Wave Motion 31 (2000) 197–207 Dispersive–dissipative solitons in nonlinear solids A.V. Porubov a,b , M.G. Velarde a,∗ a Instituto Pluridisciplinar, Universidad Complutense de Madrid, Paseo Juan XXIII, n 0 1, 28040, Madrid, Spain b A.F. Ioffe Physical Technical Institute of the Russian Academy of Sciences, St. Petersburg, 194021, Russia Received 9 November 1998; received in revised form 9 March 1999; accepted 25 August 1999 Abstract It is found that the propagation of long nonlinear longitudinal strain waves in an elastic rod embedded in a viscoelastic external medium may be governed by a nonlinear dispersive–dissipative equation with an appropriate input–output energy balance. This equation admits exact traveling solitary wave solutions. When dissipation is small enough an initially dissipa- tionless strain solitary wave is shown to transform to a dissipative solitary wave with prescribed parameters. ©2000 Elsevier Science B.V. All rights reserved. 1. Introduction Recently, the theory has been developed to account for long longitudinal localized strain waves propagating in a free elastic rod [1–3]. The nonlinearity, caused by both the finite stress values and elastic material properties, and the dispersion resulting from the finite transverse size of the rod, when in balance allow the propagation of such localized strain waves in the dissipationless elastic medium. The equation governing this process is of Boussinesq type, namely, a double dispersive equation as called in [2]. It admits, in particular, an exact solitary wave solution and possesses some conservation laws. Numerical simulations have illustrated the time dependent generation of the long strain solitary wave [3,4]. Furthermore, there has been successful experimental generation of such strain solitons in beautiful agreement with the theory [5,6]. In this paper we turn our attention to the role played by dissipation always present in a realistic case. Dissipative effects may be caused by internal features of the elastic material, hence, an irreversible part should be included into the stress tensor in addition to the reversible one depending only upon the density of the Helmholtz energy [7]. Accordingly, the governing equations for strains will contain viscoelastic terms. Dissipation may also occur in an elastic waveguide through phenomena occurring at or across its lateral surface as shown, e.g., by Kerr [8]. Based on the results of footing load tests performed on a snow base he proposed a viscoelastic model for the interaction between an elastic body and the external snow (or permafrost) medium. Here we consider the role of Kerr’s model in the propagation of long bulk strain waves in a nonlinearly elastic rod. This approach may help in setting up a valuable ∗ Corresponding author. E-mail address: mvelarde@eucmax.sim.ucm.es (M.G. Velarde). 0165-2125/00$ – see front matter ©2000 Elsevier Science B.V. All rights reserved. PII:S0165-2125(99)00032-3