The goal of this thesis consisting of articles is to study certain geometric aspects of supermanifolds associated with integrable suspersymmetric systems. This work is contained in four published articles and one currently submitted article in international peer-reviewed journals. In the first article, two supersymmetric extensions of the Gauss–Weingarten and Gauss–Codazzi equations for surfaces immersed in Euclidean superspaces were constructed. This allowed us to provide a geometric characterization of such surfaces with linearly independent tangent vectors oriented in the directions of the infinitesimal displacement of the fermionic covariant derivatives. In addition, a study of the symmetries of the supersymmetric versions of the Gauss–Codazzi equations led to the construction of invariant solutions, involving bosonic and fermionic variables, through the symmetry reduction method, which led to nontrivial surfaces, e.g. vanishing Gauss curvature surfaces. In the second article, a conjecture stating the necessary conditions for a system to be integrable in the sense of soliton theory was extended to the supersymmetric cases. This was accomplished by introducing a new projection operator and by comparing the symmetries of the original system to those of the associated linear problem. This conjecture was applied to some examples and a fermionic “spectral” parameter was introduced in one of the systems. In the third article, two supersymmetric versions of the Fokas–Gel’fand formula for the immersion of soliton surfaces in Lie superalgebras were constructed. The geometric characterization of the immersion function presented in this article allowed us to investigate the behavior of the associated surfaces. These theoretical considerations were applied to the supersymmetric sine-Gordon equation, for which constant Gaussian curvature surfaces and nonlinear-type surfaces were obtained. The fourth article was devoted to integrability properties of the supersymmetric sine-Gordon equation and to the construction of explicit multisoliton solutions. Two types of linear spectral problems, a set of coupled super-Riccati equations and the auto-Bäcklund transformation, all equivalent to the supersymmetric sine-Gordon equation, were studied. In addition, a detailed analysis of the nth Darboux transformations allowed us to find nontrivial multisoliton solutions of the supersymmetric sine-Gordon equation. These solutions were used to investigate the bosonic supersymmetric version of the Sym–Tafel immersion formula. In the fifth article, a new geometric characterization of the Fokas–Gel’fand immersion formula was presented. In order to do this, three different types of linear spectral problems were studied, one involving the covariant fermionic derivatives, one involving the bosonic variable derivatives and one involving the fermionic variable derivatives. This geometric characterization involves eight linearly independent coefficients for both the first and second fundamental forms, in constrast with three such coefficients in the third article, which leads to a richer geometry in the sense that curve-like supermanifolds in the third article are of higher dimensions in the fifth article.

The objects under consideration in this thesis are systems of first-order quasilinear equations. In the first part of the thesis, a study is made of an ideal plasticity model from the point of view of the classical Lie point symmetry group. Planar flows are investigated in both the stationary and non-stationary cases. Two new vector fields are obtained. They complete the Lie algebra of the stationary case, and the subalgebras are classified into conjugacy classes under the action of the group. In the non-stationary case, a classification of the Lie algebras admissible under the chosen force is performed. For each type of force, the vector fields are presented. For monogenic forces, the algebra is of the highest possible dimension. Its classification into conjugacy classes is made. The symmetry reduction method is used to obtain explicit and implicit solutions of several types. Some of them can be expressed in terms of one or two arbitrary functions of one variable. Others can be expressed in terms of Jacobi elliptic functions. Many solutions are interpreted physically in order to determine the shape of realistic extrusion dies. In the second part of the thesis, we examine solutions expressed in terms of Riemann invariants for first-order quasilinear systems. The generalized method of characteristics, along with a method based on conditional symmetries for Riemann invariants are extended so as to be applicable to systems in their elliptic regions. The applicability of the methods is illustrated by examples such as non-stationary ideal plasticity for an irrotational flow as well as fluid mechanics equations. A new approach is developed, based on the introduction of rotation matrices which satisfy certain algebraic conditions. It is directly applicable to non-homogeneous and non-autonomous systems. Its efficiency is illustrated by examples which include a system governing the non-linear superposition of waves and particles. The general solution is constructed in explicit form.

In this work, the conditional symmetry method is adapted in order to construct solutions expressed in terms of Riemann invariants. Nonelliptic quasilinear homogeneous systems of multidimensional partial differential equations of hydrodynamic type are considered. A detailed description of the necessary and sufficient conditions for the local existence of these types of solutions is given. The relationship between the structure of integral elements and the possibility of constructing certain classes of rank-k solutions is discussed. These classes of solutions include nonlinear superpositions of Riemann waves and multisolitonic solutions. This approach is generalized to first-order inhomogeneous hyperbolic quasilinear systems. These methods are applied to the equations describing an isentropic fluid flow in (3+1) dimensions. Several new classes of rank-2 and rank-3 solutions are obtained which contain double and triple solitonic solutions. New nonlinear and linear superpositions of Riemann waves are described. Finally, certain aspects of the construction of rank-2 solutions through an application to the dispersionless Kadomtsev-Petviashvili equation are discussed.

Mémoire numérisé par la Direction des bibliothèques de l'Université de Montréal.

Thèse numérisée par la Direction des bibliothèques de l'Université de Montréal.