Research project is (co) funded by the Slovenian Research Agency.

**UL Member:** Faculty of Mathematics and Physics

**Code:** N1-0137

**Project:** Nonlinear Waves and Spectral Theory

**Period:** 1. 6. 2020 - 31. 5. 2023

**Range per year:** 1,09 FTE, **category:** C

**Head:** Aleksey Kostenko

**Research activity**: Natural sciences and mathematics

**Research Organisations**

**Researchers**

**Citations for bibliographic records**

**Project description:**

Spectral theory of linear operators finds numerous applications in the study of nonlinear wave equations. For example, the inverse scattering transform (IST) was introduced in 1967 to solve the celebrated Korteweg—de Vries equation and its discovery is regarded as one of major achievements of the 20th century connecting different branches of pure mathematics and theoretical physics. The IST approach has numerous advantages (e.g., construction of explicit solutions and tools to analyse behavior of solutions at large times), however, it is applicable to a very limited class of nonlinear equations. Another example is provided by the study of stability of soliton-type solutions, which usually leads to a thorough investigation of spectral properties of associated linearized equations.

Our project has several main goals. The first aim is to develop the IST approach to the Camassa—Holm and Degasperis—Processi equations. This requires a significant progress in understanding the inverse spectral/scattering theory of the corresponding one-dimensional isospectral problems (generalized indefinite strings and cubic strings), which is very far from being complete. Another main aim is the development of spectral theory of linear operators in indefinite inner product spaces (Krein spaces) and its applications to the study of stability of soliton-type solutions for nonlinear wave equations. All these problems also have numerous applications in other areas of mathematics and physics and we plan both to enrich the existing connections and solve several related old-standing problems.