Integrability in 3 dimensions
Vladimir Novikov (Loughborough University)
Tuesday 6th November, 2018 16:00-17:00 Maths 311B
We consider the problem of detecting and classifying integrable partial differential (and difference) equations in 3D. Our approach is based on the observation that dispersionless limits of integrable systems in 3D possess infinitely many multi-phase solutions coming from the so-called hydrodynamic reductions. We consider a novel perturbative approach to the classification problem of dispersive equations. Based on the method of hydrodynamic reductions, we first classify integrable quasilinear systems which may (potentially) occur as dispersionless limits of soliton equations in 3D. To reconstruct dispersive deformations, we require that all hydrodynamic reductions of the dispersionless limit are inherited by the corresponding dispersive counterpart. This procedure leads to a complete list of integrable third and fifth order equations, which generalize the examples of Kadomtsev-Petviashvili, Veselov-Novikov and Harry Dym equations as well as integrable Davey-Stewartson type equations, some of which are apparently new. We also consider the problem of dispersive deformations on the Lax representation level and thus show that our approach allows starting from the dispersionless Lax representations to construct the fully dispersive Lax pairs representing the fully dispersive integrable systems.
We extend this approach to the fully discrete case. Based on the method of deformations of hydrodynamic reductions, we classify discrete 3D integrable Hirota-type equations within various particularly interesting subclasses as well as a number of classification results of scalar differential-difference integrable equations including that of the intermediate long wave and Toda type.