Chaos theory Applications to PDEs (geometry design) Essay

Chaos theory Applications to PDEs (geometry design) - Essay Example 55). Therefore, there has been a growing demand for the development for a much stronger theory than for the finite dimensional systems. In mathematics, there are significant challenges in the studies on the infinite dimensional systems (Taylor, 1996; p. 88). For instance, as phase spaces, the Banach spaces have many structures than in Euclidean spaces. In application, the most vital natural phenomena are explained by the partial differential equations, most of important natural phenomena are described by the Yang-Mills equations, partial differential equations, nonlinear wave equations, and Navier-Stokes equations among others. Problem Statement Chaos theory has led to profound mathematical equations and theorems that have numerous applications in different fields including chemistry, biology, physics, and engineering among other fields or professions. Problem Definition The nonlinear wave equations are usually significant class of equations especially natural sciences (Cyganowski, K loeden, and Ombach, 2002; p. 33). They usually describe a wide spectrum of phenomena including water waves, motion of plasma, vortex motion, and nonlinear optics (laser) among others (Wasow, 2002). Notably, these types of equations often describe differences and varied phenomena; particularly, similar soliton equation that describes several different situations. These types of equations can be described by the nonlinear Schrodinger equation 1 The equation 1 above has a soliton solution 2 Where the variable This leads to 3 The equation leads to the development of the soliton equations whose Cauchy problems that are solved completely through the scattering transformations. The soliton equations are similar to the integrable Hamiltonian equations that are naturally counterparts of the finite dimensionalintegrable differential systems. Setting up the systematic study of the chaos theory in the partial differential equations, there is a need to start with the perturbed soliton equations (Wasow, 2002). The perturbed soliton equations can be classified into three main categories including: 1. Perturbed (1=1) dimensional soliton equations 2. Perturbed soliton lattices 3. Perturbed (1 + n) dimensional soliton equations (n? 2). For each of the above categories, to analyze the chaos theory in the partial differential equations, there is needed to choose a candidate for study. The integrable theories are often parallel for every member within the same category (Taylor, 1996; p. 102). Moreover, members of different categories are often different substantial. Therefore, the theorem that describes the existence of chaos on each candidate can be generalized parallely to other members under the same category (Wasow, 2002). For instance; The candidate in the first category is often described by a perturbed cubic that often focuses on the nonlinear Schrodinger equation 4 Under even and periodic boundary conditions q (x+1) = q (x) and q (x) =q (x), and is a real constant. The can didates in category 2 are often considered as the perturbed discrete cubic that often focus on the nonlinear Schrodinger equation + Perturbations, 5 The above equation is only valid under even and periodic boundary conditions described by +N = The candidates falling under category 3 are perturbed Davey-Stewartson II equations 6 The equation is only satisfied under the even and periodic