Fractional effects on solitons in a 1D array of rectangular ferroelectric nanoparticles

References

• Lorenzo CF, Hartley TT. The fractional trigonometric: with applications to fractional differential equations and science. Hoboken (NJ): Wiley; 2017.
   Google Scholar
• Tarasov VE. Fractional dynamics: applications of fractional calculus to dynamic of particles field and media. Berlin: Springer-Verlag; 2010 ( Nonlinear Physical Science).
   Google Scholar
• Osler TJ. The fractional derivative of a composite function. SIAM J Math Anal. 1970;1:288–293 doi: 10.1137/0501026
   Google Scholar
• Osler TJ. Taylor's series generalized for fractional derivatives and applications. SIAM J Math Anal. 1971;2:37–48 doi: 10.1137/0502004
   Google Scholar
• Herrmann R. Fractional calculus: an introduction for physicists. Germany: World Scientific, GigaHedron; 2014.
   Google Scholar
• Zheng B. A new fractional Jacobi elliptic equation method for solving fractional partial differential equation. Adv Differ Equ. 2014;2014:288.
   Google Scholar
• Ye ZG. Handbook of advanced dielectric, piezoelectric and ferroelectric materials: synthesis, properties and applications. Cambridge: Woodhead Publishing Ltd; 2008.
   Google Scholar
• Kobune M. Ferrolectric materials and their applications. Jpn J Appl Phys. 2015;54(N10):01.
   Google Scholar
• Ertug B. The Overview of the electrical properties of Barium Titanate. Am J Eng Res. 2013;2(8):1–7.
   Google Scholar
• Meyer V, Sallese J-M, Fazan P, et al. Modeling the polarization in ferroelectric materials: a novel analytical approach. Solid State Electron. 2003;47:1479–1486. doi: 10.1016/S0038-1101(03)00104-7
   Web of Science ®Google Scholar
• Matsuo H. Heavy Mn-doping effect on spontaneous polarisation in ferroelectric BiFeO3 thin films. Jpn J Appl Phys. 2015;54(N10):03.
   Google Scholar
• Kalinin SV, Morozovska AN, Chen LQ, et al. Local polarization dynamics in ferroelectric materials. Rep Prog Phys. 2010;73:056502. doi: 10.1088/0034-4885/73/5/056502
   Web of Science ®Google Scholar
• Fellberg W, Mantese J, Schubring N, et al. Origin of the “up,” “down” hysteresis offsets observed from polarization-graded ferroelectric materials. Appl Phys Lett. 2001;78:524–526. doi: 10.1063/1.1342045
   Web of Science ®Google Scholar
• Petrovsky V, Petrovsky T, Kamlapurkar S, et al. Dielectric constant of barium titanate powders near curie temperature. J Am Ceram Soc. 2008;91(N11):3590–3592. doi: 10.1111/j.1551-2916.2008.02693.x
   Google Scholar
• Dayal K, Bhattacharya K. Active tuning of photonic device characteristics during operation by ferroelectric domain switching. J Appl Phys. 2007;102:064102. doi: 10.1063/1.2779261
   Web of Science ®Google Scholar
• Bottcher CJF. Theory of electric polarization. New York: Elsevier Science Publishing; 1993.
   Google Scholar
• Agrawal OP. Formulation of Euler–Lagrange equations for fractional variational problems. J Math Anal Appl. 2002;272:368–379. doi: 10.1016/S0022-247X(02)00180-4
   Web of Science ®Google Scholar
• Garbovskiy Y, Zribi O, Glushchenko A. Emmerging applications of ferroelectric nanoparticles in materials technologies, biology and medicine. Intech. 2013. doi:10.5772/52516
   Google Scholar
• Zhen QC, Kyeong HK, Panki K. Fractional time stochastic partial differential equations. Stoch Proc Appl. 2015;125(4):1470–1499. doi: 10.1016/j.spa.2014.11.005.
   Web of Science ®Google Scholar
• Tarasov VE. On chain rule for fractional derivatives. Commun Nonlinear Sci Numer Simul. 2016;30:1–4 doi: 10.1016/j.cnsns.2015.06.007
   Web of Science ®Google Scholar
• Aslan I. Exact solutions for a local fractional DDE associated with a nonlinear transmission line. Commun Theor Phys. 2016;66:315–320. doi: 10.1088/0253-6102/66/3/315
   Web of Science ®Google Scholar
• Lu H, Bates PW, Lu SN. Dynamics of the 3-D fractional complex Ginzburg–Landau equation. J Differ Equ. 2015;259:5276–5301. doi: 10.1016/j.jde.2015.06.028
   Web of Science ®Google Scholar
• Jumarie G. On the derivative chain-rules in fractional calculus via fractional difference and their application to systems modeling. Cent. Eur. J. Phys. 2013. doi:10.2478/s11534-013-0256-7
   Google Scholar
• Jumarie G. Stochastic differential equations with fractional Brownian motion input. Int J Syst Sci. 1993;24(N6):1113–1131. doi: 10.1080/00207729308949547
   Google Scholar
• Guyomar D, Ducharne B, Sebald G. The use of fractional derivation in modeling ferroelectric dynamic hysteresis behavior over large frequency bandwidth. J Appl Phys. 2010;107:114108. doi: 10.1063/1.3393814
   Web of Science ®Google Scholar
• Guyomar D, Ducharne B, Sebald G, et al. Fractional derivative operators for modeling the dynamic polarisation behavior as a function of frequency and electrc field amplitude. IEEE. 2009;56:437.
   Google Scholar
• Ozkan G, Omer U, Ahmet B, et al. Soliton solutions and other solutions to a nonlinear fractional differential equation. Ame. Ins. Phys. 2017;1798,020169. doi:10.1063/1.4972761
   Google Scholar
• Miller KS, Ross B. An introduction to the fractional calculus and fractional differential equations. New York (NY): Wiley; 1993.
   Google Scholar
• Jumarie G. Modified Riemann–Liouville derivative and fractional Taylor series of nondifferentiable functions further results. Comput Math Appl. 2006;51:1367–1376. doi: 10.1016/j.camwa.2006.02.001
   Web of Science ®Google Scholar
• Eslami M, Vajargah BF, Mirzazadeh M, et al. Application of first integral method to fractional partial differential equations. Indian J Phys. 2014;88:177–184. doi: 10.1007/s12648-013-0401-6
   Web of Science ®Google Scholar
• Zhen W. Discrete tanh method for nonlinear difference–differential equations. Comput Phys Commun. 2009;180:1104–1108 doi: 10.1016/j.cpc.2009.01.010
   Web of Science ®Google Scholar
• Amadou Y, Betchewe G, Douvagai D, et al. Discrete exact solutions for the double-well potential model through the discrete tanh method. Eur Phys J Plus. 2015;130:52. doi: 10.1140/epjp/i2015-15013-9
   Web of Science ®Google Scholar
• He JH. The variational iteration method for eighth-order initial-boundary value problems. Phys Scr. 2007;76:680–682. doi: 10.1088/0031-8949/76/6/016
   Web of Science ®Google Scholar
• Mai Z, Fu S, Wu J, et al. Discrete solitons in waveguide arrays with long-range linearly coupled effect. J Phys Soc Jpn. 2014;83:034404. doi: 10.7566/JPSJ.83.034404
   Google Scholar
• Guner O, Bekir A. Bright and dark soliton solutions for some nonlinear fractional differential equations. Chin Phys B. 2016;25:030203. doi: 10.1088/1674-1056/25/3/030203
   Web of Science ®Google Scholar
• Lin Z, Xu W, Yue X, et al. Study on the effect of environmental pollution based on a fractional derivative resource depletion model. Chaos Solitons Fractals. 2017;104:705–715. doi: 10.1016/j.chaos.2017.09.019
   Web of Science ®Google Scholar
• Pooseh S, Almeida R, Torres DFM, et al. Discrete direct methods in the fractional calculus of variations. Comput Math Appl. 2013;66:668–676. doi: 10.1016/j.camwa.2013.01.045
   Web of Science ®Google Scholar
• Rizvi STR, Ali K, Bashir S, et al. Exact soliton of (2+1)-dimensional fractional Schrödinger equation. Super Micro. 2017;107:234–239. doi: 10.1016/j.spmi.2017.04.029
   Web of Science ®Google Scholar
• Zhou Q, Sonmezoglu A, Ekici M, et al. Optical solitons of some fractional differential equations in nonlinear optics. J Mod Opt. 2017;64:2345–2349. doi: 10.1080/09500340.2017.1357856
   Web of Science ®Google Scholar
• Reyna AS, Bergmann E, Brevet P-F, et al. Nonlinear polarization instability in cubic-quintic plasmonic nanocomposites. Opt Express. 2017;25:21049. doi: 10.1364/OE.25.021049
   PubMed Web of Science ®Google Scholar