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The main objective of this series of papers is to explore the entirelandscape of numerical methods for fast nonlinear Fourier transformation (NFT)within the class of integrators known as the exponential integrators. In thispaper, we explore the theoretical aspects of exponential Runge-Kutta (RK) andlinear multistep (LM) methods, in particular, the stability and convergence ofthese methods via the transfer matrix formulation. The analysis carried out inthe paper shows that while the exponential LM methods are naturally amenable toFFT-based fast polynomial arithmetic, the RK methods require equispaced nodesto achieve that. Therefore, each these family of methods is capable of yieldinga family of fast NFT algorithms such that the scattering coefficients can becomputed with a complexity of $\mathscr{O}(N\log^2N)$ and a rate of convergencegiven by $\mathscr{O}(N^{-p})$ where $N$ is the number of samples of the signaland $p$ is order of the underlying discretization scheme. Further, while RKmethods can accommodate vanishing as well as periodic boundary conditions, theLM methods can only handle the former type of boundary conditions withoutrequiring a starting procedure. The ideas presented in this paper extendnaturally to the family of integrators known as general linear methods whichwill be explored in a forthcoming paper.

Numerical Methods for Fast Nonlinear Fourier Transformation, Part I: Exponential Runge-Kutta and Linear Multistep Methods