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Abstract We consider the Toda lattice with exponentially repulsive interactions between the units and view these units as Brownian elements capable of pumping energy from a surrounding heat bath or reservoir, and we show that solitons can be excited and maintained in the presence of dissipation. Then, we endow these Toda lattice units with electric charge, i.e., we make them positive ions and add free electrons to the system. We use this to show that, in the presence of an external electric field, following an instability of the base linear Ohm conduction state, the electromechanical Toda lattice is able to maintain a non-Ohmic soliton-driven supersonic electric current, and we then discuss its striking characteristics. Thus the lattice appears very much like a versatile neural transmission line.

1 Introduction The soliton concept, and the coinage of the word soliton, originates from the work of Zabusky and Kruskal [1] (see also [2, 3, 4]). They dealt with the dynamics of one-dimensional (1D) anharmonic lattices and their (quasi) continuum approximation [5] provided by the Boussinesq–Korteweg-de Vries (BKdV) equation [6, 7]. That work followed research done by Fermi et al. [8] (see also [9]) who tried to understand equi-partition in a lattice by adding anharmonic forces. They used 1D lattices with 16, 32 and 64 units interacting with springs obeying x2 and x3 anharmonic M.G. Velarde Instituto Pluridisciplinar, Universidad Complutense de Madrid, Paseo Juan XXIII, 1, E-28040 Madrid, Spain, [email protected] W. Ebeling Institut f¨ur Physik, Humboldt-Universit¨at Berlin, Newtonstr. 15, D-12489 Berlin, Germany, [email protected] A.P. Chetverikov Faculty of Physics, Chernychevsky State University, Astrakhanskaya 83, 410012 Saratov, Russia, [email protected]

Velarde, M.G. et al.: Anharmonic Oscillations, Dissipative Solitons and Non-Ohmic Supersonic Electric Transport. Lect. Notes Phys. 751, 321–335 (2008) c Springer-Verlag Berlin Heidelberg 2008 DOI 10.1007/978-3-540-78217-9 12 

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forces and another described by a nonlinear but “piecewise linear” function. The significant achievements of Visscher and collaborators are also worth mentioning. While trying to understand heat transfer, they used the Lennard-Jones potential [10] to explore the role of anharmonicity and of impurities (i.e., doping a given lattice with different masses, thus generating isotopically disordered lattices). More recently, Heeger, Schrieffer and collaborators have used solitons to explain the electrical conductivity of polymers [11]. Finally, we ought to highlight the work done by Toda (1967) on the lattice (which he invented) with a peculiar exponential interaction [12, 13], since here we build upon the results obtained by Toda. [N.B. We make no claim of completeness in the list of references offered here. For a thorough, albeit now a bit old, review of solitons in condensed matter, see Bishop et al. [14]; for an in-depth discussion of heat transfer see Toda [15].] In one limit, the Toda interacti