Solitons and shock waves in molecular chains

In this field the dynamical properties of one dimensional discrete systems which describe molecular chains have been studied. In the paper [43] a new discrete version of the nonlinear Schrödinger equation (NLS) was introduced. This equation was reported as "Salerno's equation" in the book Nonlinear Science, by A.C.Scott. From a physical point of view it represents a nonlinear generalization of the tight-binding Schrödinger model for the dynamics of a quasiparticle in a molecular crystal, with the nonlinear terms modeling the interaction of the quasiparticle with the lattice. Among the interesting properties of this equation there is the fact that it is a q-deformation of the usual discrete, non integrable, version of the NLS, which reduce, for a particular value of the deformation parameter to the integrable discrete version. In the work [49] the modulational instability of Salerno's equation has been studied, while in [56] it has been demonstrated that the presence of an electrical field in the system induces spatial oscillations of the soliton which give rise to dynamical localization. This result suggests an interesting analogy between the soliton dynamics and the one of an electron moving in a perfect crystal in presence of electric fields (Bloch oscillations). In the paper [56] it has been shown that Bloch oscillations of a soliton in such system are possible also in the presence of time dependent electric fields. In [61] the interaction of solitons with electric fields and impurity centers (both of dissipative and conservative nature), has been investigated. In particular, it was shown that depending on the intensity of the impurity, the soliton can be either pinned or executing oscillations around the impurity leading to the dynamical localization phenomenon.

The existence of shock waves in discrete systems such as chains of two-level atoms, Heisenberg ferromagnets, nonlinear Schrödinger chains, etc., has been investigated. In particular, in the work [64] the existence of solitons of bright and dark type in Salerno's equation have been considered. The stability regions of these solutions have been studies as a function of the parameters of the system, finding, quite surprisingly, that on the borders of these regions anomalous dispersion is possible leading to shock waves formation. In the papers [65,P12] the dynamics of these shocks have been characterized as a function of the background wave-number inside the Brillouin zone. It was shown that shock waves with rectangular wave front followed by a train of solitons and by background radiation, can exist. In [66] this analysis was extended to chains of two-level atoms in the presence of exchange and dipole-dipole interaction describing Frenkel excitons. In particular it was found that the exchange interaction helps the formation of the shocks (both of bright and dark type) while the dipole-dipole interaction works against it, making unstable the background field. The possibility of shock waves in anisotropic Heisenberg quantum chains was investigated in [70]. The analysis has been done by using the coherent state representation and the stationary phase approximation to derive quasiclassical equation of motion. It was shown that in this approximation values of the anisotropy parameter exist for which anomalous dispersion and shock wave formation is possible. From a physical point of view these shock waves separate different magnetization regions and therefore are relevant for the formation magnetic domains. This analysis has been extended to the case of the pure quantum regime in [75].

In [12] a mechanical model for a class of "multiple" sine-Gordon equations was introduced and the quasi-soliton solution of the double sine-Gordon system investigated. In particular, in the papers [19, 20] the anharmonicity effects in the small oscillations around the 4p-kink solutions of the double sine-Gordon system were considered. In the papers [4,11, 14] it has been investigated, both from an analytical and numerical point of view, the effect of adiabatic perturbations on the sine-Gordon system.

Back to Mario Salerno home page