Scientific Project

We aim to analyze diverse aspects in the study of dispersive equations under the scrutiny of microlocal analysis and other techniques, including some applications. As some partial differential equations appear in models for several physical phenomena, we plan to study well-posedness, control, and stabilization properties for Benney-Luke, Boussinesq KdV-KdV type systems which describe the propagation of bidirectional water waves, for Schrödinger on exterior domains which is a model of wave propagation in fiber optics and for biharmonic Schrödinger equation (or fourth order Schrödinger equation) on different domains which physically represents the propagation of intense laser beams.

The main objective of this proposal is to contribute to the strengthening of the research capacity in South America in the area of partial differential equations related to models that represent a physical phenomenon, through the establishment of research collaboration networks with first-level researchers in France, Chile, Colombia, and Brazil.   

From a scientific point of view, the purpose of this joint project is to study mathematical models of nonlinear dispersive waves that occur in theories of fluids, plasmas, nonlinear optics, Bose-Einstein condensates, and other branches of physical science. Several aspects such as unique continuation, control theory (internal and boundary), determination of coefficients, the study of pseudo-differential operators, stability analysis of special solutions, and the analysis of well-posedness for the Cauchy problem in some of the models considered. In addition, with the microlocal analysis, we will study some aspects of various partial differential equations (PDEs) concerning control theory in domains like manifolds, torus, and nonstandard domains.   

The proposed project is not only of scientific and engineering significance but also mutually advantageous to the participating institutions and countries, e.g., the young postdocs, and Ph.D. students will be benefited from this interchange. The accomplishment of this project will put together leading researchers from France and South America's mathematical community at the forefront of this mathematical area.   

The participants of this proposal offer a very good combination of expertise and extensive experience in modeling fluid mechanics and plasma, mathematical analysis, and its applications that certainly guarantee the achievement of the objectives of this project. 

So we expect that, at the end of the proposed support period, we shall have accomplished specific analysis and numerical simulations. Moreover, the techniques developed in carrying out this project are expected to be useful for other nonlinear dispersive equations. We also believe that more research problems will arise in the course of pursuing our goal, which will further enrich our future research agenda. 

1. Improve the understanding of Microlocal analysis methods in the analysis of PDEs and control problems. 

2. Study the unique continuation for Benney-Luke, Benjamin-Ono, Boussinesq systems type equation and systems.  

3. Obtain spectral properties for Schrödinger and other dispersive equations in the framework of Microlocal analysis.  

4. Study controllability and stability for Benney-Luke, Benjamin-Ono, Boussinesq systems type equations and systems, Schödinger-like equations and its generalizations.  

5. Study the well-posedness and some basic properties of partial differential equations modeling physical and biological phenomena: the propagation of bidirectional water waves and enzymatic reactions. 

6. Develop convergence and error estimation of numerical methods for dispersive equations.