Saturation effects in ac + dc driven Frenkel–Kontorova model Petar Mali a,⇑ , Jasmina Tekic ´ b , Milan Pantic ´ a a Department of Physics, Faculty of Sciences, University of Novi Sad, Trg Dositeja Obradovic ´a 4, Novi Sad, Serbia b ‘‘Vinc ˇa’’ Institute of Nuclear Sciences, Laboratory for Theoretical and Condensed Matter Physics – 020, University of Belgrade, PO Box 522, 11001 Belgrade, Serbia article info Article history: Received 16 December 2013 Accepted 17 March 2014 Available online 22 March 2014 Keywords: Frenkel–Kontorova model Josephson junctions arrays Charge density waves Shapiro steps abstract Properties of the Shapiro steps in different commensurate structures of the ac + dc driven Frenkel–Kontorova model with asymmetric deformable substrate potential have been examined. Deviation from the sinusoidal potential produces appearance of large subhar- monic steps. In general, different commensurate structures respond differently to the applied force and have different velocities. The results have shown that after shape param- eter gets some critical value, there is no difference in the behavior of Shapiro steps for dif- ferent commensurate structures. The step size and the critical depinning force for different commensurate structures saturate to the same values, due to reduction of degrees of free- dom with the large deformation of potential. Ó 2014 Elsevier B.V. All rights reserved. 1. Introduction Dissipative Frenkel–Kontorova (FK) model has been often used as one of the most suitable models for description of dif- ferent kinds of phenomena in many fields of physics, such as charge or spin density wave systems [1–7], system of Josephson junctions arrays [8–10], and in recent years superconducting nanowires [11,12]. The one-dimensional FK model represents a simple classical model which describes a chain of particles, usually identical, coupled to their nearest neighbors and sub- jected to the periodic or quasiperiodic on-site substrate potential [13,14]. In the standard FK model, interaction among the particles is harmonics, and the on-site substrate potential is sinusoidal [15]. Although very simple, the one-dimensional standard FK model is still complex enough to describe many physical phenomena, and exhibits very rich dynamics when it is submitted under external forces. Dynamics of the ac + dc driven FK model is characterized by the appearance of dynamical mode locking or the Shapiro steps, in the plot of average velocity as a function of the average external driving force tð FÞ. These steps are due to the locking (resonance or interference) of the internal frequency that comes from the motion of particles over the periodic substrate po- tential with the frequency of an external ac force. The dynamical mode locking is only possible if the set of ground states is discrete, otherwise t should be strictly increasing function of F . The steps are called harmonic if the locking appears for inte- ger values of frequency, while if steps appear for the rational noninteger values of frequency they are called subharmonic. Usually, the standard FK model could be only used for description of harmonic mode locking. Although it has been shown that the standard FK model exhibits subharmonic steps [16], they are negligibly small that they can be hardly seen in the plot tð F Þ. However, contrary to the standard case, we have shown in our previous works that very large subharmonic steps might appear in the case of the FK model with deformable substrate potential [17,18]. Potential that we used comes from the large http://dx.doi.org/10.1016/j.cnsns.2014.03.017 1007-5704/Ó 2014 Elsevier B.V. All rights reserved. ⇑ Corresponding author. Tel.: +381 214852810; fax: +381 21459367. E-mail address: petar.mali@df.uns.ac.rs (P. Mali). Commun Nonlinear Sci Numer Simulat 19 (2014) 3469–3475 Contents lists available at ScienceDirect Commun Nonlinear Sci Numer Simulat journal homepage: www.elsevier.com/locate/cnsns