Hamiltonian Hopf bifurcations and dynamics of NLS/GP standing-wave modes (original) (raw)

Hamiltonian Hopf bifurcations and chaos of NLS/GP standing-wave modes

We examine the dynamics of solutions to nonlinear Schrödinger/Gross-Pitaevskii equations that arise due to Hamiltonian Hopf (HH) bifurcations-the collision of pairs of eigenvalues on the imaginary axis. To this end, we use inverse scattering to construct localized potentials for this model which lead to HH bifurcations in a predictable manner. We perform a formal reduction from the partial differential equations (PDE) to a small system of ordinary differential equations (ODE). We show numerically that the behavior of the PDE is well-approximated by that of the ODE and that both display Hamiltonian chaos. We analyze the ODE to derive conditions for the HH bifurcation and use averaging to explain certain features of the dynamics that we observe numerically.

Exact states in waveguides with periodically modulated nonlinearity

EPL (Europhysics Letters), 2017

We introduce a one-dimensional model based on the nonlinear Schrödinger/Gross-Pitaevskii equation where the local nonlinearity is subject to spatially periodic modulation in terms of the Jacobi dn function, with three free parameters including the period, amplitude, and internal form-factor. An exact periodic solution is found for each set of parameters and, which is more important for physical realizations, we solve the inverse problem and predict the period and amplitude of the modulation that yields a particular exact spatially periodic state. Numerical stability analysis demonstrates that the periodic states become modulationally unstable for large periods, and regain stability in the limit of an infinite period, which corresponds to a bright soliton pinned to a localized nonlinearity-modulation pattern. Exact dark-bright soliton complex in a coupled system with a localized modulation structure is also briefly considered. The system can be realized in planar optical waveguides and cigar-shaped atomic Bose-Einstein condensates.

Radiation from a cut-off point in a two layer nonlinear TE mode waveguide

Wave Motion, 2003

In this work, the propagation of a nonlinear transverse electric (TE) mode in an optical two layer waveguide is considered for the case in which the layers are slowly varying. For a semi-infinite straight boundary between the layers, it is known that trapped modes exist which travel close to the interface. In the present work the upper layer light channel is taken to be of finite extent, while the lower layer is taken to be semi-infinite. The lateral stratification causes trapped modes to cutoff , so that energy is then beamed into the lower layer. In the present work a canonical nonlinear Schrödinger (NLS) equation is obtained which describes, together with an appropriate boundary condition, the radiation beamed into the lower light channel (material layer). It is found from numerical solutions that the radiating mode in the lower layer propagates as a soliton. Approximate solutions for this radiation are found using two methods. The first assumes that the radiating mode is a soliton whose amplitude and width are constant, but whose velocity can vary. The equation governing the soliton velocity is derived using conservation of energy. The second method allows the amplitude, width and velocity all to vary and the equations governing these parameters are obtained from an averaged Lagrangian for the NLS equation. Solutions obtained from the second approximate method are in much better agreement with numerical solutions since the amplitude of the soliton undergoes significant variation in the lower layer (light channel). Since the equation is canonical, it is apparent that nonlinearity induces coherent propagation in the wave radiated into the lower layer (light channel).

Wave motion in infinite inhomogeneous waveguides

Advances in Engineering Software, 2003

The analysis of wave motion in infinite homogeneous waveguides, having a complicated cross-section and/or an irregular inclusion, is a rather difficult task for the majority of available methods, especially when striving for accurate results. In contrast, this presented procedure performed in the frequency domain, is simple to apply. It yields correct results because the radiation conditions are considerably accurately satisfied, and it offers a clear parametric insight into wave motion. This procedure uses the FE modelling of an analysed section of the waveguide. It is based on the decomposition of wave motion, distinguishing propagating and non-propagating wavemodes by solving the eigenvalue problem. The presented examples demonstrate the effectiveness of this procedure, whilst a comparison between computed and analytical results demonstrates its accuracy. q

Quantum anticentrifugal potential in a bent waveguide

Annalen der Physik, 2011

We show the existence of an anticentrifugal force for a quantum particle in a bent waveguide. This counterintuitive force due to dimensionality was shown to exist in a flat R 2 space but there it needs an additional δ-like potential at the origin in order to brake the translational invariance and to exhibit localized states. In the case of the bent waveguide there is no need of any additional potential since here the boundary conditions break the symmetry. The effect may be observed in interference experiments which are sensitive to the additional phase of the wavefunction gained in the bent regions and can find application in distinguishing between straight and bent geometries.

Trapping dynamics in nonlinear wave scattering by local guiding defects

Optics Express, 2008

We study numerically the trapping dynamics of nonlinear waves scattered by local guiding photonic centers with normal eigenmodes which are embedded in uniform nonlinear Kerr waveguides. The linear and nonlinear scattering from a local defect may be treated from either a wave optics approach or a ray optics approach. The former provides a better understanding of the wave dynamics while the latter enables one to perform quasi-analytical estimates of the extent of trapping in a given structure. In the presence of a single-site multi-mode local scattering center, power may localize in a certain normal mode of the center, or periodically oscillate between different normal modes. The degree of trapping and mode of localization can be controlled as function of both the input power and the angle of incidence. With multi-site local scattering centers, the trapping dynamics are strongly dependent on the degree of coupling between the neighboring sites. In the scattering by a local multi-site defect with strongly coupled adjacent sites, two possibilities for nonlinear trapping arise. At intermediate nonlinear powers, periodic tunneling of power between adjacent sites and their normal modes is observed. At highly nonlinear powers, but still within experimental feasibility, the radiation can become strongly localized in a single site. The scattering and trapping dynamics are also described in the context of nonlinear Fabry-Perot etalons, as a function of the local defect's refractive index. induced relaxation to the ground state in a two-level system," Phys. Rev. Lett. 95, 073902 (2005). 23. A. Soffer and M. I. Weinstein, "Theory of nonlinear dispersive waves and selection of the ground state," Phys. Rev. Lett. 95, 213905 (2005). 24. J. Meier, J. Hudock, D. Christodoulides, G. Stegeman, Y. Silberberg, R. Morandotti, and J. S. Aitchison, "Discrete vector solitons in kerr nonlinear waveguide arrays," Phys. Rev. Lett. 91, 143907 (2003). 25. Y. Linzon, Y. Shavit, M. Elazar, R. Morandotti, M. Volatier-Ravat, V. Aimez, R. Ares, and S. Bar-Ad, "Single beam mapping of nonlinear phase shift profiles in planar waveguides with an embedded mirror," Opt. Express 15, 12068-12075 (2007). 26.

Transmission of nonlinear localized modes through waveguide bends

2005

In a recent work, a model for a bend of a nonlinear waveguide in planar geometry was introduced [Yu.S. Kivshar, P.G. Kevrekidis, S. Takeno, Phys. Lett. 307 (2003) 287]. Motivated by photonic-crystal waveguides, we examine transmission of localized pulses through the bend, and identify outcomes of the interaction of a moving pulse with the bend, as a function of the bend's strength and the initial velocity of the pulse. Comparisons with the linear counterpart of the model are also discussed. Some features, such as transition from capture to reflection, may be explained by an analytical perturbation theory based on the quasi-continuum approximation.

Nonlinear localized modes at phase-slip defects in waveguide arrays

Optics Letters, 2008

We study light localization at a phase-slip defect created by two semi-infinite mismatched identical arrays of coupled optical waveguides. We demonstrate that the nonlinear defect modes possess the specific properties of both nonlinear surface modes and discrete solitons. We analyze stability of the localized modes and their generation in both linear and nonlinear regimes.

Geometry and transport in a model of two coupled quadratic nonlinear waveguides

Chaos: An Interdisciplinary Journal of Nonlinear Science, 2008

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Forces on solitons in finite, nonlinear, planar waveguides

Microwave and Optical Technology Letters, 1994

The forces acting on and the energies of solitons governed by the nonlinear Schrödinger equation in finite, planar waveguides with periodic and with homogeneous Dirichlet, Neumann and Robin boundary conditions are determined by means of a quantum analogy. It is shown that these densities have S-shape profiles and increase as the hardness of the boundary conditions is increased.