A new integrable two-component system with cubic nonlinearityFollowing the considerable progress in nanoribbon technology, we propose to model the nonlinear Frenkel-like excitations on a triangular-lattice ribbon by the integrable nonlinear ladder system with the background-controlled intersite resonant coupling. The system of interest arises as a proper reduction of first general semidiscrete integrable system from an infinite hierarchy. The most significant local conservation laws related to the first general integrable system are found explicitly in the framework of generalized recursive approach. The obtained general local densities are equally applicable to any general semidiscrete integrable system from the respective infinite hierarchy. Using the recovered second densities, the Hamiltonian formulation of integrable nonlinear ladder system with background-controlled intersite resonant coupling is presented. In doing so, the relevant Poisson structure turns out to be essentially nontrivial. The Darboux transformation scheme as applied to the first general semidiscrete system is developed and the key role of Bäcklund transformation in justification of its self-consistency is pointed out. The spectral properties of Darboux matrix allow to restore the whole Darboux matrix thus ensuring generation one more soliton as compared with a priori known seed solution of integrable nonlinear system. The power of Darboux-dressing method is explicitly demonstrated in generating the multicomponent one-soliton solution to the integrable nonlinear ladder system with background-controlled intersite resonant coupling. C 2015 AIP Publishing LLC.[http://dx.doi.org/10.1063/1.4914510] under the proper choice of spectral L(n|z) and evolution A(n|z) operators. Here, the integer n standing for the discrete spatial coordinate runs from minus to plus infinity, the letter z marks the time-independent complex-valued spectral parameter, while the overdot assumes the differentiation over the time variable τ:ḟ (n) ≡ d f (n)/dτ. In its original formulation, 1 the system of our interest has been derived relying upon certain Laurent-type form of 4 × 4 square matrices L(n|z) and A(n|z). In doing so, the analysis of a limiting spectral problem dictated by the auxiliary spectral operator a) vakhnenko@bitp.kiev.ua

We developed and thoroughly examined a model of longitudinal vibrational resonance in bar-shaped sedimentary rocks; these materials exhibit memory that originates from an essential asymmetry in processes of rupture and recovery of intergrain and interlamina cohesive bonds. The theory relies on an appropriate isolation and an adequate formalization of two mutually dependent subsystems, namely, a subsystem of ruptured bonds and a subsystem of internal longitudinal displacements. The subsystem of ruptured bonds is shown to be of a soft-ratchet type, so that its response to an alternating internal stress is characterized by broken symmetry and appears as nonzero long-term temporal and spatial changes in the concentration of ruptured bonds. The internal stress is generated by an alternating external drive acting both directly through the subsystem of longitudinal displacements and indirectly through temporal and spatial modifications of Young's modulus due to changes in concentration of ruptured bonds. The scheme reproduces the main experimental effects by using the simplest linear form of attenuation in an elastic subsystem and realistic assumptions about the stress-strain relation. In particular, it correctly describes: hysteretic behavior of a resonance curve on both its upward and downward slopes; linear softening of resonant frequency with increase of driving level; gradual ͑almost logarithmic͒ recovery ͑increase͒ of resonant frequency at low dynamical strains after the sample was conditioned by high strains; and temporal relaxation of response acceleration amplitude at fixed frequency. These are the most interesting observations typical of forced longitudinal oscillations of sandstone bars in the nonlinear regime. Further, we are able to trace how water saturation enhances the hysteresis and simultaneously decreases the quality factor because of an increase in equilibrium concentration of ruptured cohesive bonds. We also predict theoretically a dynamical effect analogous to the widely known quasistatic effect of hysteresis with discrete ͑end-point͒ memory.

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A closed-form description is proposed to explain nonlinear and slow dynamics effects exhibited by sandstone bars in longitudinal resonance experiments. Along with the fast subsystem of longitudinal nonlinear displacements we examine the strain-dependent slow subsystem of broken intergrain and interlamina cohesive bonds. We show that even the simplest but phenomenologically correct modeling of their mutual feedback elucidates the main experimental findings typical for forced longitudinal oscillations of sandstone bars, namely, (i) hysteretic behavior of a resonance curve on both its upward and downward slopes, (ii) linear softening of resonant frequency with an increase of driving level, and (iii) gradual recovery (increase) of resonant frequency at low dynamical strain after the sample was conditioned by high strain. In order to reproduce the highly nonlinear elastic features of sandstone grained structure a realistic nonperturbative form of stress-strain relation was adopted. In our theory slow dynamics associated with the experimentally observed memory of peak strain history are attributed to strain-induced kinetic changes in concentration of ruptured intergrain and interlamina cohesive bonds, causing a net hysteretic effect on the elastic Young's modulus. Finally, we explain how enhancement of hysteretic phenomena originates from an increase in equilibrium concentration of ruptured cohesive bonds that are due to water saturation.

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