Integrability in Multidimensions and Boundary Value Problems

There exist certain distinctive nonlinear equations called integrable. The impact of the mathematical analysis of such special equations cannot be overestimated. For example, firstly, for integrable equations we have learned detailed aspects of solution behaviour, which includes the long-time asymptotics of solutions and the central role played by solitons. Secondly, it has become apparent that some of the lessons taught by integrable equations have applicability even in non-integrable situations. Indeed, many investigations in the last decade regarding well-posedness of PDEs in appropriate Sobolev spaces have their genesis in results coming from the theory of integrable systems, albeit in non-integrable settings. Perhaps the two most important open problems in the theory of integrable equations have been (a) the solution of initial-boundary as opposed to initial value problems and (b) the derivation and solution of integrable nonlinear PDEs in 3+1 dimensions. In the last 12 years, the PI has made significant progress towards the solution of both of these problems. Namely, regarding (a) he has introduced a unified approach for analysing boundary value problems in two dimensions and regarding (b) he has derived and solved integrable nonlinear PDEs in 4+2. However, many fundamental problems remain open. The proposal aims to investigate several such problems among which the most significant are: (a) the extension of the method for solving boundary value problems from two to three dimensions and (b) the reduction of the new integrable PDEs from 4+2 to 3+1. In addition, the Camassa-Holm analogue of the celebrated sine Gordon equation, several boundary value problems of the elliptic version of the Ernst equation, and the KdV equation on the half-line with time periodic boundary conditions will also be investigated.

This is mainly a mathematical analysis project: Its successful implementation will lead to the emergence of efficient mathematical techniques for: (a) the analysis of boundary (as opposed to initial) value problems (b) the investigation of PDEs in 4+2 and 3+1. These techniques will be directly applicable only to the restricted class of integrable PDEs. However, earlier analytical results obtained for integrable PDEs, have had a significant impact on analytical results obtained by classical PDE techniques. Thus, it is natural to expect that the results obtained in this project will have a significant impact on the analysis of nonlinear PDEs beyond integrability. In addition, the formulae obtained using the new method for solving boundary value problems, have certain advantages which lead to efficient numerical computations. For linear and integrable nonlinear evolution PDEs in 1+1 this has already been demonstrated in [17] and in the work of Zhang [34]. Thus, it is natural to expect that the results obtained in this project will have an impact on the development of novel numerical techniques. The PI in collaboration with P. Barbano and J. Lennels will attempt to implement XFCT. This is of course adventurous, since different modalities present different challenges, and also because we have to find sources of real data. However, if successful, this project will have significant practical implications.