Recently, Professor Tian Shoufu from the School of Mathematics of our university, together with Greek mathematical physics expert Andreas Chatziafratis and Professor Tohru Ozawa from Waseda University, Japan, discovered a previously unknown long-term instability phenomenon of the one-dimensional linear Schrödinger equation on the quarter-plane of vacuum spacetime. The relevant results were published in the international top mathematics journal Mathematische Annalen.

This work takes the inhomogeneous linear Schrödinger equation as an illustrative example. In the context of classical smoothness, it solves the half-line problem with general initial data, boundary conditions, and forcing terms through the formula proposed by the linear Fokas unified transform method. First, by proposing a new appropriate decomposition of this formula on the complex plane, convergence is ensured in a strictly defined sense. This novel analysis allows, for the first time, the necessary rigorous posteriori verification of the complete initial-boundary value problem. The work then conducts a thorough study of the behavior of solutions near the boundary of the spacetime domain, proving that the integrals in the formula proposed by the linear Fokas unified transform method converge uniformly to the "prescribed" values, and that solutions can be smoothly extended to the boundary only if the data satisfy certain compatibility conditions. In addition, this work performs an effective asymptotic study of far-field dynamics, derives new explicit asymptotic formulas to characterize the properties of solutions, and discovers that the asymptotic behavior of solutions is sensitive to perturbations of data at the "corner" of the quadrant. However, in all cases, even if the initial data are assumed to belong to the Schwartz class, the solution loses this property as soon as time is positive. Thus, by re-examining a well-known low-order linear evolutionary partial differential equation, a newly discovered type of long-range instability effect in the Stokes equation is further confirmed and clarified. Meanwhile, the rigorous analytical methods of this work can be directly extended to other Schrödinger-like evolutionary equations and more general problems with dispersive representations on domains with quasi-infinite boundaries, playing an important role in promoting the research of long-term instability phenomena.

It is understood that in 1997, Academician of the European Academy of Sciences and Professor Fokas of the University of Cambridge proposed a unified method that can apply the Riemann-Hilbert method to solve initial-boundary value problems of integrable systems, later known as the Fokas method. This method uses the Riemann-Hilbert method, synergetic theory, and relaxation pair form to solve integrable nonlinear partial differential equations. Unlike classical methods such as Laplace, Mellin, and (co)sine transforms, which are only applicable to very special cases, this new method has been extended to solve linear partial differential equations for some inseparable and non-adjoint problems. At the same time, this new method has achieved success in solving nonlinear linear partial differential equations of arbitrary order on various types of domains, including elliptic problems and cases with variable coefficients or even moving boundaries. Furthermore, it has been extended to the solution of mixed derivatives, nonlocal/inseparable conditions, interface problems, systems of equations, and fractional evolutionary equations. For a long time, although many experts and scholars have made numerous extensions to this method and obtained a series of excellent results, there have been no mature research results on its posteriori and the properties of functions defined by this method. The research by Professor Tian Shoufu and his collaborators fills the gap in this field of research.