Auto-Bcklund transformation and exact solutions of Wick-type stochastic Burgers equation

Burgers equation in random environment is studied. In order to give the exact solutions of random Burgers equation, we only consider the Wick-type stochastic Burgers equation which is the perturbation of the Burgers equation with variable coefficients by white noise W（t）=Bt, where Bt is a Brown motion. The auto-Baecklund transformation and stochastic soliton solutions of the Wick-type stochastic Burgers equation are shown by the homogeneous balance and Hermite transform. The generalization of the Wick-type stochastic Burgers equation is also studied.

Predicting Nonlinear Wave Trough Distributions Utilizing A Transformed Linear Simulation Method

This paper first proposes a new approach for predicting the nonlinear wave trough distributions by utilizing a transformed linear simulation method. The linear simulation method is transformed based on a Hermite transformation model where the transformation is chosen to be a monotonic cubic polynomial and calibrated such that the first four moments of the transformed model match the moments of the true process. The proposed new approach is applied for calculating the wave trough distributions of a nonlinear sea state with the surface elevation data measured at the coast of Yura in the Japan Sea, and its accuracy and efficiency are convincingly validated by comparisons with the results from two theoretical distribution models, from a linear simulation model and a secondorder nonlinear simulation model. Finally, it is further demonstrated in this paper that the new approach can be applied to all the situations characterized by similar nondimensional spectrum.

Prediction of Height and Period Joint Distributions for Stochastic Ocean Waves

This article proposes a new methodology to predict the wave height and period joint distributions by utilizing a transformed linear simulation method. The proposed transformed linear simulation method is based on a Hermite transformation model where the transformation is chosen to be a monotonic cubic polynomial, calibrated such that the first four moments of the transformed model match the moments of the true process. The proposed new approach is applied for calculating the wave height and period joint distributions of a sea state with the surface elevation data measured at an offshore site, and its accuracy and efficiency are favorably validated by using comparisons with the results from an empirical joint distribution model, from a linear simulation model and from a second-order nonlinear simulation model.

The fractional coupled KdV equations: Exact solutions and white noise functional approach

Variable coefficients and Wick-type stochastic fractional coupled KdV equations are investigated. By using the mod- ified fractional sub-equation method, Hermite transform, and white noise theory the exact travelling wave solutions and white noise functional solutions are obtained, including the generalized exponential, hyperbolic, and trigonometric types.

Abundant solutions of Wick-type stochastic fractional 2D KdV equations

A modified fractional sub-equation method is applied to Wick-type stochastic fractional two-dimensional （2D） KdV equations. With the help of a Hermit transform, we obtain a new set of exact stochastic solutions to Wick-type stochastic fractional 2D KdV equations in the white noise space. These solutions include exponential decay wave solutions, soliton wave solutions, and periodic wave solutions. Two examples are explicitly given to illustrate our approach.

Soliton-like Solutions to Wick-type Stochastic mKdV Equation

In this paper, the Wick-type stochastic mKdV equation is researched. Many Wick-type stochastic solitonlike solutions are given via Hermite transformation and further generalized projective Riccati equation method.

The Mehler Formula for the Generalized Clifford-Hermite Polynomials

The Mehler formula for the Hermite polynomials allows for an integral representation of the one-dimensional Fractional Fourier transform. In this paper, we introduce a multi-dimensional Fractional Fourier transform in the framework of Clifford analysis. By showing that it coincides with the classical tensorial approach we are able to prove Mehler＇s formula for the generalized Clifford-Hermite polynomials of Clifford analysis.

New Exact Solutions for the Wick-Type Stochastic Kudryashov–Sinelshchikov Equation

In this article, exact solutions of Wick-type stochastic Kudryashov–Sinelshchikov equation have been obtained by using improved Sub-equation method. We have used Hermite transform for transforming the Wick-type stochastic Kudryashov–Sinelshchikov equation to deterministic partial differential equation. Also we have applied inverse Hermite transform for obtaining a set of stochastic solutions in the white noise space.