Data-driven identification of the spectral operator in AKNS Lax pairs using conserved quantities

Lax-integrable partial differential equations (PDEs) can by definition be described through a compatibility condition between two linear operators. These operators are said to form a Lax pair for the PDE, which itself is usually nonlinear. Lax pairs are a very useful tool, but unfortunately finding them is a difficult problem in practice. In this paper, we propose a method that determines the spectral operator of an AKNS-type Lax pair such that the corresponding PDE fits given measurement data as well as possible. The spectral operator then enables practitioners to solve or analyze the underlying PDE using the induced nonlinear Fourier transform. The underlying PDE only has to be approximately Lax-integrable; the method will find the spectral operator that explains the data best. Together with the dispersion relation, the spectral operator of AKNS type completely determines an integrable PDE that approximates the true underlying PDE. We identify the most suitable spectral operator by matching PDE-dependent quantities that should be conserved during evolution. The method is automatic and only requires recordings of solutions at two different values of the evolution variable, which do not have to be close.