We propose amodel of a kink bearing Hamiltonianwith a new non-linear potential V ( q , p ) whose double-well shape can be vaned continuously as a function of the parameter p and which has the q 4 potential as a particular case. Exact classical kink solutions that dependonp areobtained. The rest masses and restenergiesof the kinksare also determined, In recent years non-linear monatomic chain models have been extensively used in condensed-matter physics because they provide a non-perturbation approach to strongly anharmonic systems. The governing equations frequently admit large-amplitude localized field profiles that are physically distinct from those obtainable by superposition of small-amplitude or linearized profiles. These localized large-amplitude excitations canpropagate through thesystem without distortionofshaFeandarecommonly referred to as solitary waves. They exhibit remarkable stability and other particle-like properties.Because of their localized nature, they have found widespread use as one-dimensional models of extended particles in non-linear quantum-field theories, dislocations in crystals, planar domain walls in ferromagnets and ferroelectrics, propagatingflux quanta in Josephson transmission lines, disgyration planes in superfluid He, charge carriers in weakly pinned charge-density-wavecondensates, and charged dislocations in superionic conductors, to mention only a few examples.Various kinds of non-linear potential have been proposed to explain these phenomena. As examples, we may retain the Toda potential (1967), the Lennard-Jones potential, the Morse potential, the sineGordon and double sine-Gordon potentials, the q4, @and @fields, adouble quadratic potential, the Schmidt potential (1979), the Magyari potential (1981), the Behera and Khare potential (1981) and the Remoissenet-Peyrardpotential(l981).Consider a general class of non-linear solitary wave bearing one-dimensional lattice Hamiltonians (see Currie er al1980).where pi is a one-component dimensionless field defined on a one-dimensional lattice of points with lattice constant 1. The first term represents the kinetic energy carried by the field, the second represents harmonic coupling between field values at neighbouring

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Soliton crystals are periodic patterns of multi-spot optical fields formed from either time or space entanglements of equally separated identical high-intensity pulses. These specific nonlinear optical structures have gained interest in recent years with the advent and progress in nonlinear optical fibers and fiber lasers, photonic crystals, wave-guided wave systems and most recently optical ring microresonator devices. In this work an extensive analysis of characteristic features of soliton crystals is carried out, with emphasis on their one-to-one correspondance with Elliptic solitons. In this purpose we examine their formation, their stability and their dynamics in ring-shaped nonlinear optical media within the framework of the Lugiato-Lefever equation. The stability analysis deals with internal modes of the system via a 2 × 2-matrix Lamé type eigenvalue problem, the spectrum of which is shown to possess a rich set of boundstates consisting of stable zero-fequency modes and unstable decaying as well as growing modes. Turning towards the dynamics of Elliptic solitons in ring-shaped fiber resonators with Kerr nonlinearity, first of all we propose a collective-coordinate approach, based on a Lagrangian formalism suitable for Elliptic-soliton solutions to the nonlinear Schródinger equation with an arbitrary perturbation. Next we derive time evolutions of Ellipticsoliton parameters in the specific context of ring-shaped optical fiber resonators, where the optical field evolution is tought to be governed by the Lugiato-Lefever equation. By solving numerically the collective-coordinate equations an analysis of the amplitude, the position, the phase of internal oscillations, the phase velocity and the energy is carried out and reveals a complex dynamics of the Elliptic soliton in ring-shaped optical microresonators.

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During the generation and transmission of nerve impulses, the cytoplasm behaves like an excitable medium that self-regulates the shapes and magnitudes of the output excitation. In connection with this self-regulatory function, one can readily think of the plasma membrane as a nerve organ holding the key role in the mechanisms of generation and transmission of the transmembrane potential, namely, it is expected to provide the essential feedback that stabilizes the stimulus. Here, a simple and coherent picture of self-regulation of the nerve impulse is proposed in terms of one single feedback associated with the main excitable biological organ of the nervous system. In this purpose, an electrodynamic theory is developed within the framework of a cable model in which the membrane capacitor is regarded as a charge-management electrical component with a defined capacity-voltage characteristic. It is found that in both myelinated and myelin-free nerve fiber contexts, the transmembrane excitations are well-localized short impulses whose shape and stability are determined by the capacity-voltage characteristic assumed to govern the self-excitability properties of the nerve membrane.