(m)KdV Solitons on the Background of Quasi-Periodic Finite-Gap Solutions


Book Description

In the introductory section, we review the formulation of the Korteweg-de Vries (KdV) equation and of the modified KdV (mKdV) equation as a compatibility condition for a Lax pair of linear operators. We then illustrate Miura's transformation, which maps solutions of the mKdV into solutions of the KdV. We then give a general overview of the concept of soliton solutions relative to general backgrounds, and of the single and double commutation methods. Finally, we present the main results of the article. To avoid the clutter of too many technical details, the paper is organized in four sections and five appendices.




Algebro-Geometric Quasi-Periodic Finite-Gap Solutions of the Toda and Kac-van Moerbeke Hierarchies


Book Description

In this work, the authors provide a self-contained discussion of all real-valued quasi-periodic finite-gap solutions of the Toda and Kac-van Moerbeke hierarchies of completely integrable evolution equations. The approach utilizes algebro-geometric methods, factorization techniques for finite difference expressions, as well as Miura-type transformations. Detailed spectral theoretic properties of Lax pairs and theta function representations of the solutions are derived. Features: Simple and unified treatment of the topic. Self-contained development. Novel results for the Kac-van Moerbeke hierarchy and its algebro-geometric solutions.




Nonlinear Partial Differential Equations and Hyperbolic Wave Phenomena


Book Description

This volume presents the state of the art in several directions of research conducted by renowned mathematicians who participated in the research program on Nonlinear Partial Differential Equations at the Centre for Advanced Study at the Norwegian Academy of Science and Letters, Oslo, Norway, during the academic year 2008-09. The main theme of the volume is nonlinear partial differential equations that model a wide variety of wave phenomena. Topics discussed include systems of conservation laws, compressible Navier-Stokes equations, Navier-Stokes-Korteweg type systems in models for phase transitions, nonlinear evolution equations, degenerate/mixed type equations in fluid mechanics and differential geometry, nonlinear dispersive wave equations (Korteweg-de Vries, Camassa-Holm type, etc.), and Poisson interface problems and level set formulations.




Operator Theory in Harmonic and Non-commutative Analysis


Book Description

This book contains the proceedings of the 23rd International Workshop on Operator Theory and its Applications (IWOTA 2012), which was held at the University of New South Wales (Sydney, Australia) from 16 July to 20 July 2012. It includes twelve articles presenting both surveys of current research in operator theory and original results.




Spectral Analysis, Differential Equations and Mathematical Physics: A Festschrift in Honor of Fritz Gesztesy's 60th Birthday


Book Description

This volume contains twenty contributions in the area of mathematical physics where Fritz Gesztesy made profound contributions. There are three survey papers in spectral theory, differential equations, and mathematical physics, which highlight, in particu




Large Time Behavior of Solutions for General Quasilinear Hyperbolic-Parabolic Systems of Conservation Laws


Book Description

We are interested in the time-asymptotic behavior of solutions to viscous conservation laws. Through the pointwise estimates for the Green's function of the linearized system and the analysis of coupling of nonlinear diffusion waves, we obtain explicit expressions of the time-asymptotic behavior of the solutions. This yields optimal estimates in the integral norms. For most physical models, the viscosity matrix is not positive definite and the system is hyperbolic-parabolic, and not uniformly parabolic. This implies that the Green's function may contain Dirac [lowercase Greek]Delta-functions. When the corresponding inviscid system is non-strictly hyperbolic, the time-asymptotic state contains generalized Burgers solutions. These are illustrated by applying our general theory to the compressible Navier-Stokes equations and the equations of magnetohydrodynamics.




Jacobi Operators and Completely Integrable Nonlinear Lattices


Book Description

This volume serves as an introduction and reference source on spectral and inverse theory of Jacobi operators and applications of these theories to the Toda and Kac-van Moerbeke hierarchy.




The Finite Irreducible Linear 2-Groups of Degree 4


Book Description

This memoir contains a complete classification of the finite irreducible 2-subgroups of GL(4, C). Specifically, the author provides a parametrized list of representatives for the conjugacy classes of such groups, where each representative is defined by generating a set of monomial matrices. The problem is treated by a variety of techniques, including: elementary character theory; a method for describing Hasse diagrams of submodule lattices; and calculation of 2-cohomology by means of the Lyndon-Hochschild-Serre spectral sequence. Related questions concerning isomorphism between the listed groups and Schur indices of their defining characters are also considered




Nonlinear Evolution Equations And Dynamical Systems - Proceedings Of The 8th International Workshop (Needs '92)


Book Description

NEEDs '92 was held in Dubna, Russia in July 1992. This set of proceedings compiles the lectures and short contributions on the soliton theory and its applications presented during the conference. The topics covered included the most recent results on relevant problems of nonlinear evolution systems such as: Multidimensional Integrable Systems, Geometric and Algebraic Methods, Painleve Property, Lie-Backlund Symmetries, Spectral Methods, Solitons and Coherent Structures, Computational Methods, Quantum Field and String Theories, Nonlinear Optics and Hydrodynamics, Condensed Matter etc. The extent of coverage for these important topics makes this book useful, informative and insighful for the mathematics and theoretical physics community, both the senior researches and those just entering the field.




Mathematical Physics And Stochastic Analysis: Essays In Honour Of Ludwig Streit


Book Description

In October 1998 a conference was held in Lisbon to celebrate Ludwig Streit's 60th birthday. This book collects some of the papers presented at the conference as well as other essays contributed by the many friends and collaborators who wanted to honor Ludwig Streit's scientific career and personality.The contributions cover many aspects of contemporary mathematical physics. Of particular importance are new results on infinite-dimensional stochastic analysis and its applications to a wide range of physical domains.List of Contributors: S Albeverio, T Hida, L Accardi, I Ya Aref'eva, I V Volovich; A Daletskii, Y Kondratiev, W Karwowski, N Asai, I Kubo, H-H Kuo, J Beckers, Ph Blanchard, G F Dell'Antonio, D Gandolfo, M Sirugue-Collin, A Bohm, H Kaldass, D Bollé, G Jongen, G M Shim, J Bornales, C C Bernido, M V Carpio-Bernido, G Burdet, Ph Combe, H Nencka, P Cartier, C DeWitt-Morette, H Ezawa, K Nakamura, K Watanabe, Y Yamanaka, R Figari, F Gesztesy, H Holden, R Gielerak, G A Goldin, Z Haba, M-O Hongler, Y Hu, B Oksendal, A Sulem, J R Klauder, C B Lang, V I Man'ko, H Ouerdiane, J Potthoff, E Smajlovic, M Röckner, E Scacciatelli, J L Silva, J Stochel, F H Szafraniec, L Vázquez, D N Kozakevich, S Jiménez, V R Vieira, P D Sacramento, R Vilela Mendes, D Volný, P Samek.