In this talk, I will give an overview of spectral theory as a nonlinear analog of Fourier analysis. There are several classes of nonlinear differential equations (among which are KdV and nonlinear Schroedinger equations) that could be solved with the help of spectral theory. As an illustration, I will discuss a system of exponentially interacting particles known as the Toda lattice whose solution is based on the spectral theory of Jacobi matrices.
In addition, I will discuss recent developments in nonlinear analysis and, in particular, present a nonlinear analog of
Parseval's identity and a nonlinear version of the Riemann--Lebesgue lemma.