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.