Nonlinear Random Fields Research Articles

Nonlinear random wave fields within a Boussinesq system

The dynamics of nonlinear random fields is important for understanding wave turbulence. In this work, we use a Boussinesq system to examine the distinctions between unidirectional and bidirectional waves. Our study demonstrates that in both scenarios, the wave spectra reach a stationary state. Moreover, we show that the occurrence of rogue waves is more probable in the unidirectional case. In the unidirectional case, the probability distribution of wave crests exceeds the one predicted by the Rayleigh distribution once the spectra reach the stationary state. Conversely, in the bidirectional case, the opposite trend is observed. The discovery of various types of rogue waves, including massive wave trains commonly known in the literature as “two sisters” and “three sisters” are found.

Informational assessment of large scale self-similarity in nonlinear random field models

Large-scale behavior of a wide class of spatial and spatiotemporal processes is characterized in terms of informational measures. Specifically, subordinated random fields defined by nonlinear transformations on the family of homogeneous and isotropic Lancaster–Sarmanov random fields are studied under long-range dependence (LRD) assumptions. In the spatial case, it is shown that Shannon mutual information between random field components for infinitely increasing distance, which can be properly interpreted as a measure of large scale structural complexity and diversity, has an asymptotic power law decay that depends on the underlying LRD parameter scaled by the subordinating function rank. Sensitivity with respect to distortion induced by the deformation parameter under the generalized form given by divergence-based Rényi mutual information is also analyzed. In the spatiotemporal framework, a spatial infinite-dimensional random field approach is adopted. The study of the large-scale asymptotic behavior is then extended under the proposal of a functional formulation of the Lancaster–Sarmanov random field class, as well as of divergence-based mutual information. Results are illustrated, in the context of geometrical analysis of sample paths, considering some scenarios based on Gaussian and Chi-Square subordinated spatial and spatiotemporal random fields.

Estimation in nonlinear random fields models of autoregressive type with random parameters

In this article, we present new original theoretical results on estimation in nonlinear random field models. We focus on two dimensionally indexed random coefficients autoregressive model with order for short. We first develop a maximum likelihood estimation procedure for estimating the unknown parameters of Moreover, we prove that the estimates are strongly consistent. Finally, these results are then applied to construct efficient estimates in 2D-RCAR model of order (0, 1). Then, a simulation part is given to illustrate the effectiveness and accuracy of the estimates.

On the Quenched Central Limit Theorem for Stationary Random Fields Under Projective Criteria

Motivated by random evolutions which do not start from equilibrium, in a recent work, Peligrad and Volný (J Theor Probab, 2018. arXiv:1802.09106 ) showed that the central limit theorem (CLT) holds for stationary ortho-martingale random fields when they are started from a fixed past trajectory. In this paper, we study this type of behavior, also known under the name of quenched CLT, for a class of random fields larger than the ortho-martingales. We impose sufficient conditions in terms of projective criteria under which the partial sums of a stationary random field admit an ortho-martingale approximation. More precisely, the sufficient conditions are of the Hannan’s projective type. We also discuss some aspects of the functional form of the quenched CLT. As applications, we establish new quenched CLTs and their functional form for linear and nonlinear random fields with independent innovations.

Quenched Invariance Principles for Orthomartingale-Like Sequences

In this paper, we study the central limit theorem and its functional form for random fields which are started not from their equilibrium, but rather under the measure conditioned by the past sigma field. The initial class considered is that of orthomartingales and then the result is extended to a more general class of random fields by approximating them, in some sense, with an orthomartingale. We construct an example which shows that there are orthomartingales which satisfy the CLT but not its quenched form. This example also clarifies the optimality of the moment conditions used for the validity of our results. Finally, by using the so-called orthomartingale-coboundary decomposition, we apply our results to linear and nonlinear random fields.

Martingale approximations for random fields

In this paper we provide necessary and sufficient conditions for the mean square approximation of a random field by an ortho-martingale. The conditions are formulated in terms of projective criteria. Applications are given to linear and nonlinear random fields with independent innovations.

On the normal approximation for random fields via martingale methods

We prove a central limit theorem for strictly stationary random fields under a sharp projective condition. The assumption was introduced in the setting of random sequences by Maxwell and Woodroofe. Our approach is based on new results for triangular arrays of martingale differences, which have interest in themselves. We provide as applications new results for linear random fields and nonlinear random fields of Volterra-type.

Scaling transition for nonlinear random fields with long-range dependence

We obtain a complete description of anisotropic scaling limits and the existence of scaling transition for nonlinear functions (Appell polynomials) of stationary linear random fields on Z2 with moving average coefficients decaying at possibly different rate in the horizontal and the vertical direction. The paper extends recent results on scaling transition for linear random fields in Puplinskaitė and Surgailis (2015, 2016).

On stationarity and second-order properties of bilinear random fields

One-dimensional indexed bilinear (BL) models are widely used for modeling non Gaussian time series. Extending BL models to multidimensional indexed (spatial) SBL one, yields a novel class of models which are capable of taking into account the important characteristic of non Gaussianity and spatiality behavior. Our main contribution here is to extend various results on BL processes to SBL one. Our attention is thus focussed on the probabilistic structure of some SBLmodels. So, we establish necessary and sufficient conditions for the existence of regular stationary and ergodic solutions in term of their transfer functions. As a consequence, we observe that the second order structure is similar to a spatial ARMA model with some uncorrelated white noise. So, it is necessary to look into higher-order moments in order to distinguish between linear and nonlinear random fields. Our study may be applied to GARCH or ARMA random fields.

Long time interaction of envelope solitons and freak wave formations

This paper concerns long time interaction of envelope solitary gravity waves propagating at the surface of a two-dimensional deep fluid in potential flow. Fully nonlinear numerical simulations show how an initially long wave group slowly splits into a number of solitary wave groups. In the example presented, three large wave events are formed during the evolution. They occur during a time scale that is beyond the time range of validity of simplified equations like the nonlinear Schrödinger (NLS) equation or modifications of this equation. A Fourier analysis shows that these large wave events are caused by significant transfer to side-band modes of the carrier waves. Temporary downshiftings of the dominant wavenumber of the spectrum coincide with the formation large wave events. The wave slope at maximal amplifications is about three times higher than the initial wave slope. The results show how interacting solitary wave groups that emerge from a long wave packet can produce freak wave events. Our reference numerical simulation are performed with the fully nonlinear model of Clamond and Grue [D. Clamond, J. Grue, A fast method for fully nonlinear water wave computations, J. Fluid Mech. 447 (2001) 337–355]. The results of this model are compared with that of two weakly nonlinear models, the NLS equation and its higher-order extension derived by Trulsen et al. [K. Trulsen, I. Kliakhandler, K.B. Dysthe, M.G. Velarde, On weakly nonlinear modulation of waves on deep water, Phys. Fluids 12 (10) (2000) 2432–2437]. They are also compared with the results obtained with a high-order spectral method (HOSM) based on the formulation of West et al. [B.J. West, K.A. Brueckner, R.S. Janda, A method of studying nonlinear random field of surface gravity waves by direct numerical simulation, J. Geophys. Res. 92 (C11) (1987) 11 803–11 824]. An important issue concerning the representation and the treatment of the vertical velocity in the HOSM formulation is highlighted here for the study of long-time evolutions.

Relaxor ferroelectrics: from Cross superparaelectric model to random field theory

A brief review is presented of recent results derived from the random field theory approach to describing relaxor ferroelectric properties. The key point in this approach is that the random field originating f rom chemical disorder of relaxor ferroelectrics destroys long range polar order, which could otherwise exist at temperatures below the Burns temperature Td. As a result, polar clusters appear of size Rc (the correlation radius of Burns reference phase), corresponding to the Cross superparaelectric phase. The relaxor properties can be calculated by averaging the electric field dependence of the paraelectric Burns reference phase with the random field distribution function of a suitable property. The distribution functions for the cases of linear and non-linear random field contributions are calculated using the statistical physics framework for electric dipoles, point charges and dilatational centres as random field sources. These calculations allow explanation of the observed Vogel–Fulcher law and non-Debye character of the dynamic dielectric response; of stiff and soft response to dc field of non-linear susceptibility; of unusual Rc temperature dependence; as well as of dielectric response anomalies in 1:1 family relaxors. The correlation radius and relaxation time distribution functions are calculated using the random field distribution function. Calculated phase diagrams for mixed relaxors, allowing for non-linear and spatial correlation effects, give a good fit to the experimental results. Comparison of calculated and observed properties leads to the conclusion that a mixed ferroglass phase with coexisting short and long polar order is the most probable state for all relaxors, the degree of mixing being the essential factor for a relaxor.

Physical mechanisms responsible for the relaxation time distribution in disordered dielectrics

The relaxation time distribution function F(τ) is calculated in the framework of the random-field theory. The function F(τ) is expressed through the distribution function f(E) of a random electric field E with due regard for the derived dependence of the relaxation time τ on the electric field. The distribution function F(τ) is calculated in terms of the statistical theory within the random-field approximation. The nonlinear random-field contributions and spatial effects of correlations between randomly distributed electric dipoles are taken into account. The calculations are performed for a mixed ferroelectric glassy phase in which the short-range and long-range polar orders coexist. It is demonstrated that the inclusion of nonlinear contributions of the random field leads to an asymmetric relaxation time distribution function F(τ), whereas allowance made only for the linear random-field contributions results in a symmetric function F(τ). A comparison of the calculated functions F(τ) with empirical functions derived from the Cole-Cole (CC), Davidson-Cole (DC), Kohlrausch-Williams-Watts (KWW), and Havriliak-Negami (HN) laws for the dielectric response shows that these laws correspond to disordered systems in which the long-range and short-range orders coexist. Different forms of the function F(τ) are determined by either linear (the CC law) or nonlinear (the DC, KWW, and HN laws) contributions of the random field.

A method of studying nonlinear random field of surface gravity waves by direct numerical simulation

A method of studying nonlinear random fields of surface gravity waves by direct numerical simulation is developed by combining the “high-order spectral method” of West et al. (J. Geophys. Res. 92 (1987) 11,803–11,824) and the complex amplitude function b(k) introduced by Zakharov (J. Appl. Mech. Tech. Phys. (Engl. Transl.) 2 (1968) 190–194) in his Hamiltonian formalism of the water wave problem. By computing the rate of change of b(k) for a given wave field by both the high-order spectral method and Zakharov's equation, it is confirmed that the high-order spectral method, when the order of nonlinearity is appropriately chosen, gives exactly the same result as Zakharov's equation within a much shorter computation time. Methods of evaluating the frequency spectrum, directional spectrum and the component phase speed are given. Results of some preliminary computations are also presented to prove the great potentialities of the present approach as a tool for the examination of various characteristics of nonlinear random wave fields with continuous spectra, such as ocean waves.

Relaxation time distribution function

The distribution function of relaxation time in disordered dielectrics has been calculated in the random field theory framework. For this purpose, we first consider the dynamics of single two-orientable electric dipole in a random electric field E in a disordered ferroelectric. The obtained dependence τ(E) made it possible to express the relaxation time distribution function F(τ) via the random field distribution function f(E). This function has been calculated self-consistently in the random field theory framework. Nonlinear random field contribution and effects of spatial correlations between the dipoles have also been taken into account. It was shown that nonlinear contribution of random field gives asymmetric shape of F(τ), while in linear case it is symmetric. Comparison of calculated F(τ) curves with those extracted from empirical Cole-Cole (CC), Davidson-Cole (DC), Kohlrausch-William-Watts (KWW) and Havriliak-Negami (HN) functions had shown, that they correspond to mixed ferroelectric-dipole glass (ferro-glass) phase with coexistence of short and long-range order. Different forms of F(τ) are determined by linear (CC) or nonlinear (DC, KWW, HN) contributions of random field.