Applications Of Analytic And Geometric Methods To Nonlinear Differential Equations

In the study of integrable systems, two different approaches in particular have attracted considerable attention during the past twenty years. (1) The inverse scattering transform (IST), using complex function theory, which has been employed to solve many physically significant equations, the `soliton' equations. (2) Twistor theory, using differential geometry, which has been used to solve the self-dual Yang--Mills (SDYM) equations, a four-dimensional system having important applications in mathematical physics. Both soliton and the SDYM equations have rich algebraic structures which have been extensively studied. Recently, it has been conjectured that, in some sense, all soliton equations arise as special cases of the SDYM equations; subsequently many have been discovered as either exact or asymptotic reductions of the SDYM equations. Consequently what seems to be emerging is that a natural, physically significant system such as the SDYM equations provides the basis for a unifying framework underlying this class of integrable systems, i.e. `soliton' systems. This book contains several articles on the reduction of the SDYM equations to soliton equations and the relationship between the IST and twistor methods. The majority of nonlinear evolution equations are nonintegrable, and so asymptotic, numerical perturbation and reduction techniques are often used to study such equations. This book also contains articles on perturbed soliton equations. Painlevé analysis of partial differential equations, studies of the Painlevé equations and symmetry reductions of nonlinear partial differential equations. (ABSTRACT) In the study of integrable systems, two different approaches in particular have attracted considerable attention during the past twenty years; the inverse scattering transform (IST), for `soliton' equations and twistor theory, for the self-dual Yang--Mills (SDYM) equations. This book contains several articles on the reduction of the SDYM equations to soliton equations and the relationship between the IST and twistor methods. Additionally, it contains articles on perturbed soliton equations, Painlevé analysis of partial differential equations, studies of the Painlevé equations and symmetry reductions of nonlinear partial differential equations.

This book contains a collection of twelve papers that reflect the state of the art of nonlinear differential equations in modern geometrical theory. It comprises miscellaneous topics of the local and nonlocal geometry of differential equations and the applications of the corresponding methods in hydrodynamics, symplectic geometry, optimal investment theory, etc. The contents will be useful for all the readers whose professional interests are related to nonlinear PDEs and differential geometry, both in theoretical and applied aspects.

In this book, fundamental methods of nonlinear analysis are introduced, discussed and illustrated in straightforward examples. Each method considered is motivated and explained in its general form, but presented in an abstract framework as comprehensively as possible. A large number of methods are applied to boundary value problems for both ordinary and partial differential equations. In this edition we have made minor revisions, added new material and organized the content slightly differently. In particular, we included evolutionary equations and differential equations on manifolds. The applications to partial differential equations follow every abstract framework of the method in question. The text is structured in two levels: a self-contained basic level and an advanced level - organized in appendices - for the more experienced reader. The last chapter contains more involved material and can be skipped by those new to the field. This book serves as both a textbook for graduate-level courses and a reference book for mathematicians, engineers and applied scientists

The analysis of PDEs is a prominent discipline in mathematics research, both in terms of its theoretical aspects and its relevance in applications. In recent years, the geometric properties of linear and nonlinear second order PDEs of elliptic and parabolic type have been extensively studied by many outstanding researchers. This book collects contributions from a selected group of leading experts who took part in the INdAM meeting "Geometric methods in PDEs", on the occasion of the 70th birthday of Ermanno Lanconelli. They describe a number of new achievements and/or the state of the art in their discipline of research, providing readers an overview of recent progress and future research trends in PDEs. In particular, the volume collects significant results for sub-elliptic equations, potential theory and diffusion equations, with an emphasis on comparing different methodologies and on their implications for theory and applications.

Besides their intrinsic mathematical interest, geometric partial differential equations (PDEs) are ubiquitous in many scientific, engineering and industrial applications. They represent an intellectual challenge and have received a great deal of attention recently. The purpose of this volume is to provide a missing reference consisting of self-contained and comprehensive presentations. It includes basic ideas, analysis and applications of state-of-the-art fundamental algorithms for the approximation of geometric PDEs together with their impacts in a variety of fields within mathematics, science, and engineering. About every aspect of computational geometric PDEs is discussed in this and a companion volume. Topics in this volume include stationary and time-dependent surface PDEs for geometric flows, large deformations of nonlinearly geometric plates and rods, level set and phase field methods and applications, free boundary problems, discrete Riemannian calculus and morphing, fully nonlinear PDEs including Monge-Ampere equations, and PDE constrained optimization Each chapter is a complete essay at the research level but accessible to junior researchers and students. The intent is to provide a comprehensive description of algorithms and their analysis for a specific geometric PDE class, starting from basic concepts and concluding with interesting applications. Each chapter is thus useful as an introduction to a research area as well as a teaching resource, and provides numerous pointers to the literature for further reading The authors of each chapter are world leaders in their field of expertise and skillful writers. This book is thus meant to provide an invaluable, readable and enjoyable account of computational geometric PDEs

This book gathers nineteen papers presented at the first NLAGA-BIRS Symposium, which was held at the Cheikh Anta Diop University in Dakar, Senegal, on June 24–28, 2019. The four-day symposium brought together African experts on nonlinear analysis and geometry and their applications, as well as their international partners, to present and discuss mathematical results in various areas. The main goal of the NLAGA project is to advance and consolidate the development of these mathematical fields in West and Central Africa with a focus on solving real-world problems such as coastal erosion, pollution, and urban network and population dynamics problems. The book addresses a range of topics related to partial differential equations, geometrical analysis of optimal shapes, geometric structures, optimization and optimal transportation, control theory, and mathematical modeling.

This well-organized and coherent collection of papers leads the reader to the frontiers of present research in the theory of nonlinear partial differential equations and the calculus of variations and offers insight into some exciting developments. In addition, most articles also provide an excellent introduction to their background, describing extensively as they do the history of those problems presented, as well as the state of the art and offer a well-chosen guide to the literature. Part I contains the contributions of geometric nature: From spectral theory on regular and singular spaces to regularity theory of solutions of variational problems. Part II consists of articles on partial differential equations which originate from problems in physics, biology and stochastics. They cover elliptic, hyperbolic and parabolic cases.

The method of normal forms is usually attributed to Poincaré although some of the basic ideas of the method can be found in earlier works of Jacobi, Briot and Bouquet. In this book, A.D.Bruno gives an account of the work of these mathematicians and further developments as well as the results of his own extensive investigations on the subject. The book begins with a thorough presentation of the analytical techniques necessary for the implementation of the theory as well as an extensive description of the geometry of the Newton polygon. It then proceeds to discuss the normal form of systems of ordinary differential equations giving many specific applications of the theory. An underlying theme of the book is the unifying nature of the method of normal forms regarding techniques for the study of the local properties of ordinary differential equations. In the second part of the book it is shown, for a special class of equations, how the method of normal forms yields classical results of Lyapunov concerning families of periodic orbits in the neighborhood of equilibrium points of Hamiltonian systems as well as the more modern results concerning families of quasiperiodic orbits obtained by Kolmogorov, Arnold and Moser. The book is intended for mathematicians, theoretical mechanicians, and physicists. It is suitable for advanced undergraduate and graduate students.