We propose, for a period of three years, the mathematical and numerical understanding of the dynamical equations governing a tsunami arriving at an open bay with an extensive discharging river, as is the case of Valdivia coastline. This problem is very challenging, and previous numerical estimations do not consider reasonable mathematical justifications. To describe this phenomenon, as explained in the subsection above, we will focus on two mathematical models: shallow water systems (abcd systems) and nonlinear Schrödinger equations (NLS).
We plan to understand how a sudden modification of the bathymetry may provide tsunami waves. We also consider how control of the bathymetry may slow down rough and tsunami waves at the surface. We also tackle the modeling issue when we have uncertainties on the bathymetry and then stochastic dispersion. We intend to extend the analysis to systems.
For models in deep water as NLS equations, we plan to study the return to the rest of the solutions, to estimate theoretically and numerically this decay rate to zero. This requires the design of numerical methods that are valid for large-time computations. We also address stochastic equations that came from uncertainties in the bathymetry, extending the previous works.
The long-time behavior for solutions of these equations, in the nonintegrable setting, is still out of reach. However, since abcd and NLS correspond to dispersive PDEs, it is well-known that the dynamics of the solutions of dispersive systems depend on special solutions: the so-called traveling wave solutions or solitons. We intend to enhance our comprehension of the dynamics of the system by gaining a deeper insight into the qualitative properties of traveling waves for these systems, such as their stability. As a by-product of our analysis, we expect some possible applications in other areas, such as applications of the propagation of light in optical fibers, governed by NLS equations.
