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Barbara Prinari

Bio: Barbara Prinari is an academic researcher from University at Buffalo. The author has contributed to research in topics: Inverse scattering transform & Soliton. The author has an hindex of 21, co-authored 79 publications receiving 1863 citations. Previous affiliations of Barbara Prinari include University of Colorado Boulder & University of Salento.


Papers
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Book
01 Jan 2004
TL;DR: In this article, the authors present a detailed mathematical analysis of the scattering theory; soliton solutions are obtained and soliton interactions, both scalar and vector, are analyzed, and the analysis is extended to the Toda lattice and nonlinear ladder network.
Abstract: In recent years there have been important and far reaching developments in the study of nonlinear waves and a class of nonlinear wave equations which arise frequently in applications. The wide interest in this field comes from the understanding of special waves called 'solitons' and the associated development of a method of solution to a class of nonlinear wave equations termed the inverse scattering transform (IST). Before these developments, very little was known about the solutions to such 'soliton equations'. The IST technique applies to both continuous and discrete nonlinear Schrodinger equations of scalar and vector type. Also included is the IST for the Toda lattice and nonlinear ladder network, which are well-known discrete systems. This book, first published in 2003, presents the detailed mathematical analysis of the scattering theory; soliton solutions are obtained and soliton interactions, both scalar and vector, are analyzed. Much of the material is not available in the previously-published literature.

462 citations

Journal ArticleDOI
TL;DR: In this paper, the inverse scattering transform for the vector defocusing nonlinear Schrodinger (NLS) equation with nonvanishing boundary values at infinity is constructed, and the discrete spectrum, bound states and symmetries of the direct problem are discussed.
Abstract: The inverse scattering transform for the vector defocusing nonlinear Schrodinger (NLS) equation with nonvanishing boundary values at infinity is constructed. The direct scattering problem is formulated on a two-sheeted covering of the complex plane. Two out of the six Jost eigenfunctions, however, do not admit an analytic extension on either sheet of the Riemann surface. Therefore, a suitable modification of both the direct and the inverse problem formulations is necessary. On the direct side, this is accomplished by constructing two additional analytic eigenfunctions which are expressed in terms of the adjoint eigenfunctions. The discrete spectrum, bound states and symmetries of the direct problem are then discussed. In the most general situation, a discrete eigenvalue corresponds to a quartet of zeros (poles) of certain scattering data. The inverse scattering problem is formulated in terms of a generalized Riemann-Hilbert (RH) problem in the upper/lower half planes of a suitable uniformization variable. Special soliton solutions are constructed from the poles in the RH problem, and include dark-dark soliton solutions, which have dark solitonic behavior in both components, as well as dark-bright soliton solutions, which have one dark and one bright component. The linear limit is obtained from the RH problem and is shown to correspond to the Fourier transform solution obtained from the linearized vector NLS system.

151 citations

Journal ArticleDOI
TL;DR: In this article, a rigorous theory of the inverse scattering transform for the defocusing nonlinear Schrodinger equation with nonvanishing boundary values is presented, where the direct problem is well posed for potentials q such that, for which analyticity properties of eigenfunctions and scattering data are established.
Abstract: A rigorous theory of the inverse scattering transform for the defocusing nonlinear Schrodinger equation with nonvanishing boundary values as is presented. The direct problem is shown to be well posed for potentials q such that , for which analyticity properties of eigenfunctions and scattering data are established. The inverse scattering problem is formulated and solved both via Marchenko integral equations, and as a Riemann-Hilbert problem in terms of a suitable uniform variable. The asymptotic behavior of the scattering data is determined and shown to ensure the linear system solving the inverse problem is well defined. Finally, the triplet method is developed as a tool to obtain explicit multisoliton solutions by solving the Marchenko integral equation via separation of variables.

111 citations

Journal ArticleDOI
TL;DR: In this article, the inverse scattering transform for an integrable discretization of the defocusing nonlinear Schrodinger equation with nonvanishing boundary values at infinity is constructed, and a suitable transformation of the scattering problem is introduced to address the open issue of analyticity of eigenfunctions and scattering data.
Abstract: The inverse scattering transform for an integrable discretization of the defocusing nonlinear Schrodinger equation with nonvanishing boundary values at infinity is constructed. This problem had been previously studied, and many key results had been established. Here, a suitable transformation of the scattering problem is introduced in order to address the open issue of analyticity of eigenfunctions and scattering data. Moreover, the inverse problem is formulated as a Riemann–Hilbert problem on the unit circle, and a modification of the standard procedure is required in order to deal with the dependence of asymptotics of the eigenfunctions on the potentials. The discrete analog of Gel’fand–Levitan–Marchenko equations is also derived. Finally, soliton solutions and solutions in the small-amplitude limit are obtained and the continuum limit is discussed. (Some figures in this article are in colour only in the electronic version)

100 citations

Journal ArticleDOI
TL;DR: In this article, it was shown that the multisoliton interaction process is pairwise and the net result of the interaction is independent of the order in which such collisions occur, which is related to the fact that the map determining the interaction of two solitons with nontrivial internal degrees of freedom satisfies the Yang-Baxter relation.
Abstract: Collisions of solitons for two coupled and N-coupled NLS equation are investigated from various viewpoints. By suitably employing Manakov's well-known formulae for the polarization shift of interacting vector solitons, it is shown that the multisoliton interaction process is pairwise and the net result of the interaction is independent of the order in which such collisions occur. Further, this is shown to be related to the fact that the map determining the interaction of two solitons with nontrivial internal degrees of freedom (e.g. vector solitons) satisfies the Yang–Baxter relation. The associated matrix factorization problem is discussed in detail. Soliton interactions are also described in terms of linear fractional transformations, and the problem of existence of a solution for a basic three-collision gate, which has recently been introduced, is analysed.

86 citations


Cited by
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Proceedings Article
14 Jul 1996
TL;DR: The striking signature of Bose condensation was the sudden appearance of a bimodal velocity distribution below the critical temperature of ~2µK.
Abstract: Bose-Einstein condensation (BEC) has been observed in a dilute gas of sodium atoms. A Bose-Einstein condensate consists of a macroscopic population of the ground state of the system, and is a coherent state of matter. In an ideal gas, this phase transition is purely quantum-statistical. The study of BEC in weakly interacting systems which can be controlled and observed with precision holds the promise of revealing new macroscopic quantum phenomena that can be understood from first principles.

3,530 citations

Journal ArticleDOI
TL;DR: To the best of our knowledge, there is only one application of mathematical modelling to face recognition as mentioned in this paper, and it is a face recognition problem that scarcely clamoured for attention before the computer age but, having surfaced, has attracted the attention of some fine minds.
Abstract: to be done in this area. Face recognition is a problem that scarcely clamoured for attention before the computer age but, having surfaced, has involved a wide range of techniques and has attracted the attention of some fine minds (David Mumford was a Fields Medallist in 1974). This singular application of mathematical modelling to a messy applied problem of obvious utility and importance but with no unique solution is a pretty one to share with students: perhaps, returning to the source of our opening quotation, we may invert Duncan's earlier observation, 'There is an art to find the mind's construction in the face!'.

3,015 citations

DatasetDOI
TL;DR: When a gas of bosonic particles is cooled below a critical temperature, it condenses into a Bose-Einstein condensate as mentioned in this paper, which is the state of the art.
Abstract: When a gas of bosonic particles is cooled below a critical temperature, it condenses into a Bose-Ei…

591 citations

Journal ArticleDOI
TL;DR: In this paper, a detailed study of the inverse scattering transform of the non-local nonlinear Schrodinger (NLS) equation is carried out and key symmetries of the eigenfunctions and scattering data and conserved quantities are obtained.
Abstract: A nonlocal nonlinear Schrodinger (NLS) equation was recently introduced and shown to be an integrable infinite dimensional Hamiltonian evolution equation. In this paper a detailed study of the inverse scattering transform of this nonlocal NLS equation is carried out. The direct and inverse scattering problems are analyzed. Key symmetries of the eigenfunctions and scattering data and conserved quantities are obtained. The inverse scattering theory is developed by using a novel left–right Riemann–Hilbert problem. The Cauchy problem for the nonlocal NLS equation is formulated and methods to find pure soliton solutions are presented; this leads to explicit time-periodic one and two soliton solutions. A detailed comparison with the classical NLS equation is given and brief remarks about nonlocal versions of the modified Korteweg–de Vries and sine-Gordon equations are made.

463 citations

MonographDOI
01 Sep 2011
TL;DR: The field of nonlinear dispersive waves has developed enormously since the work of Stokes, Boussinesq and Korteweg-de Vries (KdV) in the nineteenth century.
Abstract: The field of nonlinear dispersive waves has developed enormously since the work of Stokes, Boussinesq and Korteweg–de Vries (KdV) in the nineteenth century. In the 1960s, researchers developed effective asymptotic methods for deriving nonlinear wave equations, such as the KdV equation, governing a broad class of physical phenomena that admit special solutions including those commonly known as solitons. This book describes the underlying approximation techniques and methods for finding solutions to these and other equations. The concepts and methods covered include wave dispersion, asymptotic analysis, perturbation theory, the method of multiple scales, deep and shallow water waves, nonlinear optics including fiber optic communications, mode-locked lasers and dispersion-managed wave phenomena. Most chapters feature exercise sets, making the book suitable for advanced courses or for self-directed learning. Graduate students and researchers will find this an excellent entry to a thriving area at the intersection of applied mathematics, engineering and physical science.

411 citations