See my curriculum vitae for an up-to-date list of publications. You can also find me on google scholar.
My research is partially supported by the U.S. National Science Foundation (grant NSF-DMS 2206270).
I study partial differential equations (PDEs) that are motivated from the modeling of physical phenomena and real-world problems in general. As most of these PDEs cannot be solved explicitly, it is important to know what kind of data guarantee the existence and uniqueness of a solution and, furthermore, how this solution depends on the data. Together, these three fundamental questions are referred to as Hadamard well-posedness.
PDEs that involve space and time derivatives are typically supplemented with initial data, giving rise to initial value problems (IVPs). Such problems are formulated on infinite spatial domains like the infinite line, or in periodic domains like the circle. However, it is often the case in applications that the spatial domain involves a boundary (e.g the half-line or the finite interval). Then, in addition to the initial data, boundary data must also be prescribed, giving rise to initial-boundary value problems (IBVPs).
A significant component of my work focuses on the Hadamard well-posedness of both IVPs and IBVPs for nonlinear evolution PDEs in one as well as in higher spatial dimensions. Famous examples of such PDEs include the nonlinear Schrödinger, Korteweg-de Vries and Camassa-Holm families of equations. While these equations has been studied extensively in the IVP setting, their corresponding IBVPs remain largely unexplored. Together with collaborators, we have introduced a new approach for studying such IBVPs using functional-analytic techniques, which takes advantage of the unified transform method of Fokas. The novel solution formulae produced by this method for linear IBVPs are used in order to establish the linear estimates necessary for proving Hadamard well-posedness of the associated nonlinear IBVPs via a contraction mapping argument.
Some prominent members of the aforementioned families of nonlinear PDEs are "special" in that they can be expressed as systems of linear equations known as Lax pairs. Hence, although nonlinear, they can be "integrated" by analyzing their Lax pairs via the powerful inverse scattering transform method. Such PDEs are known as integrable and, although extremely rare, they appear as central models in many important applications. For example, the cubic nonlinear Schrödinger and the Korteweg-de Vries equations are the two prototypical examples of integrable PDEs in one spatial dimension. A substantial component of my research is devoted to the study of integrable nonlinear equations in one and two spatial dimensions via inverse scattering techniques. Apart from the solution of these equations in both the IVP and the IBVP setting, I am especially interested in the study of their long-time asymptotic behavior via the steepest descent method of Deift and Zhou for oscillatory Riemann-Hilbert problems.