# Analysis and Partial Differential Equations

Analysis grew from the processes of passage to limit in Differential and Integral Calculus. One of its main objectives is the solution of differential and integral equations, the characterization of the space of solutions and the convergence of processes of solution by approximation.

Many of its techniques were later unified in Functional Analysis, in which functional spaces are viewed in abstract form. Analysis is greatly used in Physics, Geometry, Engineering and in practically all of Applied Mathematics.

Research in Analysis at IMPA has developed in the following sectors:

**Partial Differential Equations in Mathematical Physics**

Non linear evolutionary equations are studied, for example, Korteweg-de Vries’, Benjamin-Ono’s, Navier-Stokes’ and Euler’s, and certain aspects are broached, such as the existence of solutions, uniqueness, dependence on initial data and asymptotic behavior. Another important theme is Schrödinger’s equation with time dependent Hamiltonian, which is studied through the spectral properties of the operators involved.

###
**Inverse Problems and Applications**

Generally speaking, mathematical models are analyzed for the propagation of radiation in general environments. Its potential applications lie in computerized tomography, in preventive medicine and in geophysics.

**Solitons and Non Linear Analysis**

Solitons are great amplitude waves that are propagated in non-linear media and interact without substantial changes in form. This theory developed greatly as of the 1970’s, in an attempt to understand the extraordinary robustness of this phenomenon and develop its countless applications, from optical engineering to signal transmission.