KdV Equations with Some Applications of White Noise Analysis

Applied functional analysis has many applications in other branches of mathematics, such as differential equations, numerical analysis, stochastic calculus and quantum field theory. Among of these applications, we interest in stochastic differential equations (SDEs). In particular, if we allow for some randomness in some of the coefficients of a partial differential equation (PDE) we often obtain a more realistic mathematical model of the situation. Gaussian white noise analysis (WNA) is an important subject of stochastic partial differential equations (SPDEs). There were many authors studied this subject. Wadati first introduced and studied the stochastic KdV equations and gave the diffusion of soliton of the KdV equations under Gaussian noise. Xie first introduced Wick-type stochastic KdV equations on white noise space. The KdV equation is one of the essential nonlinear equations in mathematical physics. Recently, Okb El Bab, Zabel, Ghany, Hyder and Zakarya studied some important subjects related to Gaussian calculu. So the main objective of this thesis is to use a round applied functional analysis in order to obtain exact solutions for some systems of KdV equations.