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Author: Jeremy Upsal Publisher: ISBN: Category : Languages : en Pages : 101
Book Description
Stability analysis for solutions of partial differential equations (PDEs) is important for determining the applicability of a model to the physical world. Establishing stability for PDE solutions is often significantly more challenging than for ordinary differential equation solutions. This task becomes tractable for PDEs possessing a Lax pair. In this dissertation, I provide a general framework for computing large parts of the Lax spectrum for periodic and quasiperiodic solutions of a general class of PDEs possessing a Lax pair. This class consists of the AKNS hierarchy admitting a common reduction and generalizations. I then relate the Lax spectrum to the stability spectrum using the squared-eigenfunction connection. Using this, I demonstrate that the subset of the real line which is part of the Lax spectrum maps to stable elements of the linearization. Several examples that demonstrate the direct applicability of this work are provided. One example is worked out in detail: the stability analysis for the elliptic solutions of the focusing nonlinear Schrödinger (NLS) equation. For the NLS equation, I go further by establishing orbital stability of the elliptic solutions with respect to a class of perturbations of integer multiples of the period of the solution.
Author: Jeremy Upsal Publisher: ISBN: Category : Languages : en Pages : 101
Book Description
Stability analysis for solutions of partial differential equations (PDEs) is important for determining the applicability of a model to the physical world. Establishing stability for PDE solutions is often significantly more challenging than for ordinary differential equation solutions. This task becomes tractable for PDEs possessing a Lax pair. In this dissertation, I provide a general framework for computing large parts of the Lax spectrum for periodic and quasiperiodic solutions of a general class of PDEs possessing a Lax pair. This class consists of the AKNS hierarchy admitting a common reduction and generalizations. I then relate the Lax spectrum to the stability spectrum using the squared-eigenfunction connection. Using this, I demonstrate that the subset of the real line which is part of the Lax spectrum maps to stable elements of the linearization. Several examples that demonstrate the direct applicability of this work are provided. One example is worked out in detail: the stability analysis for the elliptic solutions of the focusing nonlinear Schrödinger (NLS) equation. For the NLS equation, I go further by establishing orbital stability of the elliptic solutions with respect to a class of perturbations of integer multiples of the period of the solution.
Author: Victor A. Galaktionov Publisher: Springer Science & Business Media ISBN: 1461220505 Category : Mathematics Languages : en Pages : 388
Book Description
* Introduces a state-of-the-art method for the study of the asymptotic behavior of solutions to evolution partial differential equations. * Written by established mathematicians at the forefront of their field, this blend of delicate analysis and broad application is ideal for a course or seminar in asymptotic analysis and nonlinear PDEs. * Well-organized text with detailed index and bibliography, suitable as a course text or reference volume.
Author: Philippe Guyenne Publisher: Springer ISBN: 149392950X Category : Mathematics Languages : en Pages : 453
Book Description
This book is a unique selection of work by world-class experts exploring the latest developments in Hamiltonian partial differential equations and their applications. Topics covered within are representative of the field’s wide scope, including KAM and normal form theories, perturbation and variational methods, integrable systems, stability of nonlinear solutions as well as applications to cosmology, fluid mechanics and water waves. The volume contains both surveys and original research papers and gives a concise overview of the above topics, with results ranging from mathematical modeling to rigorous analysis and numerical simulation. It will be of particular interest to graduate students as well as researchers in mathematics and physics, who wish to learn more about the powerful and elegant analytical techniques for Hamiltonian partial differential equations.
Author: Victor A. Galaktionov Publisher: CRC Press ISBN: 1420011626 Category : Mathematics Languages : en Pages : 530
Book Description
Exact Solutions and Invariant Subspaces of Nonlinear Partial Differential Equations in Mechanics and Physics is the first book to provide a systematic construction of exact solutions via linear invariant subspaces for nonlinear differential operators. Acting as a guide to nonlinear evolution equations and models from physics and mechanics, the book
Author: Saïd Abbas Publisher: Walter de Gruyter GmbH & Co KG ISBN: 311055318X Category : Mathematics Languages : en Pages : 359
Book Description
This book deals with the existence and stability of solutions to initial and boundary value problems for functional differential and integral equations and inclusions involving the Riemann-Liouville, Caputo, and Hadamard fractional derivatives and integrals. A wide variety of topics is covered in a mathematically rigorous manner making this work a valuable source of information for graduate students and researchers working with problems in fractional calculus. Contents Preliminary Background Nonlinear Implicit Fractional Differential Equations Impulsive Nonlinear Implicit Fractional Differential Equations Boundary Value Problems for Nonlinear Implicit Fractional Differential Equations Boundary Value Problems for Impulsive NIFDE Integrable Solutions for Implicit Fractional Differential Equations Partial Hadamard Fractional Integral Equations and Inclusions Stability Results for Partial Hadamard Fractional Integral Equations and Inclusions Hadamard–Stieltjes Fractional Integral Equations Ulam Stabilities for Random Hadamard Fractional Integral Equations
Author: Joseph Wiener Publisher: World Scientific ISBN: 9814505110 Category : Mathematics Languages : en Pages : 425
Book Description
The need to investigate functional differential equations with discontinuous delays is addressed in this book. Recording the work and findings of several scientists on differential equations with piecewise continuous arguments over the last few years, this book serves as a useful source of reference. Great interest is placed on discussing the stability, oscillation and periodic properties of the solutions. Considerable attention is also given to the study of initial and boundary-value problems for partial differential equations of mathematical physics with discontinuous time delays. In fact, a large part of the book is devoted to the exploration of differential and functional differential equations in spaces of generalized functions (distributions) and contains a wealth of new information in this area. Each topic discussed appears to provide ample opportunity for extending the known results. A list of new research topics and open problems is also included as an update.
Author: Leszek Gawarecki Publisher: Springer Science & Business Media ISBN: 3642161944 Category : Mathematics Languages : en Pages : 300
Book Description
The systematic study of existence, uniqueness, and properties of solutions to stochastic differential equations in infinite dimensions arising from practical problems characterizes this volume that is intended for graduate students and for pure and applied mathematicians, physicists, engineers, professionals working with mathematical models of finance. Major methods include compactness, coercivity, monotonicity, in a variety of set-ups. The authors emphasize the fundamental work of Gikhman and Skorokhod on the existence and uniqueness of solutions to stochastic differential equations and present its extension to infinite dimension. They also generalize the work of Khasminskii on stability and stationary distributions of solutions. New results, applications, and examples of stochastic partial differential equations are included. This clear and detailed presentation gives the basics of the infinite dimensional version of the classic books of Gikhman and Skorokhod and of Khasminskii in one concise volume that covers the main topics in infinite dimensional stochastic PDE’s. By appropriate selection of material, the volume can be adapted for a 1- or 2-semester course, and can prepare the reader for research in this rapidly expanding area.
Author: Ioan I Vrabie Publisher: World Scientific Publishing Company ISBN: 981474980X Category : Mathematics Languages : en Pages : 529
Book Description
This book presents, in a unitary frame and from a new perspective, the main concepts and results of one of the most fascinating branches of modern mathematics, namely differential equations, and offers the reader another point of view concerning a possible way to approach the problems of existence, uniqueness, approximation, and continuation of the solutions to a Cauchy problem. In addition, it contains simple introductions to some topics which are not usually included in classical textbooks: the exponential formula, conservation laws, generalized solutions, Caratheodory solutions, differential inclusions, variational inequalities, viability, invariance, and gradient systems.In this new edition, some typos have been corrected and two new topics have been added: Delay differential equations and differential equations subjected to nonlocal initial conditions. The bibliography has also been updated and expanded.
Author: Boling Guo Publisher: World Scientific ISBN: 9814544264 Category : Languages : en Pages : 267
Book Description
Contents: Direct and Inverse Diffraction by Periodic Structures (G Bao)Weak Flow of H-Systems (Y-M Chen)Strongly Compact Attractor for Dissipative Zakharov Equations (B-L Guo et al.)C∞-Solutions of Generalized Porous Medium Equations (M Ôtani & Y Sugiyama)Cauchy Problem for Generalized IMBq Equation (G-W Chen & S-B Wang)Inertial Manifolds for a Nonlocal Kuramoto–Sivashinsky Equation (J-Q Duan et al.)Weak Solutions of the Generalized Magnetic Flow Equations (S-H He & Z-D Dai)The Solution of Hammerstein Integral Equation Without Coercive Conditions (Y-L Shu)Global Behaviour of the Solution of Nonlinear Forest Evolution Equation (D-J Wang)Uniqueness of Generalized Solutions for Semiconductor Equations (J-S Xing & Y Hu)On the Vectorial Hamilton–Jacobi System (B-S Yan)An Integrable Hamiltonian System Associated with cKdV Hierarchy (J-S Zhang et al.)and other papers Readership: Mathematicians. Keywords:Diffraction;Weak Flow;Zakharov Equations;Porous Medium Equations;Cauchy Problem;IMBq Equation;Kuramoto-Sivashinsky Equation;Magnetic Flow Equations;Hammerstein Integral Equation;Nonlinear Forest Evolution Equation;Uniqueness;Generalized Solutions;Semiconductor Equations;HamiltonâJacobi System;Hamiltonian System;cKdV Hierarchy
Author: Ivanka Stamova Publisher: Walter de Gruyter ISBN: 3110221829 Category : Mathematics Languages : en Pages : 241
Book Description
This book is devoted to impulsive functional differential equations which are a natural generalization of impulsive ordinary differential equations (without delay) and of functional differential equations (without impulses). At the present time the qualitative theory of such equations is under rapid development. After a presentation of the fundamental theory of existence, uniqueness and continuability of solutions, a systematic development of stability theory for that class of problems is given which makes the book unique. It addresses to a wide audience such as mathematicians, applied researches and practitioners.
Author: Jeremy Upsal
Author: Jeremy Upsal
Author: Victor A. Galaktionov
Author: Philippe Guyenne
Author: Victor A. Galaktionov
Author: Saïd Abbas
Author: Joseph Wiener
Author: Leszek Gawarecki
Author: Ioan I Vrabie
Author: Boling Guo
Author: Ivanka Stamova