Research Interests: System theory, Operator theory,
(Inverse) scattering theory, Completely integrable PDEs.
I am collecting CI PDEs (in order to classify them later via vessel parameters). Please send me (email@example.com) any equation that does not appear on the list of known CI equations (known to me :-)!).
Few words about my research:
I have developed a method to solve a special type of partial differential equations called Completely Integrable (CI). As in the classical case, the (inverse) scattering theory is at the heart of this solution, but I have generalized it and inserted into a formalism of a “realization theory”. Initially, in order to understand a physical object, one could study how the light (or an electromagnetic wave) is scattered if pointed on this object. It was discovered in 60th that it is possible to fully recover the object from its “scattering data” and a mathematical arsenal was developed in order to solve so called Schrodinger 1-dimensional equation. Parameters of this equation, which become a serious obstacle nowadays, must be decreasing or the method of the inverse scattering does not work. In my method, the parameters are arbitrary (analytic) which enables to study much more difficult problems of this theory. Moreover, this method can be generalized to much wider families of linear differential equations.
Linear differential equations are at the heart of estimation theory and signal processing. These last two pairs of words could be considered as the most difficult problems of the modern engineering. They are extremely complicated and only linear cases are fully studied, because we have enough mathematics for it. Non linear estimation and signal processing is the most important and complicated problem in engineering. A toy, non linear equation, on which everyone checks his tools is so called Korteweg-de Vries equation. The method of inverse scattering solution for this equation was developed in 1967, and few years later many more completely integrable PDEs were shown to match the scheme of the inverses scattering solution. All these ideas found its implementation in engineering on different levels: for example, Non Linear Schrodinger equation is at the heart of the fiber optics; Boussinesq equation is at the basis of shallow water waves. There are many more examples, related to physics (electromagnetic fields, quantum theory, etc)
In my research it was shown that already these three important equations (appearing in Wikipedia): Korteweg-de Vries, Non Linear Schrodinger, Boussinesq are solved using my method for wider families of parameters (again, in the original setting they were usually rapidly decreasing). This enables to solve at least the indicated equations in many circumstances. Moreover, the theory itself is just at its first stage and few more equations were revealed and studied, using this method. Worth to mention that equations of the KdV hierarchy were created by this method too.
See vessels link for more details on the mathematics behind this method.
Number of hits since April 2014