A method for studying extreme wave solutions of the 1+1D nonlinear Schr"{o}dinger equation (NLSE) with periodic boundary conditions is presented in this work. The existing methods for solving NLSE in the periodic case usually require information about the full period. Obtaining that information may not always be possible, when the experimental data is collected outside laboratory settings. In addition, some NLSE solutions contain fine details and have extremely long periods. As such, a very large mesh would be required in order to apply numerical methods to simulate the propagation of the wave. Finally, as some solutions only experience exponential growth once in their lifetime, the number of time steps necessary to numerically recreate an extreme or Rogue wave may be significant.

The way to determine whether a solution is stable with respect to small perturbations or not (in Benjamin-Feir sense) is available in the literature. One relies on representing a solution using Riemann theta functions that depend on a set of parameters which, in particular, can be used to determine stability. An algorithm for finding those parameters is developed and is based on wavelet representation. The existence of wavelet families with compact support allows restricting the analysis of the solution to a given interval and this approach is found to work for the incomplete sets of input data. The implementation of the algorithm requires the evaluation of the integrals of wavelet triple products (triplets). A method to evaluate the values of those triplets analytically is described, which allows one to avoid the necessity of approximating the wavelets numerically. The triplet values could be precomputed independently from the specific problem. This, in turn, allows the implemented algorithm to run on desktop computers. To demonstrate the efficiency of the method, various simulations have been performed by using data obtained by the research group. The algorithm proved to be efficient and robust, correctly processing the input data even with a small-to-moderate noise in the signal, unlike curve-fitting methods that were found to fail in the presence of noise in the input. The analytical basis and algorithms developed in this dissertation can be useful for examining extreme or freak waves that arise in a number of contexts, as well as solutions with localized features in space and time.