Abstract
“Linear Fourier analysis” is based on the application of “periodic Fourier series” to a wide range of wave simulation and data analysis problems. A key feature of the method is the direct computation of the “correlation function” and its Fourier transform which is the “power spectrum.” A key idea in these applications is the assumption that the “phases” of the periodic Fourier transform are “random numbers on zero to two pi.” The method leads to simple ways to compute the “coherence functions” and “transfer functions” between two signals. Many books have been written about the “linear model” and its myriad applications. In the present talk I address the infinite number of “nonlinear wave equations” that are solvable by the “periodic/quasiperiodic inverse scattering transform” and, amazingly, I show how such equations are solvable with “quasiperiodic Fourier series.” This means that all operations of the nonlinear problems of water waves are isomorphic to those of the linear problem! All operations that one can conduct for the linear problem with periodic Fourier series can also be conducted for nonlinear problems with quasiperiodic Fourier series: This means the correlation function, power spectrum, coherence functions, and transfer functions for the nonlinear problem can be computed in analogy with the linear problem. This further means further the nonlinear problem is just as easy to work with as the linear problem! I give a fast Fourier transform for quasiperiodic Fourier series and show how to conduct numerical simulations of nonlinear waves using the procedure. The methods discussed here provide for a new future in which nonlinear problems are just as easy to treat as linear ones. This work emphasizes one dimension in space, but two-dimensions are easily found by extension.
