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Register a new userLength scale competition in soliton bearing equations: A collective coordinate approach
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We study the phenomenon of length scale competition, an instability of solitons and other coherent structures that takes place when their size is of the same order of some characteristic scale of the system in which they propagate. Working on the framework of nonlinear Klein-Gordon models as a paradigmatic example, we show that this instability can be understood by means of a collective coordinate approach in terms of soliton position and width. As a consequence, we provide a quantitative, natural explanation of the phenomenon in much simpler terms than any previous treatment of the problem. Our technique allows to study the existence of length scale competition in most soliton bearing nonlinear models and can be extended to coherent structures with more degrees of freedom, such as breathers.
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Stochastically perturbed Korteweg-de Vries (KdV) equations are widely used to describe the effect of random perturbations on coherent solitary waves. We present a collective coordinate approach to describe the effect on coherent solitary waves in stochastically perturbed KdV equations. The collective coordinate approach allows one to reduce the infinite-dimensional stochastic partial differential equation (SPDE) to a finite-dimensional stochastic differential equation for the amplitude, width and location of the solitary wave. The reduction provides a remarkably good quantitative description of the shape of the solitary waves and its location. Moreover, the collective coordinate framework can be used to estimate the time-scale of validity of stochastically perturbed KdV equations for which they can be used to describe coherent solitary waves. We describe loss of coherence by blow-up as well as by radiation into linear waves. We corroborate our analytical results with numerical simulations of the full SPDE.
We demonstrate that soliton-plasmon bound states appear naturally as propagating eigenmodes of nonlinear Maxwells equations for a metal/dielectric/Kerr interface. By means of a variational method, we give an explicit and simplified expression for the full-vector nonlinear operator of the system. Soliplasmon states (propagating surface soliton-plasmon modes) can be then analytically calculated as eigenmodes of this non-selfadjoint operator. The theoretical treatment of the system predicts the key features of the stationary solutions and gives physical insight to understand the inherent stability and dynamics observed by means of finite element numerical modeling of the time independent nonlinear Maxwell equations. Our results contribute with a new theory for the development of power-tunable photonic nanocircuits based on nonlinear plasmonic waveguides.
We demonstrate dynamical topological phase transitions in evolving Su-Schrieffer-Heeger (SSH) lattices made of interacting soliton arrays, which are entirely driven by nonlinearity and thereby exemplify emergent nonlinear topological phenomena. The phase transitions occur from topologically trivial-to-nontrivial phase in periodic succession with crossovers from topologically nontrivial-to-trivial regime. The signature of phase transition is gap-closing and re-opening point, where two extended states are pulled from the bands into the gap to become localized topological edge states. Crossovers occur via decoupling of the edge states from the bulk of the lattice.
In this letter, we investigate the coherent tunneling process of photons between a defected circular resonator and a waveguide based on the recently developed discrete coordinate scattering methods (L. Zhou et al., Phys. Rev. Lett. 101, 100501 (2008)). We show the detailed microscopic mechanism of the tunneling and present a simple model for defect coupling in the resonator. The Finite-Difference Time-Domain(FDTD) numerical results is explored to illustrate the analysis results.
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Peter Y. P. Chen (School of Mechanical
, Manufacturing Engineering
, n University of New South Wales
2006
We consider a model of a matter-wave laser generating a periodic array of solitary-wave pulses. The system, a general version of which was recently proposed in Ref. [5], is composed of two parallel tunnel-coupled cigar-shaped traps (a reservoir and a lasing cavity), solitons being released through a valve at one edge of the cavity. We report a stable lasing mode accounted for by circulations of a narrow soliton in the cavity, which generates an array of strong pulses (with 1,000 - 10,000 atoms in each, the arrays duty cycle ~ 30%) when the soliton periodically hits the valve.
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