Asymptotic properties of coupled nonlinear langevin equations in the limit of weak noise. II: Transition to a limit cycle

We apply the singular perturbation technique, developed in the companion paper, to the study of the fluctuations at the onset of a limit cycle, both for the cases of a soft and a hard transition. The technique and results are illustrated on the Poincaré model (soft transition) and on the Van der Pol oscillator (hard transition).     

Studies in nonlinear stochastic processes. III. Approximate solutions of nonlinear stochastic differential equations excited by Gaussian noise and harmonic disturbances

A fusion of the highly successful methods of harmonic and statistical linearization is used as a first approximation in determining, either iteratively or via a nonlinear integral equation, the effects of higher harmonics and non-Gaussian distortion terms on the second-order statistics of a wide variety of nonlinear stochastic differential equations perturbed by some linear combination of Gaussian noise and a periodic deterministic/stochastic excitation. Physical a posteriori applicability criteria are presented which justify when these higher order effects may be neglected. A simple modification of this statistical-harmonic linearization procedure based upon the Fokker-Planck variance is proposed.This work was supported in part by the National Science Foundation under grant CHE75-20624.     

Onsager-Casimir symmetry properties of the Burnett equations

We demonstrate the approximate nature of the Onsager-Casimir relations for the example of the linearized Burnett equations for a dilute gas. For any discussion of Onsager relations the choice of a correct set of thermodynamic forces and fluxes is of course crucial. By retracing the Chapman-Enskog procedure, we show that the usual expressions for the thermodynamic forces require modifications at the Burnett level. However, inclusion of these terms does not remedy the violation of Onsager symmetry first noticed by McLennan. A modified version of the Onsager symmetry that involves thermodynamic forces derived from an entropy Lagrangian rather than from the entropy itself does remain valid on the Burnett level. Throughout, we allow for the presence of an external potential; the Burnett equations including potential terms are derived in an appendix for a set of variables particularly suited for our discussion. We stress that in discussing Onsager relations one should use the full thermodynamic fluxes rather than their dissipative parts only, in spite of the fact that only the latter contribute to the entropy production.     

Studies in nonlinear stochastic processes. I. Approximate solutions of nonlinear stochastic differential equations by the method of statistical linearization

A method based on a variational procedure is presented which provides simple and useful approximate solutions to a wide variety of nonlinear stochastic differential equations. This method of statistical linearization is most successful when the stochasticity of the differential equation is due to excitations which are normally distributed or harmonic with random phase. Effects due to deviations from normality can be corrected for in a systematic fashion. Comments regarding existence and uniqueness are given and some error bounds arising from the use of statistical linearization are computed.Supported by the National Science Foundation under Grants NSF GP 32031X and MPS72-04353 A03.     

Determination of nonlinear optical properties using the Voigt function: Stochastic considerations

In this work, the absorptive and dispersive profiles of a molecular system have been calculated through a methodology based on the averaging of the coherence obtained by the resolution of the optical stochastic Bloch equations. To this aim, a generalized Lorentzian approximation to the Voigt function has been used as a probability distribution, which allows a more generalized analysis of the interactions between the solvent and the molecule. This has been modeled with the parameters of the standard organic dye Green Malaquite, which exhibits a nonlinear behavior under the interaction with a high intensity electric field.     

Localized waves of the coupled cubic–quintic nonlinear Schrdinger equations in nonlinear optics

We investigate some novel localized waves on the plane wave background in the coupled cubic–quintic nonlinear Schr o¨dinger(CCQNLS) equations through the generalized Darboux transformation(DT). A special vector solution of the Lax pair of the CCQNLS system is elaborately constructed, based on the vector solution, various types of higherorder localized wave solutions of the CCQNLS system are constructed via the generalized DT. These abundant and novel localized waves constructed in the CCQNLS system include higher-order rogue waves, higher-order rogues interacting with multi-soliton or multi-breather separately. The first-and second-order semi-rational localized waves including several free parameters are mainly discussed:(i) the semi-rational solutions degenerate to the first-and second-order vector rogue wave solutions;(ii) hybrid solutions between a first-order rogue wave and a dark or bright soliton, a second-order rogue wave and two dark or bright solitons;(iii) hybrid solutions between a first-order rogue wave and a breather, a second-order rogue wave and two breathers. Some interesting and appealing dynamic properties of these types of localized waves are demonstrated, for example, these nonlinear waves merge with each other markedly by increasing the absolute value of α.These results further uncover some striking dynamic structures in the CCQNLS system.     

General initial value problem for the nonlinear shallow water equations: Runup of long waves on sloping beaches and bays

We formulate a new approach to solving the initial value problem of the shallow water-wave equations utilizing the famous Carrier–Greenspan transformation (Carrier and Greenspan (1957) [9]). We use a Taylor series approximation to deal with the difficulty associated with the initial conditions given on a curve in the transformed space. This extends earlier solutions to waves with near shore initial conditions, large initial velocities, and in more complex U-shaped bathymetries; and allows verification of tsunami wave inundation models in a more realistic 2-D setting.     

Infinite Compressibility States in the Hierarchical Reference Theory of Fluids. I. Analytical Considerations

In its customary formulation for one-component fluids, the Hierarchical Reference Theory yields a quasilinear partial differential equation (PDE) for an auxiliary quantity f that can be solved even arbitrarily close to the critical point, reproduces non-trivial scaling laws at the critical singularity, and directly locates the binodal without the need for a Maxwell construction. In the present contribution we present a systematic exploration of the possible types of behavior of the PDE\ for thermodynamic states of diverging isothermal compressibility _T as the renormalization group theoretical momentum cutoff approaches zero. By purely analytical means we identify three classes of asymptotic solutions compatible with infinite _T, characterized by uniform or slowly varying bounds on the curvature of f, by monotonicity of the build-up of diverging _T, and by stiffness of the PDE in part of its domain, respectively. These scenarios are analzyed and discussed with respect to their numerical properties. A seeming contradiction between two of these alternatives and an asymptotic solution derived earlier [Parola et al., Phys. Rev. E 48:3321 (1993)] is easily resolved.     

Chaotic states in a random world: Relationship between the nonlinear differential equations of excitability and the stochastic properties of ion channels

Excitable membranes allow cells to generate and propagate electrical signals. In the nervous system these signals transmit information, in muscle they trigger contraction, and in heart they regulate spontaneous beating. A central question in excitability theory concerns the relationship between the aggregate properties of membranes (marcoscopic) and the properties of channels in the membranes (mircoscopic). Hodgkin and Huxley (1952) laid the foundations of membrane excitability, and Neher and Sakmann (1976) developed techniques to study individual channels. This article focuses on the relationship between the macroscopic domain, in which non-linear differential equations determine the electrical properties of cells, and the microscopic domain, in which the probabilistic nature of channels establishes the pattern of activity. Using nerve cell membranes as an example, we examine how information in one domain predicts behavior in the other. We conclude that the probabilistic nature of channels generates virtually all macroscopic electrical properties, including resting potentials, action potentials, spontaneous firing, and chaos.