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The point vortex system is a system of longstanding interest in nonlinear dynamics, describing the motion of a two-dimensional inviscid fluid that is irrotational except at a discrete set of moving point vortices, at which the vorticity diverges. The leapfrogging orbit consists of two rotating pairs of like-signed vortices which, taken as a quartet, propagate at constant velocity. It is known that if the two pairs are initially widely separated, the motion is stable, while if they are closer together it becomes unstable, with this relation represented by a dimensionless parameter $\alpha$ defined in the text. We here demonstrate analytically that the transition from stability to instability happens at a critical value $\alpha = \phi^{-2}$, where $\phi$ is the golden ratio. This value had been hypothesized based on careful numerics by Toph{\o}j and Aref, and by the present authors using a semi-analytic argument but not previously demonstrated through exact analysis., Comment: 13 pages, 3 figures

When the coefficients of the cubic terms match the coefficients in the boundary conditions at a vertex of a star graph and satisfy a certain constraint, the nonlinear Schr\"{o}dinger (NLS) equation on the star graph can be transformed to the NLS equation on a real line. Such balanced star graphs have appeared in the context of reflectionless transmission of solitary waves. Steady states on such balanced star graphs can be translated along the edges with a translational parameter and are referred to as the shifted states. When the star graph has exactly one incoming edge and several outgoing edges, the steady states are spectrally stable if their monotonic tails are located on the outgoing edges. These spectrally stable states are degenerate minimizers of the action functional with the degeneracy due to the translational symmetry. Nonlinear stability of these spectrally stable states has been an open problem up to now. In this paper, we prove that these spectrally stable states are nonlinearly unstable because of the irreversible drift along the incoming edge towards the vertex of the star graph. When the shifted states reach the vertex as a result of the drift, they become saddle points of the action functional, in which case the nonlinear instability leads to their destruction. In addition to rigorous mathematical results, we use numerical simulations to illustrate the drift instability and destruction of the shifted states on the balanced star graph., Comment: 30 pages, 6 figures

The nonlinear Schrodinger/Gross–Pitaevskii (NLS/GP) equation is considered in the presence of three equally-spaced potentials. The problem is reduced to a finite-dimensional Hamiltonian system by a Galerkin truncation. Families of oscillatory orbits are sought in the neighborhoods of the system’s nine branches of standing wave solutions. Normal forms are computed in the neighborhood of these branches’ various Hamiltonian Hopf and saddle–node bifurcations, showing how the oscillatory orbits change as a parameter is increased. Numerical experiments show agreement between normal form theory and numerical solutions to the reduced system and NLS/GP near the Hamiltonian Hopf bifurcations and some subtle disagreements near the saddle–node bifurcations due to exponentially small terms in the asymptotics.

We investigate the stability of a one-parameter family of periodic solutions of the four-vortex problem known as `leapfrogging' orbits. These solutions, which consist of two pairs of identical yet oppositely-signed vortices, were known to W.\ Gr\"obli (1877) and A.\ E.\ H.\ Love (1883), and can be parameterized by a dimensionless parameter $\alpha$ related to the geometry of the initial configuration. Simulations by Acheson (2000) and numerical Floquet analysis by Toph{\o}j and Aref (2012) both indicate, to many digits, that the bifurcation occurs when $1/\alpha=\phi^2$, where $\phi$ is the golden ratio. This study aims to explain the origin of this remarkable value. Using a trick from the gravitational two-body problem, we change variables to render the Floquet problem in an explicit form that is more amenable to analysis. We then implement G. W. Hill's method of harmonic balance to high order using computer algebra to construct a rapidly-converging sequence of asymptotic approximations to the bifurcation value, confirming the value found earlier.

A dynamical system is said to be reversible if, given an output, the input can always be recovered in a well-posed manner. Nevertheless, we argue that reversible systems that have a time-reversal symmetry, such as the Nonlinear Schrodinger equation and the ϕ 4 equation can become ”physically irreversible”. By this, we mean that realistically-small experimental errors in measuring the output can lead to dramatic differences between the recovered input and the original one. The loss of reversibility reveals a natural ”arrow of time”, reminiscent of the thermodynamic one, which is the direction in which the radiation is emitted outward. Our results are relevant to imaging and reversal applications in nonlinear optics.

We analyze the fractal structure seen in kink-antikink collisions of the \(\phi ^4\) equation. The analysis is based on qualitative ODE models related to those obtained using the variational approximation. We derive a discrete-time iterated map approximation to the dynamics, from which many features of the model ODE system may be derived. We discuss the problems with Sugiyama’s variational model that have been pointed out recently by Takyi and Weigel.

An efficient and accurate numerical method is presented for computing invariant manifolds of maps which arise in the study of dynamical systems. A quasi-interpolation method due to Hering-Bertram et al. is used to decrease the number of points needed to compute a portion of the manifold. Bezier triangular patches are used in this construction, together with adaptivity conditions based on properties of these patches. Several numerical tests are performed, which show the method to compare favorably with previous approaches.

We investigate nonlinear phenomena in dispersed two-phase systems under creeping-flow conditions. We consider nonlinear evolution of a single deformed drop and collective dynamics of arrays of hydrodynamically coupled particles. To explore physical mechanisms of system instabilities, chaotic drop evolution, and structural transitions in particle arrays we use simple models, such as small-deformation equations and effective-medium theory. We find numerical and analytical solutions of the simplified governing equations. The small-deformation equations for drop dynamics are analyzed using results of dynamical systems theory. Our investigations shed new light on the dynamics of complex fluids, where the nonlinearity often stems from the evolving boundary conditions in Stokes flow.

We consider the interaction of a nonlinear Schrodinger soliton with a spatially localized (point) defect in the medium through which it travels. Using numerical simulations, we find parameter regimes under which the soliton may be reflected, transmitted, or captured by the defect. We propose a mechanism of resonant energy transfer to a nonlinear standing wave mode supported by the defect. Extending Forinash et al. [Phys. Rev. E 49 (1994) 3400], we then derive a finite-dimensional model for the interaction of the soliton with the defect via a collective coordinates method. The resulting system is a three degree-of-freedom Hamiltonian with an additional conserved quantity. We study this system both numerically and using the tools of dynamical systems theory, and find that it exhibits a variety of interesting behaviors, largely determined by the structures of stable and unstable manifolds of special classes of periodic orbits. We use this geometrical understanding to interpret the simulations of the finite-dimensional model, compare them with the nonlinear Schrodinger simulations, and comment on differences due to the finite-dimensional ansatz.

Downstream development is a term encompassing a variety of effects relating to the propagation of storm systems at midlatitude. We investigate a mechanism behind downstream development and study how wave propagation is affected by varying several physical parameters. We then develop a multiple scales modulation theory based on processes in the leading edge of propagating fronts to examine the effect of nonlinearity and weak variation in the background flow. Detailed comparisons are made with numerical experiments for a simple model system.

We study a model derived by Fei et al. [Phys. Rev. A 45 (1992) 6019] of a kink solution to the sine-Gordon equation interacting with an impurity mode. The model is a two degree of freedom Hamiltonian system. We investigate this model using the tools of dynamical systems, and show that it exhibits a variety of interesting behaviors including transverse heteroclinic orbits to degenerate equilibria at infinity, chaotic dynamics, and an extremely complex and delicate structure describing the interaction of the kink with the defect. We interpret this in terms of phase space transport theory.

We study the long-time behavior of solutions to the nonlinear Schr\"odinger / Gross-Pitaevskii equation (NLS/GP) with a symmetric double-well potential, continuing work of the 2nd and 3rd authors. NLS/GP governs nearly-monochromatic guided optical beams in weakly coupled waveguides with both linear and nonlinear (Kerr) refractive indices and zero absorption. The optical power ($L^2$ norm) is conserved with propagation distance. At low optical power, the beam energy executes beating oscillations between the two waveguides. There is an optical power threshold above which the set of guided mode solutions splits into two families of solutions. One type of solution corresponds to an optical beam which is concentrated in either waveguide, but not both. Solutions in the second family undergo tunneling oscillations between the two waveguides. NLS/GP can also model the behavior of Bose-Einstein condensates. A finite dimensional reduction (system of ODEs) well-approximates the PDE dynamics on long time scales. In particular, we derive this reduction, find a class of exact solutions and prove the very long-time shadowing of these solutions by applying the approach of the 2nd and 3rd authors., Comment: 27 pages, 3 figures, Revised version based upon referee reports

Coupled backward and forward wave amplitudes of an electromagnetic field propagating in a periodic and nonlinear medium at Bragg resonance are governed by the nonlinear coupled mode equations (NLCME). This system of PDEs, similar in structure to the Dirac equations, has gap soliton solutions that travel at any speed between 0 and the speed of light. A recently considered strategy for spatial trapping or capture of gap optical soliton light pulses is based on the appropriate design of localized defects in the periodic structure. Localized defects in the periodic structure give rise to defect modes, which persist as {\it nonlinear defect modes} as the amplitude is increased. Soliton trapping is the transfer of incoming soliton energy to {\it nonlinear} defect modes. To serve as targets for such energy transfer, nonlinear defect modes must be stable. We therefore investigate the stability of nonlinear defect modes. Resonance among discrete localized modes and radiation modes plays a role in the mechanism for stability and instability, in a manner analogous to the nonlinear Schr\"odinger / Gross-Pitaevskii (NLS/GP) equation. However, the nature of instabilities and how energy is exchanged among modes is considerably more complicated than for NLS/GP due, in part, to a continuous spectrum of radiation modes which is unbounded above and below. In this paper we (a) establish the instability of branches of nonlinear defect states which, for vanishing amplitude, have a linearization with eigenvalue embedded within the continuous spectrum, (b) numerically compute, using Evans function, the linearized spectrum of nonlinear defect states of an interesting multiparameter family of defects, and (c) perform direct time-dependent numerical simulations in which we observe the exchange of energy among discrete and continuum modes., Comment: 40 pages, 37 figures including subfigures, added references

We present a new and complete analysis of the $n$-bounce resonance and chaotic scattering in solitary-wave collisions. In these phenomena, the speed at which a wave exits a collision depends in a complicated fractal way on its input speed. We present a new asymptotic analysis of collective-coordinate ordinary differential equations (ODEs), reduced models that reproduce the dynamics of these systems. We reduce the ODEs to discrete-time iterated separatrix maps and obtain new quantitative results unraveling the fractal structure of the scattering behavior. These phenomena have been observed repeatedly in many solitary-wave systems over 25 years.

We derive a family of singular iterated maps--closely related to Poincare maps--that describe chaotic interactions between colliding solitary waves. The chaotic behavior of such solitary wave collisions depends on the transfer of energy to a secondary mode of oscillation, often an internal mode of the pulse. Unlike previous analyses, this map allows one to understand the interactions in the case when this mode is excited prior to the first collision. The map is derived using Melnikov integrals and matched asymptotic expansions and generalizes a ``multi-pulse'' Melnikov integral and allows one to find not only multipulse heteroclinic orbits, but exotic periodic orbits. The family of maps derived exhibits singular behavior, including regions of infinite winding. This problem is shown to be a singular version of the conservative Ikeda map from laser physics and connections are made with problems from celestial mechanics and fluid mechanics., Comment: 29 pages, 17 figures, submitted to Chaos, higher-resolution figures available at author's website: http://m.njit.edu/goodman/publications

We consider the interactions of two identical, orthogonally polarized vector solitons in a nonlinear optical fiber with two polarization directions, described by a coupled pair of nonlinear Schroedinger equations. We study a low-dimensional model system of Hamiltonian ODE derived by Ueda and Kath and also studied by Tan and Yang. We derive a further simplified model which has similar dynamics but is more amenable to analysis. Sufficiently fast solitons move by each other without much interaction, but below a critical velocity the solitons may be captured. In certain bands of initial velocities the solitons are initially captured, but separate after passing each other twice, a phenomenon known as the two-bounce or two-pass resonance. We derive an analytic formula for the critical velocity. Using matched asymptotic expansions for separatrix crossing, we determine the location of these "resonance windows." Numerical simulations of the ODE models show they compare quite well with the asymptotic theory., 32 pages, submitted to Physical Review E

A model of soliton-defect interactions in the sine-Gordon equations is studied using singular perturbation theory. Melnikov theory is used to derive a critical velocity for strong interactions, which is shown to be exponentially small for weak defects. Matched asymptotic expansions for nearly heteroclinic orbits are constructed for the initial value problem, which are then used to derive analytical formulas for the locations of the well known two- and three-bounce resonance windows, as well as several other phenomena seen in numerical simulations., Comment: 26 pages, 17 figures

Gap solitons are localized nonlinear coherent states which have been shown both theoretically and experimentally to propagate in periodic structures. Although theory allows for their propagation at any speed $v$, $0\le v\le c$, they have been observed in experiments at speeds of approximately 50% of $c$. It is of scientific and technological interest to trap gap solitons. We first introduce an explicit multiparameter family of periodic structures with localized defects, which support linear defect modes. These linear defect modes are shown to persist into the nonlinear regime, as {\it nonlinear defect modes}. Using mathematical analysis and numerical simulations we then investigate the capture of an incident gap soliton by these defects. The mechanism of capture of a gap soliton is resonant transfer of its energy to nonlinear defect modes. We introduce a useful bifurcation diagram from which information on the parameter regimes of gap soliton capture, reflection and transmission can be obtained by simple conservation of energy and resonant energy transfer principles., 45 pages, Submitted to Journal of the Optical Society B

We consider the nonlinear propagation of light in an optical fiber waveguide as modeled by the anharmonic Maxwell-Lorentz equations (AMLE). The waveguide is assumed to have an index of refraction which varies periodically along its length. The wavelength of light is selected to be in resonance with the periodic structure (Bragg resonance). The AMLE system considered incorporates the effects non-instantaneous response of the medium to the electromagnetic field (chromatic or material dispersion), the periodic structure (photonic band dispersion) and nonlinearity. We present a detailed discussion of the role of these effects individually and in concert. We derive the nonlinear coupled mode equations (NLCME) which govern the envelope of the coupled backward and forward components of the electromagnetic field. We prove the validity of the NLCME description and give explicit estimates for the deviation of the approximation given by NLCME from the {\it exact} dynamics, governed by AMLE. NLCME is known to have gap soliton states. A consequence of our results is the existence of very long-lived {\it gap soliton} states of AMLE. We present numerical simulations which validate as well as illustrate the limits of the theory. Finally, we verify that the assumptions of our model apply to the parameter regimes explored in recent physical experiments in which gap solitons were observed., Comment: To appear in The Journal of Nonlinear Science; 55 pages, 13 figures

We consider the model of catalysis due to Ziff, Gulari, and Barshad [Phys. Rev. Lett. 56, 2553 (1986)], as a pattern formation problem. We find that the model supports trigger waves, and we examine the dependence of the wave velocity on diffusion. In addition to the usual interface width, there is a statistical broadening of the wave front that increases in time as ${\mathit{t}}^{1/3}$. (c) 1995 The American Physical Society

We examine the dynamics of solutions to nonlinear Schrodinger/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., Comment: submitted to J. Phys. A., 32 pages, many figures

While immunosuppressive agents are used widely in the management of Crohn's disease, their efficacy has not been well established in randomised controlled trials. This study was designed to examine whether azathioprine increases remission rate when used in conjunction with a diminishing dose regimen of prednisolone over a period of 12 weeks. It further examined whether azathioprine offers any therapeutic advantage over placebo in the maintenance of remission in Crohn's disease over a period of 15 months. Sixty three patients with active Crohn's disease were treated with a 12 weeks diminishing dose of prednisolone and at the same time entered into a randomised, double blind 15 month trial of either azathioprine (2.5 mg/kg) or placebo. Remission rates between the two groups were compared at 12 weeks and at 15 months. There was no significant difference in the proportion of patients who had achieved and maintained remission by week 12 but at 15 months there was a highly significant difference in the proportion of patients in remission (42% receiving azathioprine v 7% receiving placebo), p = 0.001. Using life tables this beneficial effect was reflected as the difference in the median number of days on the trial (p = 0.02). There were significantly greater decreases over the trial period in the median erythrocyte sedimentation rate, C reactive protein, and leucocyte count in the azathioprine group. There were no cases of severe bone marrow suppression or clinical pancreatitis. In conclusion, azathioprine offers a therapeutic advantage over placebo in the maintenance of remission in Crohn's disease.