Europhys. Lett., 60 (2), pp. 174–180 (2002) EUROPHYSICS LETTERS 15 October 2002 Transport of modulated phases by pumping L. M. Flor´ ıa 1,2 , F. Falo 1,3 , P. J. Mart´ ınez 1,4 and J. J. Mazo 1,3 1 Department of Theory and Simulation of Complex Systems, ICMA CSIC-Universidad de Zaragoza - E-50009 Zaragoza, Spain 2 Mathematics Institute, University of Warwick - Coventry CV4 7AL, UK 3 Departamento de F´ ısica de la Materia Condensada, Universidad de Zaragoza E-50009 Zaragoza, Spain 4 Departamento de F´ ısica Aplicada, Universidad de Zaragoza E-50009 Zaragoza, Spain (received 16 May 2002; accepted in final form 30 July 2002) PACS. 05.45.-a – Nonlinear dynamics and nonlinear dynamical systems. PACS. 05.60.-k – Transport processes. PACS. 05.45.Yv – Solitons. Abstract. – We give here some definite answers to the question on the possibility of transport of modulated phases by homogeneous pumping of (pinning) energy. They are presented for the convex class of the Frenkel-Kontorova model (discrete sine-Gordon equation). Our approach leads to a rigorous characterization of the transport under dissipative evolution and large pumping time scales. We also show an explicit example of computed phase diagram of the flow. When combined with Discommensuration Theory, our formal approach (essentially rooted in Aubry-Mather theory) gives an exact account of the features and singularities of this complex phase diagram in generic physical terms. Modulated phases in physical systems arise from the competition between different inter- actions contributing to the system energy, each one imposing its own length scale [1–3]. For discrete modulated structures, it is often the case that they are localized (pinned) due to the existence of Peierls-energy barriers which prevent the modulated phase to slide. The height of the Peierls barrier depends on the relative strength K of the competing interactions, that can often be externally controlled (by, e.g., monitoring an external field, or pressure, ...). We are concerned here with the possibility of transport of modulated phases by (determin- istic) cyclic variation of the relative strength K(t) of the competing interactions: Is there any stationary flow of the distribution density of the modulated variable? In the affirmative, how does the flow depend on physical parameters? What specific symmetries (or lack of them) of the modulating field should be required? What kind of role discommensurations (kinks, solitons) play in the transport? In order to give some definite and physically meaningful answers to these questions, we address them for the convex class of the Frenkel-Kontorova model [4,5]. Our choice is justified c  EDP Sciences