Fall 2025 Schedule
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On the proximity between solutions to certain integrable systems and their non-integrable generalizations
The question of persistence of dynamics between integrable systems and their non-integrable counterparts is a central one in terms of both theory and applications. In this talk, we explore this direction in the context of (1) the (continuous) cubic nonlinear Schrödinger equation and (2) the (discrete) Ablowitz-Ladik equation, both of which are prime examples of completely integrable systems. Specifically, we discuss the proximity of solutions between these two models and a broad class of non-integrable generalizations in the framework of the associated initial value problem on the whole line. We consider the standard case of initial data that decay to zero at infinity, as well as the more intriguing case of nonzero boundary conditions at infinity which induces the phenomenon of modulational instability in focusing media. The talk is based on joint works with D.Hennig, N. Karachalios, J. Cuevas-Maraver, D. Mitsotakis, and I. Stratis.
3:00 pm, 306 Snow Hall
Well-posedness of the KdV-KS equation on a finite interval
The Korteweg-de Vries Kuramoto-Sivashinsky (KdV-KS) equation is a fourth-order non-linear evolution equation with a KdV-type nonlinearity. The traditional KS equation serves to model a variety of phenomena, such as disturbances in laminar flame fronts and widths of liquid films as they run down a surface. The KdV-KS equation includes a linear third derivative term in addition to the second and fourth found in KS, with a relative strength that acts as a parameter of the problem. In the liquid film example, this parameter allows the modeler to account for inclination in the surface, such as in the case of rain running down a smooth roof. In this talk, we consider the initial-boundary value problem for the one-dimensional KdV-KS on the finite interval. We establish the local well-posedness of this problem in the sense of Hadamard (existence and uniqueness of the solution as well as its continuous dependence on the data) for initial data in the Sobolev space Hs and boundary data in suitable Sobolev spaces determined by the regularity of the initial data and the KdV-KS equation. A foundational element of our proof is the linear solution operator derived through the unified transform of Fokas. We introduce new optimizations to this method which lead to a more elegant linear solution compared to the traditional approach, facilitating the proof of well-posedness all the way down to negative-order Sobolev spaces.
3:00 pm, 306 Snow Hall
Energy dissipation in ocean surface-breaking waves and two-layer stability analysis
Wave breaking profoundly influences mixing in the oceanic boundary layer, thereby modulating air–sea exchanges of momentum, heat, and freshwater that are essential to Earth’s weather and climate. This study examines wave breaking from two perspectives: (1) the physical energetics of the breaking process, and (2) the stability of interfacial flows. We first employ multiphase direct numerical simulations of deep-water surface-breaking waves by solving the incompressible Navier–Stokes equations with surface tension. With varied wave steepness and the Bond number, the energy budget is analyzed to reveal how wave energy is redistributed into turbulent dissipation, bubble generation, and droplet dynamics. We find that the fraction of energy dissipated by turbulence approaches a saturation limit even as the breaking intensity increases. In addition, we investigate the stability of two-layer superposed flows with differing densities and viscosities across a flat interface. We first consider the inviscid two-layer rotating flow as an initial-value problem and demonstrate its instability. Extending the work of Brattkus & Davis (1991), we further examine the linear stability of a two-layer viscous stagnation-point flow with a similarity-solution base state under the long-wave approximation.
3:00 pm, 306 Snow Hall
Stability and uniqueness in the large for some hyperbolic conservation laws
We consider $2 \times 2$ systems of hyperbolic conservation laws. It is classically known that such systems are well-posed in the BV class for small-BV initial data. For some systems, well-posedness in BV is known for large initial data. For some of these systems (specifically, scalar conservation laws and the isentropic Euler with $\gamma=1$), we will discuss recent work regarding well-posedness (in a non-BV class) for solutions with large-BV initial data. Possible extensions and future work will be discussed at the end.
3:00 pm, 306 Snow Hall