Professor Ihyeok Seo’s research team has proven the existence of solutions in various critical regimes of the inhomogeneous nonlinear Schrödinger equation. While the subcritical regimes had been previously studied, this research is the first to reveal the existence of solutions in critical regimes, accomplished through a new approach.

This equation is widely used to explain various physical phenomena, and its inhomogeneity reflects the complexity of real-world physical systems. Proving the existence of solutions to this equation is crucial for understanding the dynamic behavior of such systems. However, the existence of solutions in critical regimes remained an unsolved problem.

Professor Seo’s team has provided the first mathematical proof of this problem, clearly demonstrating that solutions do exist in critical regimes of the equation. During the research, a new integrability estimate of the solution, a key element of the proof, was devised, with all possible estimates presented (see figure below). Additionally, Fourier analysis techniques and partial differential equation (PDE) theory were employed. This research is expected to contribute to various fields where the inhomogeneous nonlinear Schrödinger equation is applicable. In particular, it will aid in understanding the complex behaviors of physical systems, such as optical phenomena in inhomogeneous media and wave propagation through inhomogeneous materials.

Professor Seo, who led the research, commented, 'This achievement marks a turning point in the study of the inhomogeneous nonlinear Schrödinger equation and will deepen our understanding of physical systems in critical regimes.'