Nonlinear Stochastic Partial Differential Equations Research Articles

A Posteriori Estimation for one Phase Stefan Problem with Noisy Boundary Input

In this paper, we present an a-posteriori estimator for state and free boundary of stochastic one phase Stefan problem with noisy observation. Formulating one phase Stefan problem as the stochastic nonlinear partial differential equation in Ito sense, observation mechanism is defined in the finitely additive measure set up. Studying the Onsager-Machlup functional, from which we can evaluate the most probable deterministic path to the original stocastic system, the likelihood functional of the system state is derived. Maximizing this cost functional, the a-posteriori state and free boundary estimator is derived.

Quantum-field superpositions via self-phase modulation of coherent wave packets.

With the use of a quantum theory of optical propagation, a set of nonlinear stochastic partial differential equations may be derived which describes the quantum-statistical properties of traveling waves due to self-phase modulation in a nonlinear medium. We calculate exact moments for the field, which exhibit classical self-phase modulation in the short-interaction limit and periodic quantum evolution through field-superposition states in the long-interaction limit. Reversible and irreversible behaviors of the stochastic description are reviewed. The relation of the present work to the corresponding single-mode nonlinear oscillator is discussed.

Nonlinear filtering and Riemannian scalar curvature, [formula omitted

In the present paper on high dimensional nonlinear filtering problems and stochastic partial differential equations, the Riemannian scalar curvature R is shown to play a fundamental role. The n-dimensional signal processes, n ≥ 2, and the m-dimensional observations processes, m ≥ 1, are assumed to be homogeneous strong Markov diffusions with independent noise functionals. The curvature R of the covariance of the diffusion signal explicitly enters the c ∞-density of the measure-valued solution of the Zakai equation for the problems considered here. The special class of signals required for our proofs are not necessary for n = 2. The general result under normalized conditions is that estimations are relatively better if R is positive and worse if R is negative. The results are applied to estimation problems arising in modelling starfish/coral interactions, but can readily be applied to chemically mediated plant/herbivore interactions.

On the solution of algebraic equations by the decomposition method

The decomposition method ( G. Adomian, “Stochastic Systems,” Academic Press, New York, 1983) developed to solve nonlinear stochastic differential equations has recently been generalized to nonlinear (and/or) stochastic partial differential equations, systems of equations, and delay equations and applied to diverse applications. As pointed out previously (see reference above) the methodology is an operator method which can be used for nondifferential operators as well. Extension has also been made to algebraic equations involving real or complex coefficients. This paper deals specifically with quadratic, cubic, and general higher-order polynomial equations and negative, or nonintegral powers, and random algebraic equations. Further work on this general subject appears elsewhere (G. Adomian, “Stochastic Systems II,” Academic Press, New York, in press).

Optimal Gaussian solutions of nonlinear stochastic partial differential equations

We present a linearization procedure of a stochastic partial differential equation for a vector field (X i (t,x)) (t∈[0, ∞),x∈R d ,i=l,...,n): ∂ t X i (t,x)=b i (X(t, x)) +D, ΔX i (t, x) + σ i f i (t, x). HereΔ is the Laplace-Beltrami operator inR d , and (f i (t,x)) is a Gaussian random field with 〈f i (t,x)f j (t′,x′)〉 = δ ij δ(t − t′)δ(x − x′). The procedure is a natural extension of the equivalent linearization for stochastic ordinary differential equations. The linearized solution is optimal in the sense that the distance between true and approximate solutions is minimal when it is measured by the Kullback-Leibler entropy. The procedure is applied to the scalar-valued Ginzburg-Landau model in R1 withb 1(z) =μz - vz 3. Stationary values of mean, variance, and correlation length are calculated. They almost agree with exact ones ifμ ≲ 1.24 (ν 2θ 1 4 /D 1 1/3:=μ c . Whenμ⩾μ c , there appear quasistationary states fluctuating around one of the bottoms of the potentialU(z) = ∫b 1(z)dz. The second moment at the quasistationary states almost agrees with the exact one. Transient phenomena are also discussed. Half-width at half-maximum of a structure function decays liket −1/2 for small t. The diffusion term∂ 2 X accelerates the relaxation from the neighborhood of an unstable initial stateX(0,x) ∼ 0.

Qualitative behavior of geostochastic systems

Stochastic partial differential equations with polynomial coefficients have many applications in the study of spatially distributed populations in genetics, epidemiology, ecology, and chemical kinetics. The purpose of this paper is to describe some methods for investigating the qualitative behavior of spatially homogenous solutions of nonlinear stochastic partial differential equations. The stability and bifurcation of solutions, as well as the behavior near critical points are discussed. The methods employed are the nonlinear Markov approximation, moment density inequalities for identifying invariant sets of behaviors and rescaling and quasi-linearization around critical points.