M. A. Ablowitz

Bio: M. A. Ablowitz is an academic researcher from University of Colorado Boulder. The author has contributed to research in topics: Inverse scattering problem & Quantum inverse scattering method. The author has an hindex of 2, co-authored 3 publications receiving 3980 citations.

Solitons, Nonlinear Evolution Equations and Inverse Scattering

Abstract: Solitons have been of considerable interest to mathematicians since their discovery by Kruskal and Zabusky. This book brings together several aspects of soliton theory currently only available in research papers. Emphasis is given to the multi-dimensional problems arising and includes inverse scattering in multi-dimensions, integrable nonlinear evolution equations in multi-dimensions and the ∂ method. Thus, this book will be a valuable addition to the growing literature in the area and essential reading for all researchers in the field of soliton theory.

Noncommutative field theory

Abstract: This article reviews the generalization of field theory to space-time with noncommuting coordinates, starting with the basics and covering most of the active directions of research. Such theories are now known to emerge from limits of M theory and string theory and to describe quantum Hall states. In the last few years they have been studied intensively, and many qualitatively new phenomena have been discovered, on both the classical and the quantum level.

Discretizing light behaviour in linear and nonlinear waveguide lattices.

Abstract: Light propagating in linear and nonlinear waveguide lattices exhibits behaviour characteristic of that encountered in discrete systems. The diffraction properties of these systems can be engineered, which opens up new possibilities for controlling the flow of light that would have been otherwise impossible in the bulk: these effects can be exploited to achieve diffraction-free propagation and minimize the power requirements for nonlinear processes. In two-dimensional networks of waveguides, self-localized states--or discrete solitons--can travel along 'wire-like' paths and can be routed to any destination port. Such possibilities may be useful for photonic switching architectures.

Handbook of Nonlinear Partial Differential Equations

Abstract: SOME NOTATIONS AND REMARKS PARABOLIC EQUATIONS WITH ONE SPACE VARIABLE Equations with Power-Law Nonlinearities Equations with Exponential Nonlinearities Equations with Hyperbolic Nonlinearities Equations with Logarithmic Nonlinearities Equations with Trigonometric Nonlinearities Equations Involving Arbitrary Functions Nonlinear Schrodinger Equations and Related Equations PARABOLIC EQUATIONS WITH TWO OR MORE SPACE VARIABLES Equations with Two Space Variables Involving Power-Law Nonlinearities Equations with Two Space Variables Involving Exponential Nonlinearities Other Equations with Two Space Variables Involving Arbitrary Parameters Equations Involving Arbitrary Functions Equations with Three or More Space Variables Nonlinear Schrodinger Equations HYPERBOLIC EQUATIONS WITH ONE SPACE VARIABLE Equations with Power-Law Nonlinearities Equations with Exponential Nonlinearities Other Equations Involving Arbitrary Parameters Equations Involving Arbitrary Functions Equations of the Form wxy=F(x,y,w, wx, wy ) HYPERBOLIC EQUATIONS WITH TWO OR THREE SPACE VARIABLES Equations with Two Space Variables Involving Power-Law Nonlinearities Equations with Two Space Variables Involving Exponential Nonlinearities Nonlinear Telegraph Equations with Two Space Variables Equations with Two Space Variables Involving Arbitrary Functions Equations with Three Space Variables Involving Arbitrary Parameters Equations with Three Space Variables Involving Arbitrary Functions ELLIPTIC EQUATIONS WITH TWO SPACE VARIABLES Equations with Power-Law Nonlinearities Equations with Exponential Nonlinearities Equations Involving Other Nonlinearities Equations Involving Arbitrary Functions ELLIPTIC EQUATIONS WITH THREE OR MORE SPACE VARIABLES Equations with Three Space Variables Involving Power-Law Nonlinearities Equations with Three Space Variables Involving Exponential Nonlinearities Three-Dimensional Equations Involving Arbitrary Functions Equations with n Independent Variables EQUATIONS INVOLVING MIXED DERIVATIVES AND SOME OTHER EQUATIONS Equations Linear in the Mixed Derivative Equations Quadratic in the Highest Derivatives Bellman Type Equations and Related Equations SECOND-ORDER EQUATIONS OF GENERAL FORM Equations Involving the First Derivative in t Equations Involving Two or More Second Derivatives THIRD-ORDER EQUATIONS Equations Involving the First Derivative in t Equations Involving the Second Derivative in t Hydrodynamic Boundary Layer Equations Equations of Motion of Ideal Fluid (Euler Equations) Other Third-Order Nonlinear Equations FOURTH-ORDER EQUATIONS Equations Involving the First Derivative in t Equations Involving the Second Derivative in t Equations Involving Mixed Derivatives EQUATIONS OF HIGHER ORDERS Equations Involving the First Derivative in t and Linear in the Highest Derivative General Form Equations Involving the First Derivative in t Equations Involving the Second Derivative in t Other Equations SUPPLEMENTS: EXACT METHODS FOR SOLVING NONLINEAR PARTIAL DIFFERENTIAL EQUATIONS Classification of Second-Order Semilinear Partial Differential Equations in Two Independent Variables Transformations of Equations of Mathematical Physics Traveling-Wave Solutions and Self-Similar Solutions. Similarity Methods Method of Generalized Separation of Variables Method of Functional Separation of Variables Generalized Similarity Reductions of Nonlinear Equations Group Analysis Methods Differential Constraints Method Painleve Test for Nonlinear Equations of Mathematical Physics Inverse Scattering Method Conservation Laws Hyperbolic Systems of Quasilinear Equations REFERENCES INDEX

Integrable nonlocal nonlinear Schrödinger equation.

Abstract: A new integrable nonlocal nonlinear Schrodinger equation is introduced. It possesses a Lax pair and an infinite number of conservation laws and is PT symmetric. The inverse scattering transform and scattering data with suitable symmetries are discussed. A method to find pure soliton solutions is given. An explicit breathing one soliton solution is found. Key properties are discussed and contrasted with the classical nonlinear Schrodinger equation.