White Noise Functional Solutions for Wick-Type Stochastic Fractional Mixed KdV-mKdV Equation Using Extended G ′ / G -Expansion Method

In this paper, white noise functional solutions of Wick-type stochastic fractional mixed KdV-mKdV equations have been obtained by using the extended G ′ / G -expansion method and the Hermite transform. Firstly, the Hermite transform is used to transform Wick-type stochastic fractional mixed KdV-mKdV equations into deterministic fractional mixed KdV-mKdV equations. Secondly, the exact traveling wave solutions of deterministic fractional mixed KdV-mKdV equations are constructed by applying the extended G ′ / G -expansion method. Finally, a series of white noise functional solutions are obtained by the inverse Hermite transform.

Exact Solutions for a Class of Wick-Type Stochastic (3+1)-Dimensional Modified Benjamin–Bona–Mahony Equations

In this paper, we investigate the Wick-type stochastic (3+1)-dimensional modified Benjamin–Bona–Mahony (BBM) equations. We present a generalised version of the modified tanh–coth method. Using the generalised, modified tanh–coth method, white noise theory, and Hermite transform, we produce a new set of exact travelling wave solutions for the (3+1)-dimensional modified BBM equations. This set includes solutions of exponential, hyperbolic, and trigonometric types. With the help of inverse Hermite transform, we obtained stochastic travelling wave solutions for the Wick-type stochastic (3+1)-dimensional modified BBM equations. Eventually, by application example, we show how the stochastic solutions can be given as white noise functional solutions.

Damped White Noise Diffusion with Memory for Diffusing Microprobes in Ageing Fibrin Gels

ABSTRACTFrom observations of colloidal tracer particles in fibrin undergoing gelation, we introduce an analytical framework that allows determination of the probability density function (PDF) for a stochastic process beyond fractional Brownian motion. Using passive microrheology via videomicroscopy, mean square displacements (MSD) of tracer particles suspended in fibrin at different ageing times are obtained. The anomalous diffusion is then described by a damped white noise process with memory, with analytical results closely matching experimental plots of MSD and PDF. We further show that the white noise functional stochastic approach applied to passive microrheology reveals the existence of a gelation parameterμwhich elucidates the dynamics of constrained tracer particles embedded in a time dependent soft material. This study offers experimental insights on the ageing of fibrin gels while presenting a white noise functional stochastic approach that could be applied to other systems exhibiting non-Markovian diffusive behavior.

Stochastic Soliton Solutions of the High-Order Nonlinear Schrödinger Equation in the Optical Fiber with Stochastic Dispersion and Nonlinearity

In this paper, the high-order nonlinear Schrödinger (HNLS) equation driven by the Gaussian white noise, which describes the wave propagation in the optical fiber with stochastic dispersion and nonlinearity, is studied. With the white noise functional approach and symbolic computation, stochastic one- and two-soliton solutions for the stochastic HNLS equation are obtained. For the stochastic one soliton, the energy and shape keep unchanged along the soliton propagation, but the velocity and phase shift change randomly because of the effects of Gaussian white noise. Ranges of the changes increase with the increase in the intensity of Gaussian white noise, and the direction of velocity is inverted along the soliton propagation. For the stochastic two solitons, the effects of Gaussian white noise on the interactions in the bound and unbound states are discussed: In the bound state, periodic oscillation of the two solitons is broken because of the existence of the Gaussian white noise, and the oscillation of stochastic two solitons forms randomly. In the unbound state, interaction of the stochastic two solitons happens twice because of the Gaussian white noise. With the increase in the intensity of Gaussian white noise, the region of the interaction enlarges.