This project studies the formation and evolution of the soliton waves in the 1D and 2D nonlinear Schrodinger equation (NLSE) with a random potential (also called the Gross-Pitaevskii equation), which governs the evolution of the mean-field wave function in Bose-Einstein condensate (BEC). The main focus is to investigate the impact of three parameters: the strength and the correlation length of the disorder, and the norm of the solution (i.e., the number of atoms in the condensate). But first, the random field approximation will be investigated within a more general context in the sense that the methodology can be applied in any applications involving uncertainties, not limited to the random potential approximation of the NLSE. In practical problems, the variables with uncertainty are often described as a second-order stochastic process (or random field/function), i.e., its second-order moment is finite. The marginal distribution and covariance function are typically the available information. One prominent way of discretizing a second order random field is through the Karhunen-Loeve (KL) series expansion. The approximation with truncated KL expansion is optimal in terms of mean square error, and the errors for the first two moments are fixed for any specific truncated KL expansion. But there are still many issues needed to be addressed. In particular, the proposed research will focus on the following topics. 1.) More efficient computation of the KL expansion by introducing two adaptive meshes, when the analytical formulas are unavailable (true for most cases). 2.) Minimize the error of the marginal distribution by determining the distribution of the random variables in the KL expansion through the minimization of higher order moments. 3. Compare the efficiency of the KL expansion and the 'direct sampling' technique paired with correlation control technique, when the correlation length is short.
Either due to the randomness in nature or the insufficiency of knowledge, uncertainty is nearly observed in all the disciplines to some degree. Petroleum reservoir a few miles under the earth's surface, traffic flow on state highways, measurement of gas flow in turbine, and the stock and futures market are several such examples. To gain a better understanding of the intrinsic dynamics, such uncertainty should be modeled and analyzed. The proposed research will enable faster and more efficient calculations of the involved uncertainties, provide unprecedented predictive capabilities. It will bring profound impact across all the scientific and engineering disciplines that involve uncertainty.
