P. Banerjee, Application of transfer matrix method to second-harmonic generation in nonlinear photonic bandgap structures: Oblique incidence, JOSA B, 32 (2015), 1456-1462.  doi: 10.1364/JOSAB.32.001456.

[27]

B. Lu, The first integral method for some time fractional differential equations, J. Math. Anal. Appl., 395 (2012), 684-693.  doi: 10.1016/j.jmaa.2012.05.066.

[28]

V. V. Nemytskii and V. V. Stepanov, Qualitative Theory of Differential Equations, Princeton University Press, Princeton, N.J. 1960.

[29]

O. K. Pashaev and L. Jyh-Hao, Resonance solitons as black holes in Madelung fluid, Modern Physics Letters A, 17 (2002), 1601-1619. 

[30]

K. Porsezian, Soliton models in resonant and nonresonant optical fibers, Pramana, 57 (2001), 1003-1039. 

[31]

L. Singh, S. Konar and A. K. Sharma, Resonant cross-modulation of two laser beams in a semiconductor slab, Journal of Physics D: Applied Physics, 34 (2001), 2237.

[32]

D. V. Skryabin and W. J. Firth, Dynamics of self-trapped beams with phase dislocation in saturable Kerr and quadratic nonlinear media, Phys. Rev. E, 58 (1998), 3916. doi: 10.1103/PhysRevE.58.3916.

[33]

L. Wenjun and K. Chen, The functional variable method for finding exact solutions of some nonlinear time-fractional differential equations, Pramana, 81 (2013), 377-384. 

[34]

B. Zheng, $(G'/G)$-expansion method for solving fractional partial differential equations in the theory of mathematical physics, Commun. Theor. Phys. (Beijing), 58 (2012), 623-630.  doi: 10.1088/0253-6102/58/5/02.

Figure 1.  Resonant kink-shaped structures of $ |\Psi| $ for (a) $ \lambda = 0.1 $ (red) and $ \lambda = 0.9 $ (orange), with $ t = 50 $ and $ \theta = 30^0 $, (b) $ t = 1 $ (red) and $ t = 200 $ (orange) with $ \lambda = 0.5 $ and $ \theta = 30^0 $, (c) with respect $ x $ and $ \theta $ keeping $ z $-axis constant ($ \theta = 30^0 $, $ \lambda = 0.5 $, $ t = 10 $), and (d) with respect $ z $ and $ \theta $ keeping x-axis constant ($ \theta = 30^0 $, $ \lambda = 0.5 $, $ t = 10 $). The remaining parameters are selected as $ \eta = -1 $, $ \sigma = 0.5 $, $ N = 1 $, $ \delta = 2 $, $ \rho = -0.5 $, $ k = 0.5 $ and $ d = 1 $
Figure 2.  Resonant kink-shaped structures of (a) $ |\Psi| $ for $ \theta = 5^0 $ (red) and $ \theta = 30^0 $ (orange), (b) $ |\Psi| $ for $ \theta = 55^0 $ (orange) and $ \theta = 85^0 $ (red), (c) real part of $ |\Psi| $ with $ \theta = 30^0 $ and (d) imaginary part of $ |\Psi| $ with $ \theta = 30^0 $. The remaining parameters are selected as $ \lambda = 0.5 $, $ \eta = -1 $, $ \sigma = 0.5 $, $ N = 1 $, $ \delta = 2 $, $ \rho = -0.5 $, $ k = 0.5 $, $ t = 10 $ and $ d = 1 $
Figure 3.  Resonant periodic wave structures of (a) $ |\Psi| $ for $ \theta = 45^0 $, (b) $ |\Psi| $ for $ \theta = 80^0 $ and (c) $ |\Psi| $ for $ \theta = 45^0 $ (red) and $ \theta = 80^0 $ (orange). The remaining parameters are selected as $ \eta = -0.1 $, $ \sigma = 0.5 $, $ N = 1.5 $, $ \delta = 1 $, $ \rho = 0.09 $, $ k = 1 $, $ t = 1 $ and $ d = 1 $
Figure 4.  The phase portraits and its vector fields of nonlinear dynamical system as mentioned in Eq. (12) by assuming different values of $ \rho = 0.09 $ (Figs. 4(a) and (b)), $ \rho = 0.01 $ (Figs. 4(c) and (d)) and $ \rho = -0.09 $ (Figs. 4(e) and (f)) with $ N = 0.5 $, $ \omega = 2 $, $ \eta = 0.1 $, $ k = 0.4 $, $ \delta = 1 $ and $ \sigma = 0.5 $
Figure 5.  The (a) phase portrait and its (b) vector fields of the nonlinear dynamical system as mentioned in Eq.(12) by assuming for the values of parameters $ N = 0.5 $, $ \omega = -2 $, $ \eta = 0.1 $, $ k = 0.4 $, $ \delta = 1 $, $ \sigma = 0.5 $, and $ \rho = 0.09 $
Figure 6.  The (a) phase portrait and its (b) vector fields of the nonlinear dynamical system as mentioned in Eq.(12) by assuming for the values of parameters $ N = 1.5 $, $ \omega = 1 $, $ \eta = 0.1 $, $ k = 0.4 $, $ \delta = 1 $, $ \sigma = 0.5 $, and $ \rho = 0.09 $
Figure 7.  The (a) phase portrait and its (b) vector fields of the nonlinear dynamical system as mentioned in Eq.(12) by assuming for the values of parameters $ N = 3 $, $ \omega = 1 $, $ \eta = 0.1 $, $ k = 0.4 $, $ \delta = 1 $, $ \sigma = 0.5 $, and $ \rho = 0.09 $
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