Harin, Alexander O.

### Abstract:

This thesis is devoted to two aspects of nonlinear PDEs which are fundamental
for the understanding of the order and coherence observed in the underlying physical
systems. These are symmetry properties and soliton solutions. We analyse these
fundamental aspects for a number of models arising in various branches of theoretical
physics and appli ed mathematics.
We start with a fluid model of a plasma in the case of a general polytropic
process. We propose a method of the analysis of unmagnetized travelling structures,
alternative to the conventional formalism of Sagdeev 's pseudopotential. This method
is then utilized to obtain the existence domain for compressive solitons and to establish
the absence of rarefactive solitons and monotonic double layers in a two-component
plasma.
The second class of models under consideration arises in (2+1)-dimensional condensed
matter physics. These are the Abelian gauge theories with Chern-Simons
term, which are currently considered as candidates for the description of high-Te
superconductivity and fra ctional quantum Hall effect. The emphasis here is on nonrelativistic
theories. The standard model of a self-gravitating gas of nonrelativistic
bosons coupled to the Chern-Simons gauge field is capable of describing asymptotically
vanishing field configurations , such as lump-like solitons. We formulate an
alternative model, which describes systems of repulsive particles with a background
electric charge and allows to incorporate asymptotically nonvanishing configurations,
such as condensate and its topological excitations. We demonstrate the absence of the condensate state in the standard nonrelativistic gauge theory and relate this fact
to the inadequate Lagrangian formulation of its nongauged precursor. Using an appropriate
modification of this Lagrangian as a basis for the gauge theory naturally
leads to the new model. Reformulating it as a constrained Hamiltonian system allows
us to find two self-duality limit s and construct a large variety of self-dual solutions.
We demonstrate the equivalence of the model with the background charge and the
standard model in the external magnetic field. Finally we discuss nontopological
bubble solutions in Chem-Simons-Maxwell theories and demonstrate their absence
in nonrelativistic theories.
Finally, we consider a model of a nonhomogeneous nonlinear string. We continue
the group theoretical classification of the string equations initiated by Ibragimov et
al. and present their preliminary group classification with respect to a countable dimensional subalgebra of their equivalence algebra. This subalgebra is an extension
of the 10-dimensional subalgebra considered by Ibragimov et al. Our main result here
is a table of non-equivalent equations possessing an additional symmetry.