Repository logo

Doctoral Degrees (Pure Mathematics)

Permanent URI for this collection


Recent Submissions

Now showing 1 - 20 of 24
  • Item
    Contribution to iterative algorithms for certain optimization problems and fixed-point problems in Banach spaces.
    (2018) Okeke, Chibueze Christian.; Mewomo, Oluwatosin Temitope.
    We study the convergence analysis of the xed points set of common solution of a one- parameter nonexpansive semigroup, the set of solution of constrained convex minimization problem and the set of solutions of generalized equilibrium problem in a real Hilbert space using the idea of regularized gradient-projection algorithm. Also, we look at the strong convergence of a modi ed gradient projection algorithm and forward-backward algorithm in Hilbert spaces with numerical computations. We also introduce an iterative algorithm for approximating a common solution of generalized mixed equilibrium problem and xed point problem in a real re exive Banach space. Using our algorithm, a strong convergence theorem is proved concerning an element in the intersection of set of solutions of general- ized mixed equilibrium problem and the set of solutions of xed point for a nite family of Bregman strongly nonexpansive mappings. Moreover, we study and analyze an iterative method for nding a common element of the xed points set of an in nite family of k-demicontractive mappings which is also a solution to a zero of the sum of two monotone operators, with one operator being maximal monotone and the other inverse-strongly monotone. We further extend our study from the frame work of real Hilbert spaces to more general real smooth and uniformly convex Banach spaces. In this space, we introduce an iterative algorithm with Meir-Keeler contractions for nding zeros of the sum of nite families of m-accretive operators and nite family of inverse strongly accretive operators. We apply our result to the approximation of solution of certain integro-di erential equation with generalized p-Laplacian operators. Furthermore, we study the convergence theorem for a new class of split variational inequal- ity and variational inclusion problem in Hilbert space. We further considered split equality for minimization problem and xed point sets, split xed point problem and monotone inclusion problems, split equilibrium problem and xed point set for multivalued map- pings. All these of our algorithms involve a step-size selected in such a way that their implementation does not require the computation or an estimate of the spectral radius. Again, an iterative algorithm that does not require any knowledge of the operator norm for approximating a solution of split equality equilibrium and xed point problems in the frame work of p-uniformly convex Banach spaces which are also uniformly smooth is introduced of which we studied the approximation of solution of split equality generalized mixed equilibrium problem and xed point problem for right Bregman strongly quasi- nonexpansive mappings in q-uniformly convex Banach spaces which are also uniformly smooth. We also study and analyze an iterative algorithm for nding a common element of the set of the split equality for monotone inclusion problem and xed point of a right Bregman strongly nonexpansive mapping T in the setting of p-uniformly convex uniformly smooth Banach spaces. Finally, we present numerical examples of our theorems and apply our results to study the convex minimization problems and equilibrium problems.
  • Item
    Fixed point approach for solving optimization problems in Hilbert, Banach and convex metric spaces.
    (2023) Ogwo, Grace Nnennaya.; Mewomo, Oluwatosin Temitope.; Alakoya, Timilehin Opeyemi.
    In this thesis, we study the fixed point approach for solving optimization problems in real Hilbert, Banach and Hadamard spaces. These optimization problems include the variational inequality problem, split variational inequality problem, generalized variational inequality problem, split equality problem, monotone inclusion problem, split monotone inclusion problem, minimization problem, split equilibrium problem, among others. We consider some interesting classes of mappings such as the nonexpansive semigroup in real Hilbert spaces, strict pseudo-contractive mapping in real Hilbert spaces and 2-uniformly convex real Banach spaces, nonexpansive mapping between a Hilbert space and a Banach space, and quasi-pseudocontractive mapping in Hilbert spaces and Hadamard spaces. We introduce several iterative schemes for approximating the solutions of the various aforementioned optimization problems and fixed point problems and prove their convergence results. We adopt and implement several inertial methods such as the inertial-viscosity-type algorithm, relaxed inertial subgradient extragradient, modified inertial forward-backward splitting algorithm viscosity method, among others. Furthermore, we present several novel and practical applications of our results to solve other optimization problems, image restoration problem, among others. Finally we present several numerical examples in comparison with some results in the literature to illustrate the applicability of our proposed methods.
  • Item
    Iterative approximations of certain nonlinear optimization, generalized eqilibrium and fixed point problems in Hilbert and Banach spaces.
    (2023) Godwin, Emeka Chigaemezu.; Mewomo, Oluwatosin Temitope.; Alakoya, Timilehin Opeyemi.
    In this thesis, in the framework of Banach spaces, we study several iterative methods for finding the solutions of many important problems in fixed point theory and optimization. Some of these include the equilibrium problem, monotone variational inclusion problem, variational inequality problem, split common fixed point problem and split minimization problem. In addition, we study fixed point problem for some important and interesting classes of mappings such as nonexpansive mappings, pseudocontractive mappings, asymptotically demicontractive mappings, quasi-pseudocontractive mappings, demimetric mappings and multivalued demicontractive mappings in real Hilbert spaces. Furthermore, we study some other classes of mappings which include the class of Bregman quasi-nonexpansive mappings and Bregman relatively nonexpansive mappings in real p-uniformly convex Banach spaces which are also uniformly smooth. Another important problem considered is the split equality problem. The split equality problem has gained attention from authors because of its vast applications to real life problems. This problem is known to contain several other optimization problems as special cases. Based on its numerous applications, we study a multiple set split equality equilibrium problem consisting of pseudomonotone bifunctions together with fixed point problem of certain nonlinear mappings in p-uniformly convex and uniformly smooth Banach spaces. In each case, we propose and study iterative algorithms for approximating the solutions of these problems and prove strong convergence theorems under suitable conditions on the control parameters. In most cases, we incorporate the inertial term which is known to speed up the convergence rate of iterative schemes. In addition, we employ several efficient iterative techniques which include the projection and contraction method, alternative regularization method, modified Halpern’s method, inertial Tseng’s extragradient method and viscosity approximation method. In all the cases, we design our algorithms in such a way that the step size does not depend on the knowledge of the Lipschitz constants of the cost operator or the norm of the bounded linear operators. We present some applications of our results to solve convex minimization problems, multiple set split variational inequality problem, image restoration problem, oligopolistic market equilibrium problem, among others. Also, we present several numerical experiments to demonstrate the efficiency, applicability and usefulness of our iterative schemes in comparison with several of the existing methods in the literature. The results obtained in this thesis extend and improve many existing results in the literature in a unified way.
  • Item
    Axial algebras for sporadic simple groups HS and Suz.
    (2018) Shumba, Tendai M. Mudziiri.; Rodrigues, Bernardo Gabriel.; Shpectorov, Sergey.
    Motivated by the construction of the Monster sporadic simple group as a group of automorphisms of an algebra and the recent development of ax- ial algebras as a generalization of Majorana representations, we construct axial algebras for the sporadic simple groups HS and Suz in different ways analogous to the Norton algebra construction. We study how these algebras decompose as direct sums of the adjoint action of an axis. Fusion rules, that is the rules with which eigenvectors from various eigenspaces of the adjoint action multiply, are found. This places these groups in a general framework of groups acting on algebras hence giving a common theme for their origin.
  • Item
    A study of optimization problems and fixed point iterations in Banach spaces.
    (2019) Jolaoso, Lateef Olakunle.; Mewomo, Oluwatosin Temitope.
    Abstract available in PDF.
  • Item
    A study of optimization and fixed point problems in certain geodesic metric spaces.
    (2019) Izuchukwu, Chinedu.; Mewomo, Oluwatosin Temitope.
    Abstract available in PDF.
  • Item
    A numerical study of entropy generation, heat and mass transfer in boundary layer flows.
    (2018) Almakki, Mohammed Hassan Mohammed.; Sibanda, Precious.
    This study lies at the interface between mathematical modelling of fluid flows and numerical methods for differential equations. It is an investigation, through modelling techniques, of entropy generation in Newtonian and non-Newtonian fluid flows with special focus on nanofluids. We seek to enhance our current understanding of entropy generation mechanisms in fluid flows by investigating the impact of a range of physical and chemical parameters on entropy generation in fluid flows under different geometrical settings and various boundary conditions. We therefore seek to analyse and quantify the contribution of each source of irreversibilities on the total entropy generation. Nanofluids have gained increasing academic and practical importance with uses in many industrial and engineering applications. Entropy generation is also a key factor responsible for energy losses in thermal and engineering systems. Thus minimizing entropy generation is important in optimizing the thermodynamic performance of engineering systems. The entropy generation is analysed through modelling the flow of the fluids of interest using systems of differential equations with high nonlinearity. These equations provide an accurate mathematical description of the fluid flows with various boundary conditions and in different geometries. Due to the complexity of the systems, closed form solutions are not available, and so recent spectral schemes are used to solve the equations. The methods of interest are the spectral relaxation method, spectral quasilinearization method, spectral local linearization method and the bivariate spectral quasilinearization method. In using these methods, we also check and confirm various aspects such as the accuracy, convergence, computational burden and the ease of deployment of the method. The numerical solutions provide useful insights about the physical and chemical characteristics of nanofluids. Additionally, the numerical solutions give insights into the sources of irreversibilities that increases entropy generation and the disorder of the systems leading to energy loss and thermodynamic imperfection. In Chapters 2 and 3 we investigate entropy generation in unsteady fluid flows described by partial differential equations. The partial differential equations are reduced to ordinary differential equations and solved numerically using the spectral quasilinearization method and the bivariate spectral quasilinearization method. In the subsequent chapters we study entropy generation in steady fluid flows that are described using ordinary differential equations. The differential equations are solved numerically using the spectral quasilinearization and the spectral local linearization methods.
  • Item
    Iterative algorithms for solutions of nonlinear equations in Banach spaces.
    (2019) Aibinu, Mathew Olajiire.; Pillay, Paranjothi.
  • Item
    Combined impulse control and optimal stopping in insurance and interest rate theory.
    (2015) Mgobhozi, Sivuyile Wiseman.; Chikodza, Eriyoti.; Mukwembi, Simon.
    In this thesis, we consider the problem of portfolio optimization for an insurance company with transactional costs. Our aim is to examine the interplay between insurance and interest rate. We consider a corporation, such as an insurance firm, which pays dividends to shareholders. We assume that at any time t the financial reserves of the insurance company evolve according to a generalized stochastic differential equation. We also consider that these liquid assets of the firm earn interest at a constant rate. We consider that when dividends are paid out, transaction costs are incurred. Due to the presence of transactions costs in the proposed model, the mathematical problem becomes a combined impulse and stochastic control problem. This thesis is an extension of the work by Zhang and Song [69]. Their paper considered dividend control for a financial corporation that also takes reinsurance to reduce risk with surplus earning interest at the constant force p > 0. We will extend their model by incorporating jump diffusions into the market with dividend payout and reinsurance policies. Jump-diffusion models, as compared to their diffusion counterpart, are a more realistic mathematical representation of real-life processes in finance. The extension of Zhang and Song [69] model to the jump case will require us to reduce the analytical part of the problem to Hamilton-Jacobi-Bellman Qausi-Variation Inequalities for combined impulse control in the presence of jump diffusion. This will assist us to find the optimal strategy for the proposed jump diffusion model while keeping the financial corporation in the solvency region. We will then compare our results in the jump-diffusion case to those obtained by Zhang and Song [69] in the no jump case. We will then consider models with stochastic volatility and uncertainty as a means of extending the current theory of modeling insurance reserves.
  • Item
    Approximation methods for solutions of some nonlinear problems in Banach spaces.
    (2017) Ogbuisi, Ferdinard Udochukwu.; Mewomo, Oluwatosin Temitope.
    Abstract available in PDF file.
  • Item
    Spherically symmetric charged Einstein-Maxwell solutions.
    (1999) Pasha, Muhammad Akmal.; Maharaj, Sunil Dutt.; Leach, Peter Gavin Lawrence.
    In this thesis we study spherically symmetric spacetimes with a perfect fluid source which incorporates charge. We seek explicit solutions to the Einstein- Maxwell system of equations. For nonaccelerating spherically symmetric models a charged, dust solution is found. With constant pressure the equations reduce to quadratures. Particular solutions are also found, with no acceleration, with the equation of state P =( y - 1)u. The Lie analysis is utilised to reduce the Einstein- Maxwell equations to a syst.em of ordinary differential equations. The evolution of the model depends on a Riccati equation for this general class of accelerating, expanding and shearing spacetimes with charge. Also arbitrary choices for the gravitational potentials lead to explicit solutions in particular cases. With constant gravitational potential A we generate a simple nonvacuum model. The analysis, in this case, enables us to reduce the solution to quadratures. With the value y = 2, for a stiff equation of state, we find that the solution is expressable in terms of elementary functions. Throughout the thesis we have attempted to relate our results to previously published work, and to obtain the uncharged perfect fluid limit where appropriate.
  • Item
    Lie group analysis of exotic options.
    (2013) Okelola, Michael.; Govinder, Keshlan Sathasiva.; O'Hara, John Gerard.
    Exotic options are derivatives which have features that makes them more complex than vanilla traded products. Thus, finding their fair value is not always an easy task. We look at a particular example of the exotic options - the power option - whose payoffs are nonlinear functions of the underlying asset price. Previous analyses of the power option have only obtained solutions using probability methods for the case of the constant stock volatility and interest rate. Using Lie symmetry analysis we obtain an optimal system of the Lie point symmetries of the power option PDE and demonstrate an algorithmic method for finding solutions to the equation. In addition, we find a new analytical solution to the asymmetric type of the power option. We also focus on the more practical and realistic case of time dependent parameters: volatility and interest rate. Utilizing Lie symmetries, we are able to provide a new exact solution for the terminal pay off case. We also consider the power parameter of the option as a time dependent factor. A new solution is once again obtained for this scenario.
  • Item
    Aspects of distance measures in graphs.
    (2011) Ali, Patrick Yawadu.; Dankelmann, Peter A.; Mukwembi, Simon.
    In this thesis we investigate bounds on distance measures, namely, Steiner diameter and radius, in terms of other graph parameters. The thesis consists of four chapters. In Chapter 1, we define the most significant terms used throughout the thesis, provide an underlying motivation for our research and give background in relevant results. Let G be a connected graph of order p and S a nonempty set of vertices of G. Then the Steiner distance d(S) of S is the minimum size of a connected subgraph of G whose vertex set contains S. If n is an integer, 2 ≤ n ≤ p, the Steiner n-diameter, diamn(G), of G is the maximum Steiner distance of any n-subset of vertices of G. In Chapter 2, we give a bound on diamn(G) for a graph G in terms of the order of G and the minimum degree of G. Our result implies a bound on the ordinary diameter by Erdös, Pach, Pollack and Tuza. We obtain improved bounds on diamn(G) for K3-free graphs and C4-free graphs. In Chapter 3, we prove that, if G is a 3-connected plane graph of order p and maximum face length l then the radius of G does not exceed p/6 + 5l/6 + 5/6. For constant l, our bound improves on a bound by Harant. Furthermore we extend these results to 4- and 5-connected planar graphs. Finally, we complete our study in Chapter 4 by providing an upper bound on diamn(G) for a maximal planar graph G.
  • Item
    On convection and flow in porous media with cross-diffusion.
    (2012) Khidir, Ahmed A.; Sibanda, Precious.
    In this thesis we studied convection and cross-diffusion effects in porous media. Fluid flow in different flow geometries was investigated and the equations for momentum, heat and mass transfer transformed into a system of ordinary differential equations using suitable dimensionless variables. The equations were solved using a recent successive linearization method. The accuracy, validity and convergence of the solutions obtained using this method were tested by comparing the calculated results with those in the published literature, and results obtained using other numerical methods such as the Runge-Kutta and shooting methods, the inbuilt Matlab bvp4c numerical routine and a local non-similarity method. We investigated the effects of different fluid and physical parameters. These include the Soret, Dufour, magnetic field, viscous dissipation and thermal radiation parameters on the fluid properties and heat and mass transfer characteristics. The study sought to (i) investigate cross-diffusion effects on momentum, heat and mass transport from a vertical flat plate immersed in a non-Darcy porous medium saturated with a non-Newtonian power-law fluid with viscous dissipation and thermal radiation effects, (ii) study cross-diffusion effects on vertical an exponentially stretching surface in porous medium and (iii) apply a recent hybrid linearization-spectral technique to solve the highly nonlinear and coupled governing equations. We further sought to show that this method is accurate, efficient and robust by comparing it with established methods in the literature. In this study the non-Newtonian behaviour of the fluid is characterized using the Ostwald-de Waele power-law model. Cross-diffusion effects arise in a broad range of fluid flow situations in many areas of science and engineering. We showed that cross-diffusion has a significant effect on heat and mass-transfer processes and cannot be neglected.
  • Item
    Measurement and modelling of households' demand and access to basic water in relation to the rapidly increasing household numbers in South Africa.
    (2010) Chidozie, Nnadozie Remigius.; Leach, Peter Gavin Lawrence.
    Service delivery in post-apartheid South Africa has become a topical issue both in the academia and the political arena . The rise of social movements, the xenophobic tensions of May 2008 and protest actions could be noted as the major traits of post-apartheid South Africa. Though there are divergent views on the underlying causes of these protests, lack of service delivery has most significantly been at the centre stage. In this thesis we investigate the relationship between household/population changes and the demand for piped-water connection in South Africa. There is an ample, albeit at times of questionable accuracy, supply of statistics from official and other sources. These statistics are both the source of inspiration of particular societal measures to be investigated and a gauge of the accuracy of the mathematical/statistical modelling which is the central feature of this project. We construct mathematical/statistical models which take into account demographic constituents of the problem using differential equations for modelling household dynamics and we also investigate the interaction of demographic parameters and the demand for piped-water connection using multivariate statistical techniques. The results show that with a boost in delivery the rich provinces seem to be in better standing of meeting targets and that the increasing demand in household-based services could be most significantly attributed to the fragmentation of households against other demographic processes like natural increase in population and net migration. The results imply that in as much as service delivery policies and programmes should focus on formerly disadvantaged poor communities, adequate provisions for increasing service demands in urban centres should also be a priority in view of the increasing in-migration from rural areas as households fragment. Most of the findings/results are in tabular and graphical forms for easy understanding of the reader.
  • Item
    Singularity and symmetry analysis of differential sequences.
    (2009) Maharaj, Adhir.; Leach, Peter Gavin Lawrence.; Euler, Marianna.
    We introduce the notion of differential sequences generated by generators of sequences. We discuss the Riccati sequence in terms of symmetry analysis, singularity analysis and identification of the complete symmetry group for each member of the sequence. We provide their invariants and first integrals. We propose a generalisation of the Riccati sequence and investigate its properties in terms of singularity analysis. We find that the coefficients of the leading-order terms and the resonances obey certain structural rules. We also demonstrate the uniqueness of the Riccati sequence up to an equivalence class. We discuss the properties of the differential sequence based upon the equation ww''−2w12 = 0 in terms of symmetry and singularity analyses. The alternate sequence is also discussed. When we analyse the generalised equation ww'' − (1 − c)w12 = 0, we find that the symmetry properties of the generalised sequence are the same as for the original sequence and that the singularity properties are similar. Finally we discuss the Emden-Fowler sequence in terms of its singularity and symmetry properties.
  • Item
    Clifford-Fischer theory applied to certain groups associated with symplectic, unitary and Thompson groups.
    (2012) Basheer, Ayoub Basheer Mohammed.; Moori, Jamshid.
    The character table of a finite group is a very powerful tool to study the groups and to prove many results. Any finite group is either simple or has a normal subgroup and hence will be of extension type. The classification of finite simple groups, more recent work in group theory, has been completed in 1985. Researchers turned to look at the maximal subgroups and automorphism groups of simple groups. The character tables of all the maximal subgroups of the sporadic simple groups are known, except for some maximal subgroups of the Monster M and the Baby Monster B. There are several well-developed methods for calculating the character tables of group extensions and in particular when the kernel of the extension is an elementary abelian group. Character tables of finite groups can be constructed using various theoretical and computational techniques. In this thesis we study the method developed by Bernd Fischer and known nowadays as the theory of Clifford-Fischer matrices. This method derives its fundamentals from the Clifford theory. Let G = N·G, where N C G and G/N = G, be a group extension. For each conjugacy class [gi]G, we construct a non-singular square matrix Fi, called a Fischer matrix. Once we have all the Fischer matrices together with the character tables (ordinary or projective) and fusions of the inertia factor groups into G, the character table of G is then can be constructed easily. In this thesis we apply the coset analysis technique (this is a method to find the conjugacy classes of group extensions) together with theory of Clifford-Fischer matrices to calculate the ordinary character tables of seven groups of extensions type, in which four are non-split and three are split extensions. These groups are of the forms: 21+8 + ·A9, 37:Sp(6, 2), 26·Sp(6, 2), 25·GL(5, 2), 210:(U5(2):2), 21+6 − :((31+2:8):2) and 22n·Sp(2n, 2) and 28·Sp(8, 2). In addition we give some general results on the non-split group 22n·Sp(2n, 2).
  • Item
    Algebraic properties of ordinary differential equations.
    (1995) Leach, Peter Gavin Lawrence.
    In Chapter One the theoretical basis for infinitesimal transformations is presented with particular emphasis on the central theme of this thesis which is the invariance of ordinary differential equations, and their first integrals, under infinitesimal transformations. The differential operators associated with these infinitesimal transformations constitute an algebra under the operation of taking the Lie Bracket. Some of the major results of Lie's work are recalled. The way to use the generators of symmetries to reduce the order of a differential equation and/or to find its first integrals is explained. The chapter concludes with a summary of the state of the art in the mid-seventies just before the work described here was initiated. Chapter Two describes the growing awareness of the algebraic properties of the paradigms of differential equations. This essentially ad hoc period demonstrated that there was value in studying the Lie method of extended groups for finding first integrals and so solutions of equations and systems of equations. This value was emphasised by the application of the method to a class of nonautonomous anharmonic equations which did not belong to the then pantheon of paradigms. The generalised Emden-Fowler equation provided a route to major development in the area of the theory of the conditions for the linearisation of second order equations. This was in addition to its own interest. The stage was now set to establish broad theoretical results and retreat from the particularism of the seventies. Chapters Three and Four deal with the linearisation theorems for second order equations and the classification of intrinsically nonlinear equations according to their algebras. The rather meagre results for systems of second order equations are recorded. In the fifth chapter the investigation is extended to higher order equations for which there are some major departures away from the pattern established at the second order level and reinforced by the central role played by these equations in a world still dominated by Newton. The classification of third order equations by their algebras is presented, but it must be admitted that the story of higher order equations is still very much incomplete. In the sixth chapter the relationships between first integrals and their algebras is explored for both first order integrals and those of higher orders. Again the peculiar position of second order equations is revealed. In the seventh chapter the generalised Emden-Fowler equation is given a more modern and complete treatment. The final chapter looks at one of the fundamental algebras associated with ordinary differential equations, the three element 8£(2, R), which is found in all higher order equations of maximal symmetry, is a fundamental feature of the Pinney equation which has played so prominent a role in the study of nonautonomous Hamiltonian systems in Physics and is the signature of Ermakov systems and their generalisations.
  • Item
    Extensions and generalisations of Lie analysis.
    (1995) Govinder, Keshlan Sathasiva.; Leach, Peter Gavin Lawrence.
    The Lie theory of extended groups applied to differential equations is arguably one of the most successful methods in the solution of differential equations. In fact, the theory unifies a number of previously unrelated methods into a single algorithm. However, as with all theories, there are instances in which it provides no useful information. Thus extensions and generalisations of the method (which classically employs only point and contact transformations) are necessary to broaden the class of equations solvable by this method. The most obvious extension is to generalised (or Lie-Backlund) symmetries. While a subset of these, called contact symmetries, were considered by Lie and Backlund they have been thought to be curiosities. We show that contact transformations have an important role to play in the solution of differential equations. In particular we linearise the Kummer-Schwarz equation (which is not linearisable via a point transformation) via a contact transformation. We also determine the full contact symmetry Lie algebra of the third order equation with maximal symmetry (y'''= 0), viz sp(4). We also undertake an investigation of nonlocal symmetries which have been shown to be the origin of so-called hidden symmetries. A new procedure for the determination of these symmetries is presented and applied to some examples. The impact of nonlocal symmetries is further demonstrated in the solution of equations devoid of point symmetries. As a result we present new classes of second order equations solvable by group theoretic means. A brief foray into Painleve analysis is undertaken and then applied to some physical examples (together with a Lie analysis thereof). The close relationship between these two areas of analysis is investigated. We conclude by noting that our view of the world of symmetry has been clouded. A more broad-minded approach to the concept of symmetry is imperative to successfully realise Sophus Lie's dream of a single unified theory to solve differential equations.
  • Item
    Coherent structures and symmetry properties in nonlinear models used in theoretical physics.
    (1994) Harin, Alexander O.; Leach, Peter Gavin Lawrence.; Barashenkov, I. V.
    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.