Symmetries and boundary conditions of integrable nonlinear partial differential equations

Abstract

The solutions of initial-boundary value problems for integrable nonlinear partial differential equations have been one the most important problems in integrable systems on one hand, and on the other hand these kind problems have proved to be very hard especially when considered on half or bounded lines. The proper generalization of the Inverse Spectral Transform or any other possible method in a way that they apply on half or bounded lines, is a complicated problem itself. But one of the other obstacles is the choice of suitable boundary conditions. In this direction, there is a pioneering work of Sklyanin which has motivated us in considering the problem of establishing boundary conditions for integrable partial differential equations. To this end, we try to develop a way to find boundary conditions which would, in turn, be suitable for certain solution techniques. This could have been done in many different ways depending upon what is understood from integrability. Throughout this work, we use the phrase integrability \i\ the sense of generalized symmetries which has proved to be one of the most efficient approaches. We first give a proper definition of compatibility of boundary condition with a symmetry. Then we interpret the well known tools of symmetry approach in a different manner. These tools include the recursion operators and symmetries themselves. After some technical theorems, we pass to examples and consider many integrable equations. Furthermore, we give, a generalization of the method which makes use of the non-homogeneous symmetries. Finally we finish by some discrete equations, including the 2D Toda lattice. It is crucial to note that all the boundary conditions that have been already known to be compatible with the integrability property of the original equation, pass our criterion of compatibility.