Nonlinear Evolution Models of Integrable Type

Cornelis VAN DER MEE


Over the past 45 years we have seen a growing interest in integrable linear systems and their applications. These equations include the Kortewegde Vries (KdV), nonlinear Schrödinger (NLS), sine-Gordon (SG), modified Korteweg-de Vries (mKdV), Toda lattice, integrable discrete nonlinear Schrödinger (IDNLS), Camassa-Holm (CH), and Degasperis-Procesi (DP) equations. They have important applications to surface wave dynamics, fibre optics, Josephson junction transmission lines, Alfven waves in collisionless plasmas, charge density waves, surfaces of constant Gaussian curvature, traffic congestion, nonlinearly coupled oscillators, and breaking wave dynamics. The mathematics used involves techniques from fields as diverse as functional analysis, Lie groups, differential geometry, numerical linear algebra, and linear control theory. The mathematical problems studied range from unique solvability issues in Sobolev spaces to analytical and numerical solution algorithms to the derivation of conservation laws from hamiltonian principles. The net result of this disparity in applications and mathematical techniques has been the creation of an immense research area where mathematicians, physicists, and engineers, in other words scientists of various pedigrees, can fruitfully work together towards a variety of common goals.
This book is based on a 20 hour minicourse given to graduate students at the University of Cagliari in the early Summer of 2012. The philosophy of this course was to discuss techniques to solve various integrable linear systems by means of the inverse scattering transform (IST) method, where the solution of the integrable nonlinear system is associated with the “potential" in a linear eigenvalue problem. Using the direct and inverse scattering theory of the linear eigenvalue problem the time evolution according to the integrable nonlinear system is converted into the time evolution of the scattering data. The crux of the IST is that the time evolution of the scattering data is so elementary that the IST method in principle yields an explicit method of solving the integrable linear system. In certain cases the exact solvability of the direct and inverse scattering problems allows one to derive extensive families of exact solutions. Discretization of the direct and inverse scattering problems leads to a numerical method to solve the integrable nonlinear system. It is the inverse scattering transform method that we wish to highlight in this monograph. Although the analytical and numerical aspects of the IST are an important part of the research conducted by the Numerical Analysis and Mathematical Modelling Group of the Department of Mathematics and Computer Science of the University of Cagliari, in this monograph we focus on its analytical aspects.
The monograph itself contains many well-known results, written up in a different context, as well as new results. The philosophy has been to maximize the use of linear algebra in order to arrive at more concise equations and proofs and to facilitate the development of numerical methods. On the other hand, in spite of my background in functional analysis, the philosophy has also been to minimize the use of functional analysis and to arrive at a treatment that is mathematically as elementary as possible.

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