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Wave solutions of the time-space fractional complex Ginzburg-Landau equation with Kerr law nonlinearity

Year 2023, Volume: 52 Issue: 6 - Special Issue: Nonlinear Evolution Problems with Applications, 1492 - 1512, 03.11.2023
https://doi.org/10.15672/hujms.1193122

Abstract

In this paper, the bifurcation theory of dynamical system is applied to investigate the time-space fractional complex Ginzburg-Landau equation with Kerr law nonlinearity. We mainly consider the case of $\alpha\neq 2\beta$ which is not discussed in previous work. By overcoming some difficulties aroused by the singular traveling wave system, such as bifurcation analysis of nonanalytic vector field, tracking orbits near the full degenerate equilibrium and calculation of complicated elliptic integrals, we give a total of 20 explicit exact traveling wave solutions of the time-space fractional complex Ginzburg-Landau equation and classify them into 11 categories. Some new traveling wave solutions of this equation are obtained including the compactons and the bounded solutions corresponding to some bounded manifolds.

References

  • [1] E. Aksoy, M. Kaplan and A. Bekir, Exponential rational function method for space-time fractional differential equations, Waves Random Complex Media 26 (2), 142-151, 2016.
  • [2] S. Arshed, Soliton solutions of fractional complex Ginzburg-Landau equation with Kerr law and non-Kerr law media, Optik 160, 322-332, 2018.
  • [3] A. Bekir and Ö. Güner, The (G'/G)-expansion method using modified Riemann-Liouville derivative for some space-time fractional differential equations. Ain Shams Eng. J. 5 (3), 959-965, 2014.
  • [4] M. Caputo and F. Mainardi, Linear models of dissipation in anelastic solids, Riv. Nuovo Cimento (Ser. II) 1 (2), 161-198, 1971.
  • [5] M. Caputo and F. Mainardi, A new dissipation model based on memory mechanism, Pure Appl. Geophys. 91, 134-147, 1971.
  • [6] M. Caputo, Linear models of dissipation whose Q is almost frequency independent-II, Geophys. J. R. Astr. Soc. 13 (5), 529-539, 1967.
  • [7] M.M. Djrbashian and A.B. Nersesian, Fractional derivative and the Cauchy problem for differential equations of fractional order, Izv. Acad. Nauk. Armjanskoi SSR 3 (1), 3-29, 1968.
  • [8] C. Huang and Z. Li, New exact solutions of the fractional complex Ginzburg-Landau equation, Math. Probl. Eng. 2021 (2021).
  • [9] A. Hussain, A. Jhangeer et al., Optical solitons of fractional complex Ginzburg-Landau equation with conformable, beta, and M-truncated derivatives: a comparative study, Adv. Differ. Equ. 2020 (1), 1-19, 2020.
  • [10] X. Jiang and H. Qi, Thermal wave model of bioheat transfer with modified Riemann-Liouville fractional derivative, J. Phys. A. Math. Theor. 45 (48), 485101, 2012.
  • [11] G. Jumarie, Modified Riemann-Liouville derivative and fractional Taylor series of nondifferentiable functions further results, Comput. Math. Appl. 51, 1367-1376, 2006.
  • [12] G. Jumarie, Stochastic differential equations with fractional Brownian motion input, J. Comput. System Sci. 3 (6), 1113-1132, 1993.
  • [13] G. Jumarie, Fourier’s transform of fractional order via Mittag-Leffler function and modified Riemann-Liouville derivative, J. Appl. Math. Inform. 26, 1101-1121, 2008.
  • [14] G. Jumarie, Laplace’s transform of fractional order via the Mittag-Leffler function and modified Riemann-Liouville derivative, Appl. Math. Lett. 22 (11), 1659-1664, 2009.
  • [15] E. Kengne, A. Lakhssassi et al., Exact solutions for generalized variable-coefficients Ginzburg-Landau equation: Application to Bose-Einstein condensates with multi-body interatomic interactions, J. Math. Phys. 53 (12), 123703, 2012.
  • [16] R. Khalil, M. Al Horani, A. Yousef and M. Sababheh, A new definition of fractional derivative, J. Comput. Appl. Math. 264, 65-70, 2014.
  • [17] J. Li, G. Chen and J. Song, Bifurcations and Dynamics of Traveling Wave Solutions for the Regularized Saint-Venant Equation, Int. J. of Bifurcation and Chaos 30 (7), 2050109, 2020.
  • [18] J. Li and J. Zhang, Bifurcations of travelling wave solutions in generalization form of the modified KdV equation, Chaos Solitons Fractals 20 (8), 899-913, 2004.
  • [19] J. Li, W. Zhu and G. Chen, Understanding peakons, periodic peakons and compactons via a shallow water wave equation, Int. J. of Bifurcation and Chaos 26 (12), 1650207, 2016.
  • [20] F. Mainardi, Fractional calculus: some basic problems in continuum and statistical mechanics, Springer, New York, 1997.
  • [21] M. Mainardi and E. Bonetti, The application of real-order derivatives in linear viscoelasticity, Rheol. Acta. 26, 64-67, 1988.
  • [22] M. Mirzazadeh, M. Ekici, A. Sonmezoglu et al., Optical solitons with complex Ginzburg-Landau equation, Nonlinear Dyn. 85 (3), 1979-2016, 2016.
  • [23] A.C. Newell and J.A. Whitehead, Finite bandwidth, finite amplitude convection, J. Fluid Mech. 38 (2), 279-303, 1969.
  • [24] M.D. Ortigueira and F. Coito, From differences to derivatives, Fract. Calc. Appl. Anal. 7 (4), 459-471, 2004.
  • [25] I. Podlubny, Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of their Solution and some of their Applications, Academic Press, San Diego, 1999.
  • [26] I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, 1999.
  • [27] S.D. Zeng, S. Migórski and Z.H. Liu, Well-posedness, optimal control, and sensitivity analysis for a class of differential variational-hemivariational inequalities, SIAM J. Optim. 31, 2829-2862, 2021.
  • [28] M. Sadaf, G. Akram and M. Dawood, An investigation of fractional complex Ginzburg-Landau equation with Kerr law nonlinearity in the sense of conformable, beta and M-truncated derivatives, Opt. Quantum Electron. 54 (4), 1-22, 2022.
  • [29] G. Samko, A. Kilbas and I. Marichev, Fractional Integrals and Derivatives: Theory and Applications, Gordon and Breach Science Publishers, USA, 1993.
  • [30] S.D. Zeng, N.S. Papageorgiou and V.D. Rˇadulescu, Nonsmooth dynamical systems: From the existence of solutions to optimal and feedback control, Bull. Sci. Math. 176, 103131, 2022.
  • [31] S.D. Zeng and E. Vilches, Well-Posedness of History/State-Dependent Implicit Sweeping Processes. J. Optim. Theory Appl. 186, 960-984, 2020.
  • [32] S. Migórski and S.D. Zeng, A class of differential hemivariational inequalities in Ba- nach spaces, J. Glob. Optim. 72, 761-779, 2018.
  • [33] S.R. Saratha, G.Sai Sundara Krishnan and M. Bagyalakshmi, Analysis of a fractional epidemic model by fractional generalised homotopy analysis method using modified Riemann-Liouville derivative, Appl. Math. Model. 92, 525-545, 2021.
  • [34] W. Smit and H. De Vries, Rheological models containing fractional derivatives, Rheol. Acta. 9 (4), 525-534, 1970.
  • [35] K. Stewartson and J.T. Stuart, A non-linear instability theory for a wave system in plane Poiseuille flow, J. Fluid Mech. 48 (3), 529-545, 1971.
  • [36] T.A. Sulaiman, H.M. Baskonus and H. Bulut, Optical solitons and other solutions to the conformable space-time fractional complex Ginzburg-Landau equation under Kerr law nonlinearity, Pramana-J. Phys. 91 (4), 1-8, 2018.
  • [37] B. Tang, Y. He, L. Wei, et al. A generalized fractional sub-equation method for fractional differential equations with variable coefficients, Phys. Lett. A 376, 2588- 2590, 2012.
  • [38] Y. Tang and W. Zhang, Generalized normal sectors and orbits in exceptional directions, Nonlinearity 17 (4), 1407-1426, 2004.
  • [39] Y.R. Bai, N.S. Papageorgiou and S.D. Zeng, A singular eigenvalue problem for the Dirichlet (p,q)-Laplacian, Math. Z. 300, 325-345, 2022.
  • [40] Z.F. Zhang, T.R. Ding, W.Z. Huang and Z.X. Dong, Qualitative Theory of Differential Equations, American Mathematical Society, Providence, RI, USA, 1992.
  • [41] Y. Zhou, G. Chen, and J. Li, Bifurcations, exact peakon, periodic peakons and solitary wave solutions of the modified Camassa-Holm equation, Int. J. of Bifurcation and Chaos 32 (5), 2250076, 2022.
  • [42] Y. Zhou and Q. Liu, Series solutions and bifurcation of traveling waves in the Benney- Kawahara-Lin equation. Nonlinear Dyn. 96 (3), 2055-2067, 2019.
Year 2023, Volume: 52 Issue: 6 - Special Issue: Nonlinear Evolution Problems with Applications, 1492 - 1512, 03.11.2023
https://doi.org/10.15672/hujms.1193122

Abstract

References

  • [1] E. Aksoy, M. Kaplan and A. Bekir, Exponential rational function method for space-time fractional differential equations, Waves Random Complex Media 26 (2), 142-151, 2016.
  • [2] S. Arshed, Soliton solutions of fractional complex Ginzburg-Landau equation with Kerr law and non-Kerr law media, Optik 160, 322-332, 2018.
  • [3] A. Bekir and Ö. Güner, The (G'/G)-expansion method using modified Riemann-Liouville derivative for some space-time fractional differential equations. Ain Shams Eng. J. 5 (3), 959-965, 2014.
  • [4] M. Caputo and F. Mainardi, Linear models of dissipation in anelastic solids, Riv. Nuovo Cimento (Ser. II) 1 (2), 161-198, 1971.
  • [5] M. Caputo and F. Mainardi, A new dissipation model based on memory mechanism, Pure Appl. Geophys. 91, 134-147, 1971.
  • [6] M. Caputo, Linear models of dissipation whose Q is almost frequency independent-II, Geophys. J. R. Astr. Soc. 13 (5), 529-539, 1967.
  • [7] M.M. Djrbashian and A.B. Nersesian, Fractional derivative and the Cauchy problem for differential equations of fractional order, Izv. Acad. Nauk. Armjanskoi SSR 3 (1), 3-29, 1968.
  • [8] C. Huang and Z. Li, New exact solutions of the fractional complex Ginzburg-Landau equation, Math. Probl. Eng. 2021 (2021).
  • [9] A. Hussain, A. Jhangeer et al., Optical solitons of fractional complex Ginzburg-Landau equation with conformable, beta, and M-truncated derivatives: a comparative study, Adv. Differ. Equ. 2020 (1), 1-19, 2020.
  • [10] X. Jiang and H. Qi, Thermal wave model of bioheat transfer with modified Riemann-Liouville fractional derivative, J. Phys. A. Math. Theor. 45 (48), 485101, 2012.
  • [11] G. Jumarie, Modified Riemann-Liouville derivative and fractional Taylor series of nondifferentiable functions further results, Comput. Math. Appl. 51, 1367-1376, 2006.
  • [12] G. Jumarie, Stochastic differential equations with fractional Brownian motion input, J. Comput. System Sci. 3 (6), 1113-1132, 1993.
  • [13] G. Jumarie, Fourier’s transform of fractional order via Mittag-Leffler function and modified Riemann-Liouville derivative, J. Appl. Math. Inform. 26, 1101-1121, 2008.
  • [14] G. Jumarie, Laplace’s transform of fractional order via the Mittag-Leffler function and modified Riemann-Liouville derivative, Appl. Math. Lett. 22 (11), 1659-1664, 2009.
  • [15] E. Kengne, A. Lakhssassi et al., Exact solutions for generalized variable-coefficients Ginzburg-Landau equation: Application to Bose-Einstein condensates with multi-body interatomic interactions, J. Math. Phys. 53 (12), 123703, 2012.
  • [16] R. Khalil, M. Al Horani, A. Yousef and M. Sababheh, A new definition of fractional derivative, J. Comput. Appl. Math. 264, 65-70, 2014.
  • [17] J. Li, G. Chen and J. Song, Bifurcations and Dynamics of Traveling Wave Solutions for the Regularized Saint-Venant Equation, Int. J. of Bifurcation and Chaos 30 (7), 2050109, 2020.
  • [18] J. Li and J. Zhang, Bifurcations of travelling wave solutions in generalization form of the modified KdV equation, Chaos Solitons Fractals 20 (8), 899-913, 2004.
  • [19] J. Li, W. Zhu and G. Chen, Understanding peakons, periodic peakons and compactons via a shallow water wave equation, Int. J. of Bifurcation and Chaos 26 (12), 1650207, 2016.
  • [20] F. Mainardi, Fractional calculus: some basic problems in continuum and statistical mechanics, Springer, New York, 1997.
  • [21] M. Mainardi and E. Bonetti, The application of real-order derivatives in linear viscoelasticity, Rheol. Acta. 26, 64-67, 1988.
  • [22] M. Mirzazadeh, M. Ekici, A. Sonmezoglu et al., Optical solitons with complex Ginzburg-Landau equation, Nonlinear Dyn. 85 (3), 1979-2016, 2016.
  • [23] A.C. Newell and J.A. Whitehead, Finite bandwidth, finite amplitude convection, J. Fluid Mech. 38 (2), 279-303, 1969.
  • [24] M.D. Ortigueira and F. Coito, From differences to derivatives, Fract. Calc. Appl. Anal. 7 (4), 459-471, 2004.
  • [25] I. Podlubny, Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of their Solution and some of their Applications, Academic Press, San Diego, 1999.
  • [26] I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, 1999.
  • [27] S.D. Zeng, S. Migórski and Z.H. Liu, Well-posedness, optimal control, and sensitivity analysis for a class of differential variational-hemivariational inequalities, SIAM J. Optim. 31, 2829-2862, 2021.
  • [28] M. Sadaf, G. Akram and M. Dawood, An investigation of fractional complex Ginzburg-Landau equation with Kerr law nonlinearity in the sense of conformable, beta and M-truncated derivatives, Opt. Quantum Electron. 54 (4), 1-22, 2022.
  • [29] G. Samko, A. Kilbas and I. Marichev, Fractional Integrals and Derivatives: Theory and Applications, Gordon and Breach Science Publishers, USA, 1993.
  • [30] S.D. Zeng, N.S. Papageorgiou and V.D. Rˇadulescu, Nonsmooth dynamical systems: From the existence of solutions to optimal and feedback control, Bull. Sci. Math. 176, 103131, 2022.
  • [31] S.D. Zeng and E. Vilches, Well-Posedness of History/State-Dependent Implicit Sweeping Processes. J. Optim. Theory Appl. 186, 960-984, 2020.
  • [32] S. Migórski and S.D. Zeng, A class of differential hemivariational inequalities in Ba- nach spaces, J. Glob. Optim. 72, 761-779, 2018.
  • [33] S.R. Saratha, G.Sai Sundara Krishnan and M. Bagyalakshmi, Analysis of a fractional epidemic model by fractional generalised homotopy analysis method using modified Riemann-Liouville derivative, Appl. Math. Model. 92, 525-545, 2021.
  • [34] W. Smit and H. De Vries, Rheological models containing fractional derivatives, Rheol. Acta. 9 (4), 525-534, 1970.
  • [35] K. Stewartson and J.T. Stuart, A non-linear instability theory for a wave system in plane Poiseuille flow, J. Fluid Mech. 48 (3), 529-545, 1971.
  • [36] T.A. Sulaiman, H.M. Baskonus and H. Bulut, Optical solitons and other solutions to the conformable space-time fractional complex Ginzburg-Landau equation under Kerr law nonlinearity, Pramana-J. Phys. 91 (4), 1-8, 2018.
  • [37] B. Tang, Y. He, L. Wei, et al. A generalized fractional sub-equation method for fractional differential equations with variable coefficients, Phys. Lett. A 376, 2588- 2590, 2012.
  • [38] Y. Tang and W. Zhang, Generalized normal sectors and orbits in exceptional directions, Nonlinearity 17 (4), 1407-1426, 2004.
  • [39] Y.R. Bai, N.S. Papageorgiou and S.D. Zeng, A singular eigenvalue problem for the Dirichlet (p,q)-Laplacian, Math. Z. 300, 325-345, 2022.
  • [40] Z.F. Zhang, T.R. Ding, W.Z. Huang and Z.X. Dong, Qualitative Theory of Differential Equations, American Mathematical Society, Providence, RI, USA, 1992.
  • [41] Y. Zhou, G. Chen, and J. Li, Bifurcations, exact peakon, periodic peakons and solitary wave solutions of the modified Camassa-Holm equation, Int. J. of Bifurcation and Chaos 32 (5), 2250076, 2022.
  • [42] Y. Zhou and Q. Liu, Series solutions and bifurcation of traveling waves in the Benney- Kawahara-Lin equation. Nonlinear Dyn. 96 (3), 2055-2067, 2019.
There are 42 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Mathematics
Authors

Niping Cai 0000-0002-7459-4038

Yuqian Zhou 0000-0002-5493-5966

Qian Liu 0000-0002-6413-1224

Publication Date November 3, 2023
Published in Issue Year 2023 Volume: 52 Issue: 6 - Special Issue: Nonlinear Evolution Problems with Applications

Cite

APA Cai, N., Zhou, Y., & Liu, Q. (2023). Wave solutions of the time-space fractional complex Ginzburg-Landau equation with Kerr law nonlinearity. Hacettepe Journal of Mathematics and Statistics, 52(6), 1492-1512. https://doi.org/10.15672/hujms.1193122
AMA Cai N, Zhou Y, Liu Q. Wave solutions of the time-space fractional complex Ginzburg-Landau equation with Kerr law nonlinearity. Hacettepe Journal of Mathematics and Statistics. November 2023;52(6):1492-1512. doi:10.15672/hujms.1193122
Chicago Cai, Niping, Yuqian Zhou, and Qian Liu. “Wave Solutions of the Time-Space Fractional Complex Ginzburg-Landau Equation With Kerr Law Nonlinearity”. Hacettepe Journal of Mathematics and Statistics 52, no. 6 (November 2023): 1492-1512. https://doi.org/10.15672/hujms.1193122.
EndNote Cai N, Zhou Y, Liu Q (November 1, 2023) Wave solutions of the time-space fractional complex Ginzburg-Landau equation with Kerr law nonlinearity. Hacettepe Journal of Mathematics and Statistics 52 6 1492–1512.
IEEE N. Cai, Y. Zhou, and Q. Liu, “Wave solutions of the time-space fractional complex Ginzburg-Landau equation with Kerr law nonlinearity”, Hacettepe Journal of Mathematics and Statistics, vol. 52, no. 6, pp. 1492–1512, 2023, doi: 10.15672/hujms.1193122.
ISNAD Cai, Niping et al. “Wave Solutions of the Time-Space Fractional Complex Ginzburg-Landau Equation With Kerr Law Nonlinearity”. Hacettepe Journal of Mathematics and Statistics 52/6 (November 2023), 1492-1512. https://doi.org/10.15672/hujms.1193122.
JAMA Cai N, Zhou Y, Liu Q. Wave solutions of the time-space fractional complex Ginzburg-Landau equation with Kerr law nonlinearity. Hacettepe Journal of Mathematics and Statistics. 2023;52:1492–1512.
MLA Cai, Niping et al. “Wave Solutions of the Time-Space Fractional Complex Ginzburg-Landau Equation With Kerr Law Nonlinearity”. Hacettepe Journal of Mathematics and Statistics, vol. 52, no. 6, 2023, pp. 1492-1, doi:10.15672/hujms.1193122.
Vancouver Cai N, Zhou Y, Liu Q. Wave solutions of the time-space fractional complex Ginzburg-Landau equation with Kerr law nonlinearity. Hacettepe Journal of Mathematics and Statistics. 2023;52(6):1492-51.