Skip to main content
SpringerLink
Log in

W-shaped surfaces to the nematic liquid crystals with three nonlinearity laws

  • Methodologies and Application
  • Published:
Soft Computing Aims and scope Submit manuscript

Abstract

In this work, we attempt to construct some novel solutions of nematicons within liquid crystals including three types of nonlinearity namely Kerr, parabolic, and power law, using the generalized exponential rational function method. The investigation of nematic liquid crystals, using the proposed method, shows that there is diversity between the solutions gained via this method with those obtained via different methods. Further, we use the constraint conditions to guarantee the existence of the solutions. The W-shaped surfaces, dark soliton, bright soliton, singular soliton, period singular soliton, periodic waves, and complex solutions of the studied equations are successfully constructed. Moreover, some obtained solutions are drawn to a better understanding of the characteristics of nematicons in liquid crystals.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9

Similar content being viewed by others

References

  • Abdelrahman MAE, Sohaly MA (2018) The Riccati–Bernoulli sub-ODE technique for solving the deterministic (stochastic) generalized-Zakharov system. Int J Math Syst Sci. https://doi.org/10.24294/ijmss.v1i3.810

    Article  Google Scholar 

  • Abdulkareem HH, Ismael HF, Panakhov ES, Bulut H (2019) Some novel solutions of the coupled Whitham-Broer-Kaup squations. In: International conference on computational mathematics and engineering sciences. Springer, Cham 1111, pp 200–208

  • Aghdaei MF, Manafian J (2016) Optical soliton wave solutions to the resonant davey-stewartson system. Opt Quant Electron 48(8):413

    Google Scholar 

  • Aksoy E, Çevikel AC, Bekir A (2016) Soliton solutions of (2+1)-dimensional time-fractional Zoomeron equation. Optik 127(17):6933–6942

    Google Scholar 

  • Alberucci A, Assanto G (2007) Dissipative self-confined optical beams in doped nematic liquid crystals. J Nonlinear Opt Phys Mater 16(03):295–305

    Google Scholar 

  • Ali KK, Ismael HF, Mahmood BA, Yousif MA (2017) MHD Casson fluid with heat transfer in a liquid film over unsteady stretching plate. Int J Adv Appl Sci 4(1):55–58

    Google Scholar 

  • Ali KK, Yilmazer R, Yokus A, Bulut H (2020a) Analytical solutions for the (3+ 1)-dimensional nonlinear extended quantum Zakharov–Kuznetsov equation in plasma physics. Physica A 548(15):124327

    MathSciNet  Google Scholar 

  • Ali KK, Yilmazer R, Baskonus HM, Bulut H (2020b) Modulation instability analysis and analytical solutions to the system of equations for the ion sound and Langmuir waves. Phys Scr 95(6):065602

    Google Scholar 

  • Ali KK, Dutta H, Yilmazer R, Noeiaghdam S (2020c) On the new wave behaviors of the Gilson–Pickering equation. Front Phys 8:54

    Google Scholar 

  • Ali KK, Yilmazer R and Bulut H (2020d) Analytical solutions to the coupled Boussinesq–Burgers equations via Sine–Gordon expansion method. In International conference on computational mathematics and engineering sciences (CMES-2019). Springer, Cham 1111, pp 233–240

  • Arnous AH, Ullah MZ, Asma M, Moshokoa SP, Mirzazadeh M, Biswas A, Belic M (2017) Nematicons in liquid crystals by modified simple equation method. Nonlinear Dyn 88(4):2863–2872

    MathSciNet  Google Scholar 

  • Assanto G, Peccianti M, Brzdakiewicz KA, de Luca A, Umeton C (2003a) Nonlinear wave propagation and spatial solitons in nematic liquid crystals. J Nonlinear Opt Phys Mater 12(02):123–134

    Google Scholar 

  • Assanto G, Peccianti M, Conti C (2003b) Spatial optical solitons in bulk nematic liquid crystals. Acta Phys Polonica A 2(103):161–167

    Google Scholar 

  • Baskonus HM, Bulut H (2015) On the numerical solutions of some fractional ordinary differential equations by fractional Adams–Bashforth–Moulton method. Open Math 13:547–556

    MathSciNet  MATH  Google Scholar 

  • Biswas A, Yıldırım Y, Yaşar E, Alqahtani RT (2018a) Optical solitons for Lakshmanan–Porsezian–Daniel model with dual-dispersion by trial equation method. Optik 168:432–439

    Google Scholar 

  • Biswas A, Ekici M, Sonmezoglu A, Triki H, Majid FB, Zhou Q, Moshokoa SP, Mirzazadeh M, Belic M (2018b) Optical solitons with Lakshmanan–Porsezian–Daniel model using a couple of integration schemes. Optik 158:705–11

    Google Scholar 

  • Bulut H, Baskonus HM, Pandir Y (2013) The modified trial equation method for fractional wave equation and time fractional generalized burgers equation. Abstr Appl Anal. https://doi.org/10.1155/2013/636802

    Article  MathSciNet  MATH  Google Scholar 

  • Dubrovsky VG, Lisitsyn YV (2002) The construction of exact solutions of two-dimensional integrable generalizations of Kaup–Kuperschmidt and Sawada–Kotera equations via \(\partial \)-ressing method. Phys Lett A 295(4):198–207

    MathSciNet  MATH  Google Scholar 

  • Dutta H, Günerhan H, Ali KK, Yilmazer R (2020) Exact soliton solutions to the cubic-quartic nonlinear Schrödinger equation with conformable derivative. Front Phys 8:62

    Google Scholar 

  • Ekici M, Mirzazadeh M, Sonmezoglu A, Ullah MZ, Zhou Q, Moshokoa SP, Biswas A, Belic M (2017) Nematicons in liquid crystals by extended trial equation method. J Nonlinear Opt Phys Mater 26(01):1750005

    Google Scholar 

  • Eskitaşçıoğlu Eİ, Aktaş MB, Baskonus HM (2019) New complex and hyperbolic forms for Ablowitz–Kaup–Newell–Segur wave equation with fourth order. Appl Math Nonlinear Sci 4(1):105–112

    MathSciNet  Google Scholar 

  • Gao W, Ismael HF, Mohammed SA, Baskonus HM, Bulut H (2019a) Complex and real optical soliton properties of the paraxial nonlinear Schrödinger equation in Kerr media with M-fractional. Front Phys 7:197

    Google Scholar 

  • Gao W, Ismael HF, Husien AM, Bulut H, Baskonus HM (2019b) Optical soliton solutions of the cubic–quartic nonlinear Schrödinger and Resonant nonlinear Schrödinger equation with the parabolic law. Appl Sci 10(1):219

    Google Scholar 

  • Gao W, Ismael HF, Bulut H, Baskonus HM (2020) Instability modulation for the (2+1)-dimension paraxial wave equation and its new optical soliton solutions in Kerr media. Phys Scr 95(3):035207

    Google Scholar 

  • Ghanbari B (2019) Abundant soliton solutions for the Hirota–Maccari equation via the generalized exponential rational function method. Mod Phys Lett B 33(9):1950106

    MathSciNet  Google Scholar 

  • Gonzalez-Gaxiola O, Biswas A, Belic MR (2019) Optical soliton perturbation of Fokas-Lenells equation by the Laplace–Adomian decomposition algorithm. J Eur Opt Soc-Rapid Publ 15(1):13

    Google Scholar 

  • Guo L, Zhang Y, Xu S, Wu Z, He J (2014) The higher order rogue wave solutions of the Gerdjikov–Ivanov equation. Phys Scr 89(3):035501

    Google Scholar 

  • Hammouch Z, Mekkaoui T, Agarwal P (2018) Optical solitons for the Calogero–Bogoyavlenski-i-Schiff equation in (2 + 1) dimensions with time-fractional conformable derivative. Eur Phys J Plus 133(7):248

    Google Scholar 

  • Hosseini K, Korkmaz A, Bekir A, Samadani F, Zabihi A, Topsakal M (2019) New wave form solutions of nonlinear conformable time-fractional Zoomeron equation in (2 + 1)-dimensions. Waves Random Complex Media 19:1–11

    Google Scholar 

  • Ilhan OA, Bulut H, Sulaiman TA, Baskonus HM (2018) Dynamic of solitary wave solutions in some nonlinear pseudoparabolic models and Dodd–Bullough–Mikhailov equation. Indian J Phys 92(8):999–1007

    Google Scholar 

  • Ihan OA, Esen A, Bulut H, Baskonus HM (2019) Singular solitons in the pseudo-parabolic model arising in nonlinear surface waves. Results Phys 12:1712–1715

    Google Scholar 

  • Ilhan OA, Manafian J, Alizadeh AA, Baskonus HM (2020) New exact solutions for nematicons in liquid crystals by the \(\tan (\phi /2) \)-expansion method arising in fluid mechanics. Eur Phys J Plus 135(3):1–19

    Google Scholar 

  • Ismael HF (2017) Carreau–Casson fluids flow and heat transfer over stretching plate with internal heat source/sink and radiation. Int J Adv Appl Sci J 6(2):81–86

    Google Scholar 

  • Ismael HF, Ali KK (2017) MHD casson flow over an unsteady stretching sheet. Adv Appl Fluid Mech 20(4):533–41

    Google Scholar 

  • Ismael HF, Arifin NM (2018) Flow and heat transfer in a maxwell liquid sheet over a stretching surface with thermal radiation and viscous dissipation. J Heat Mass Transf 15(4):491–498

    Google Scholar 

  • Ismael HF, Bulut H (2019) On the Solitary Wave Solutions to the (2+ 1)-Dimensional Davey-Stewartson Equations. In: International conference on computational mathematics and engineering sciences. Springer, Cham, vol 1111, pp 156–165

  • Ismael HF, Bulut H, Baskonus HM (2020) Optical soliton solutions to the Fokas–Lenells equation via sine-Gordon expansion method and \((m+ (G^{\prime }/ G))\)-expansion method. Pramana-J Phys 94(1):35

    Google Scholar 

  • Kumar D, Joardar AK, Hoque A, Paul GC (2019) Investigation of dynamics of nematicons in liquid crystals by extended sinh-Gordon equation expansion method. Opt Quant Electron 51(7):212

    Google Scholar 

  • Ling L, Feng BF, Zhu Z (2018) General soliton solutions to a coupled Fokas–Lenells equation. Nonlinear Anal Real World Appl 40:185–214

    MathSciNet  MATH  Google Scholar 

  • Manafian J, Foroutan M, Guzali A (2017) Applications of the ETEM for obtaining optical soliton solutions for the Lakshmanan–Porsezian–Daniel model. Eur Phys J Plus 132(11):494

    Google Scholar 

  • Manafian J, Lakestani M, Bekir A (2016) Study of the analytical treatment of the (2+1)-Dimensional Zoomeron, the duffing and the SRLW equations via a new analytical approach. Int J Appl Comput Math 2(2):243–68

    MathSciNet  MATH  Google Scholar 

  • Osman MS, Ghanbari B (2018) New optical solitary wave solutions of Fokas–Lenells equation in presence of perturbation terms by a novel approach. Optik 175:328–33

    Google Scholar 

  • Owolabi KM, Atangana A (2019) On the formulation of Adams–Bashforth scheme with Atangana–Baleanu–Caputo fractional derivative to model chaotic problems. Chaos Interdiscip J Nonlinear Sci 29(2):23111

    MathSciNet  MATH  Google Scholar 

  • Pandey PK, Jaboob SSA (2018) A finite difference method for a numerical solution of elliptic boundary value problems. Appl Math Nonlinear Sci 3(1):311–320

    MathSciNet  Google Scholar 

  • Raza N, Afzal U, Butt AR, Rezazadeh H (2019) Optical solitons in nematic liquid crystals with Kerr and parabolic law nonlinearities. Opt Quant Electron 51(4):107

    Google Scholar 

  • Raza N, Zubair A (2018) Bright, dark and dark-singular soliton solutions of nonlinear Schrödinger’s equation with spatio-temporal dispersion. J Mod Opt 65(17):1975–1982

    Google Scholar 

  • Seadawy AR, Lu D (2017) Bright and dark solitary wave soliton solutions for the generalized higher order nonlinear Schrödinger equation and its stability. Results Phys 7:43–48

    Google Scholar 

  • Sulaiman TA, Bulut H, Yokus A, Baskonus HM (2019) On the exact and numerical solutions to the coupled Boussinesq equation arising in ocean engineering. Indian J Phys 93(5):647–56

    Google Scholar 

  • Vakhnenko VO, Parkes EJ, Morrison AJ (2003) A Bäcklund transformation and the inverse scattering transform method for the generalised Vakhnenko equation. Chaos Solitons Fractals 17(4):683–92

    MathSciNet  MATH  Google Scholar 

  • Yang X, Yang Y, Cattani C, Zhu M (2017) A new technique for solving the 1-D Burgers equation. Thermal Sci 21(suppl. 1):129–36

    Google Scholar 

  • Yang XF, Deng ZC, Wei Y (2015) A Riccati–Bernoulli sub-ODE method for nonlinear partial differential equations and its application. Adv Differ Equ 1:1–7

    MathSciNet  MATH  Google Scholar 

  • Ye Y, Zhou Y, Chen S, Baronio F, Grelu P (2019) General rogue wave solutions of the coupled Fokas-Lenells equations and non-recursive Darboux transformation. Proc R Soc A Math Phys Eng Sci 475(2224):20180806

    MathSciNet  Google Scholar 

  • Yokus A, Baskonus HM, Sulaiman TA, Bulut H (2018) Numerical simulation and solutions of the two-component second order KdV evolutionarysystem. Numer Methods Partial Differ Equ 34(1):211–27

    MATH  Google Scholar 

  • Yokus A, Gülbahar S (2019) Numerical solutions with linearization techniques of the fractional Harry Dym equation. Appl Math Nonlinear Scie 4(1):35–42

    MathSciNet  Google Scholar 

  • Zeeshan A, Ismael HF, Yousif MA, Mahmood T, Rahman SU (2018) Simultaneous effects of slip and wall stretching/shrinking on radiative flow of magneto nanofluid through porous medium. J Magn 23(4):491–8

    Google Scholar 

  • Wan P, Manafian J, Ismael HF, Mohammed SA (2020) Investigating one-, two-, and triple-wave solutions via multiple exp-function method arising in engineering sciences. Adv Math Phys 2020:8018064

    MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Faculty of Science, University of Zakho, Zakho, Iraq

    Hajar Farhan Ismael

  2. Department of Mathematics, Faculty of Science, Firat University, Elazig, Turkey

    Hajar Farhan Ismael & Hasan Bulut

  3. Department of Mathematics and Science Education, Harran University, Şanliurfa, Turkey

    Haci Mehmet Baskonus

Authors
  1. Hajar Farhan Ismael
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Hasan Bulut
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Haci Mehmet Baskonus
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Hajar Farhan Ismael.

Ethics declarations

Conflict of interest

The authors declare that they have no conflict of interest.

Additional information

Communicated by V. Loia.

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Ismael, H.F., Bulut, H. & Baskonus, H.M. W-shaped surfaces to the nematic liquid crystals with three nonlinearity laws. Soft Comput 25, 4513–4524 (2021). https://doi.org/10.1007/s00500-020-05459-6

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00500-020-05459-6

Keywords

Search

Navigation