University of Southern Queensland, Australia

Many important problems of applied mathematics and physics may be reduced to analyses of single evolution PDE with dissipation (like the Kuramoto-Sivashinsky equation in chemical physics and fluid mechanics). In order to balance internal energy influx such equations include high-order () spatial derivatives responsible for the energy dissipation. We consider two high-order dissipative models of that kind, one of which describes spinning combustion front, and the other describes extended elementary particle. The equations have similar properties. They are invariant with respect to constant shift of function of interest and contain nonlinear internal source of energy. As a result of a balance between the source and dissipation, nonlinear dissipative structures emerge in the form of pulses. The first equation in question, which is fourth-order, leads to stable standing pulse representing extended elementary particle, that is particle with finite size (not just physical point as in classical quantum mechanics). This model develops the work of Sivashinsky (Nuovo Cimento, 77A, 1983, 21) where the classical Hamilton-Jacobi (HJ) equation was extended using fourth-order dissipation. That approach, however, led to unstable pulse. We show that the Sivashinsky equation can be modified further to include nonlinear terms in order to ensure stability of the pulse. Remarkably, the original aim of his model was to put forward possible explanation of quantum randomness by deterministic law. This aim was achieved, because the extended HJ equation had global chaotic solutions. Whether or not chaos exists within our model is not clear yet. The second equation in question, which is sixth-order, models spinning combustion waves. It has fundamental solution in the form of travelling pulse. We present and discuss the results of numerical analyses of the models.