Data-driven identification of Lax-integrable partial differential equations

Abstract

Nonlinear partial differential equations (PDEs) are generally hard to solve, but over the past 65 years the notions of Lax integrability and \textit{nonlinear Fourier transforms} (NFTs) have been developed to solve a large class of so called \textit{Lax-integrable} partial differential equations.
If we are able to find a suitable pair of linear operators, consisting of a spectral operator and a propagation operator, a so-called \textit{Lax pair}, we are able to analytically solve the nonlinear PDE.
If this pair of linear operators is a correct Lax pair for the PDE, then the spectral operator can be used to define an NFT, which transforms the signal from the physical domain to a PDE-specific spectral domain, in which all spectral components are independent, similar to the regular Fourier transform for linear problems.
In the spectral domain, the transformed signal can be propagated using the linear propagation operator, after which the propagated signal can be transformed back into the physical domain, hence solving the nonlinear PDE.

The presence of a Lax pair can thus be used to solve a nonlinear PDE, but it also yields insight into the underlying dynamics of the system. Therefore, the identification of a Lax pair for a system is of great value. However, there is no general method to find a Lax pair given an evolution equation. Systems may also be non-integrable, and therefore a corresponding Lax pair may not even exist. To the best of our knowledge, no practical general methods are known in the literature to determine whether or not a system is Lax-integrable.
Since the general problem is very complex, one may focus on more specific cases for practical purposes, in which a Lax-integrable PDE is sought using measurement data.

In this thesis, we therefore focus on data-driven identification of Lax-integrable partial differential equations (PDEs), specifically those of the AKNS-type.
The primary objective is to find a Lax-integrable PDE that best explains given experimental measurement data, enabling a comprehensive analysis using the nonlinear Fourier transform (NFT).
While real-world systems may not be exactly be Lax-integrable due to imperfections, many are well approximated by such equations, including scenarios such as fiber optical wave propagation, surface wave propagation in shallow water canals, and mechanical wave propagation in coupled pendulums.