An instability mechanism for radially symmetric standing waves of a nonlinear Schrödinger equation

AbstractA condition is proved for the spectrum of nonlinear Schrödinger equations linearised at a standing wave to have a positive eigenvalue. The standing waves considered are radially symmetric ones in higher space dimensions. The instability result is applied to show that if there are multiple asymptotically positive, nondegenerate waves with a fixed number of zeroes then there is an unstable one. The techniques used are dynamical systems arguments and involve a shooting argument in the space of Lagrangian planes

A coherent structure approach for parameter estimation in Lagrangian Data Assimilation

We introduce a data assimilation method to estimate model parameters with observations of passive tracers by directly assimilating Lagrangian Coherent Structures. Our approach differs from the usual Lagrangian Data Assimilation approach, where parameters are estimated based on tracer trajectories. We employ the Approximate Bayesian Computation (ABC) framework to avoid computing the likelihood function of the coherent structure, which is usually unavailable. We solve the ABC by a Sequential Monte Carlo (SMC) method, and use Principal Component Analysis (PCA) to identify the coherent patterns from tracer trajectory data. Our new method shows remarkably improved results compared to the bootstrap particle filter when the physical model exhibits chaotic advection

Geometric phase in the Hopf bundle and the stability of non-linear waves

We develop a stability index for the traveling waves of non-linear reaction–diffusion equations using the geometric phase induced on the Hopf bundle . This can be viewed as an alternative formulation of the winding number calculation of the Evans function, whose zeros correspond to the eigenvalues of the linearization of reaction–diffusion operators about the wave. The stability of a traveling wave can be determined by the existence of eigenvalues of positive real part for the linear operator. Our for locating and counting eigenvalues is inspired by the numerical results in Way’s Way (2009). We provide a detailed proof of the relationship between the phase and eigenvalues for dynamical systems defined on and sketch the proof of the method of geometric phase for and its generalization to boundary-value problems. Implementing the numerical method, modified from Way (2009), we conclude with open questions inspired from the results

Solitary Waves of the Regularized Short Pulse and Ostrovsky Equations

We derive a model for the propagation of short pulses in nonlinear media. The model is a higher-order regularization of the short-pulse equation (SPE). The regularization term arises as the next term in the expansion of the susceptibility in derivation of the SPE. Without the regularization term there do not exist traveling pulses in the class of piecewise smooth functions with one discontinuity. However, when the regularization term is added, we show, for a particular parameter regime, that the equation supports smooth traveling waves which have structure similar to solitary waves of the modified Korteweg-deVries equation. The existence of such traveling pulses is proved via the Fenichel theory for singularly perturbed systems and a Melnikov-type transversality calculation. Corresponding statements for the Ostrovsky equations are also included

Soliton broadening under random dispersion fluctuations: Importance sampling based on low-dimensional reductions

We demonstrate that dispersion-managed solitons are less likely to experience critical broadening under the influence of random dispersion fluctuations than are solitons of the integrable nonlinear Schrödinger equation, and that this robustness increases with map strength from the constant-dispersion (integrable) limit to the large-map-strength limit. To achieve this, we exploit a separation of scales in dispersion-managed soliton dynamics to implement an importance-sampled Monte Carlo approach that determines the probability of rare broadening events. This approach reconstructs the tails (i.e., the regions of practical importance) of probability distribution functions with an efficiency that is several orders of magnitude greater than conventional Monte Carlo simulations. We further show that the variational approach with an appropriately scaled ansatz is surprisingly good at capturing the effect of random dispersion on pulse broadening; where it fails, it can still be used to guide very efficient simulation of the original equation

Approximation of Random Slow Manifolds and Settling of Inertial Particles Under Uncertainty

A method is provided for approximating random slow manifolds of a class of slow-fast stochastic dynamical systems. Thus approximate, low dimensional, reduced slow systems are obtained analytically in the case of sufficiently large time scale separation. To illustrate this dimension reduction procedure, the impact of random environmental fluctuations on the settling motion of inertial particles in a cellular flow field is examined. It is found that noise delays settling for some particles but enhances settling for others. A deterministic stable manifold is an agent to facilitate this phenomenon. Overall, noise appears to delay the settling in an averaged sense

Stable Pulse Solutions for the Nonlinear Schrödinger Equation with Higher Order Dispersion Management

The evolution of optical pulses in fiber optic communication systems with strong, higher order dispersion management is modeled by a cubic nonlinear Schrödinger equation with periodically varying linear dispersion at second and third order. Through an averaging procedure, we derive an approximate model for the slow evolution of such pulses and show that this system possesses a stable ground state solution. Furthermore, we characterize the ground state numerically. The results explain the experimental observation of higher order dispersion managed solitons, providing theoretical justification for modern communication systems design

Viscous perturbations of marginally stable Euler flow and finite-time Melnikov theory

The effect of small viscous dissipation on Lagrangian transport in two-dimensional vorticity-conserving fluid flows motivates this work. If the inviscid equation admits a base flow in which different fluid regions are divided by separatrices, then transport between these regions is afforded by the splitting of separatrices caused by viscous dissipation. Finite-time Melnikov theory allows us to measure the splitting distance of separatrices provided the perturbed velocity field of the viscous fluid flow stays sufficiently close to vorticity-conserving base flow over sufficiently long time intervals. In this paper, we derive the necessary long-term estimates of solutions to Euler’s equation and to the barotropic vorticity equation upon adding viscous perturbations and forcing. We discover that a certain stability condition on the unperturbed flow is sufficient to guarantee these long time estimates

Existence of Multi-Pulses of the Regularized Short-Pulse and Ostrovsky Equations

The existence of multi-pulse solutions near orbit-flip bifurcations of a primary single-humped pulse is shown in reversible, conservative, singularly perturbed vector elds. Similar to the nonsingular case, the sign of a geometric condition that involves the first integral decides whether multi-pulses exist or not. The proof utilizes a combination of geometric singular perturbation theory and Lyapunov-Schmidt reduction through Lin's method. The motivation for considering orbit flips in singularly perturbed systems comes from the regularized short-pulse equation and the Ostrovsky equation, which both fi t into this framework and are shown here to support multi-pulses