Nonlinear Wave Equations and Krein-de Branges theory

The main objects of our project are the most important 1+1 completely integrable nonlinear wave equations (the Kortewegde Vries equation, the nonlinear Schrödinger equation, and the CamassaHolm equation) and the corresponding isospectral problems (1-D Schrödinger equation, 1-D Dirac equation, and strings, respectively). The inverse scattering transform (IST) approach is a very powerful tool to treat these equations (the IST approach enables us to construct explicit solutions and to perform a rather detailed analysis of initial value problems). However, it has a rather restrictive range of applicability. Indeed, the key ingredient is a solution of the corresponding inverse spectral/scattering problem and this immediately explains all the restrictions. Our goal is twofold. First, we are going to make a progress in understanding the inverse spectral/scattering theory of one-dimensional spectral problems. Our second goal is to develop the IST approach to these nonlinear problems with low regularity initial data.

One of the major outcomes of the project is Szegö-type theorems for generalized indefinite strings (Invent. Math. 238 (2024)). More specifically, for several classes of coefficients that can be regarded as Hilbert-Schmidt perturbations of model problems, we provide a complete characterization of the corresponding set of spectral measures. In his famous book on the Szegö Theorem, Barry Simon termed this sort of results as spectral gems. On the one hand, this advancement indicates that the theory of generalized indefinite strings, the object introduced in our earlier work with J. Eckhardt, is extremely well developed now. On the other hand, a well developed direct and inverse spectral theory of a Lax operator is an indispensable tool in the study of a related nonlinear completely integrable system (the Camassa-Holm equation in our case). In particular, these results were applied in our work to study the conservative Camassa-Holm flow under the assumptions on initial data which were out of reach earlier. Moreover, using spectral theory of generalized indefinite strings we were able to derive the set of new conservation laws, which cannot be derived from the corresponding bi-Hamiltonian structure, integrate the conservative Camassa-Holm flow with step-like initial data, construct and investigate several interesting classes of infinite peakon solutions, that is, solutions constructed as a superposition of an infinite number of peaked solitons.