Coherent structures and symmetry properties in nonlinear models used in theoretical physics.

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.