Abstract

While there exist several powerful classical techniques for the solution of boundary value problems for linear PDEs, such as integral transforms (Fourier, Laplace and variations), Green’s functions and the method of images, there does not exist a systematic solution method in the case of nonlinear PDEs. Moreover, even in the case of the linear problems one often comes across complicated boundary conditions which are not separable and thus cannot be addressed by the aforementioned classical techniques. This talk will be on a new, unified approach to boundary value problems for linear PDEs, with various types of boundary conditions which are not necessarily separable, as well as on its extension to nonlinear PDEs using the concept of Lax pairs and an appropriate Riemann-Hilbert or d-bar formalism.