Some Recent Trends of Nonlinear Partial Differential Equations

It is a great honor for me to be awarded the Silver Professorship and to have this opportunity to tell you about some of the central aspects of the research field in which I have been involved. I was told that the readers of this article will be mostly non-mathematicians, including some non-scientists. This is very difficult for me; all of my experience as a speaker or writer has been addressing specialist in a particular research field—and so this will most likely be a dismal attempt. Though I will try to be as non-technical as possible, I will have to use some mathematical language and notation and I apologize in advance (also to specialists who happen to read this article for whom the matters explained here will be rather familiar). My work is mostly concerned with partial differential equations (PDEs). A partial differential equation for an unknown function is a simply a relationship between the unknown and its various partial derivatives with respect to several variables. Like algebraic equations, such a PDE could be linear, quadratic, or genuinely nonlinear in the unknown function. Naturally, nonlinear PDEs are often much harder to understand than the linear ones. So in this research field, we all have problems...... few wil