Book•
Theory of Solitons: The Inverse Scattering Method
01 Jan 1984-
About: The article was published on 1984-01-01 and is currently open access. It has received 1883 citations till now. The article focuses on the topics: Inverse scattering problem & Inverse scattering transform.
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TL;DR: A detailed overview of the physics and applications of optical dark solitons can be found in this article, where the authors discuss the instability-induced dynamics of dark-solitons in the models of generalized (i.e., non-Kerr) optical nonlinearities.
1,076 citations
Cites methods from "Theory of Solitons: The Inverse Sca..."
...Therefore, nonlinearity in fibers is always weak and it is well modeled by the cubic NLS equation, which is known to be integrable by means of the inverse scattering transform [Zakharov and Shabat (1971, 1973); see also Zakharov et al. (1980)]....
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TL;DR: In this article, a unified approach to the long wavelength Bethe equations, the classical ferromagnet and the classical string solutions in the SU(2) sector is presented, governed by complex curves endowed with meromorphic differentials with integer periods.
Abstract: We discuss the AdS/CFT duality from the perspective of integrable systems and establish a direct relationship between the dimension of single trace local operators composed of two types of scalar fields in = 4 super Yang-Mills and the energy of their dual semiclassical string states in AdS5 × S5. The anomalous dimensions can be computed using a set of Bethe equations, which for ``long'' operators reduces to a Riemann-Hilbert problem. We develop a unified approach to the long wavelength Bethe equations, the classical ferromagnet and the classical string solutions in the SU(2) sector and present a general solution, governed by complex curves endowed with meromorphic differentials with integer periods. Using this solution we compute the anomalous dimensions of these long operators up to two loops and demonstrate that they agree with string-theory predictions.
707 citations
TL;DR: A new integrable nonlocal nonlinear Schrödinger equation is introduced that possesses a Lax pair and an infinite number of conservation laws and is PT symmetric.
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.
682 citations
Book•
01 Jan 2001
TL;DR: Inverse spectral problems for Sturm-Liouville differential operators are studied in this paper, where the authors present the main results and methods on inverse spectral problems and their applications.
Abstract: This book presents the main results and methods on inverse spectral problems for Sturm-Liouville differential operators and their applications. Inverse problems of spectral analysis consist in recovering operators from their spectral characteristics. Such problems often appear in mathematics, mechanics, physics, electronics, geophysics, meteorology and other branches of natural sciences. Inverse problems also play an important role in solving non-linear evolution equations in mathematical physics. Interest in this subject has been increasing permanently because of the appearance of new important applications, resulting in intensive study of inverse problem theory all over the world.
616 citations
TL;DR: The Bogolyubov-Whitham averaging method for field-theoretic systems and soliton lattices was introduced in this paper, and the results of Whitham and Hayes for Lagrangian systems were shown to preserve the Hamiltonian structure under averaging.
Abstract: CONTENTS Introduction Chapter I. Hamiltonian theory of systems of hydrodynamic type § 1. General properties of Poisson brackets § 2. Hamiltonian formalism of systems of hydrodynamic type and Riemannian geometry § 3. Generalizations: differential-geometric Poisson brackets of higher orders, differential-geometric Poisson brackets on a lattice, and the Yang-Baxter equation § 4. Riemann invariants and the Hamiltonian formalism of diagonal systems of hydrodynamic type. Novikov's conjecture. Tsarev's theorem. The generalized hodograph method Chapter II. Equations of hydrodynamics of soliton lattices § 5. The Bogolyubov-Whitham averaging method for field-theoretic systems and soliton lattices. The results of Whitham and Hayes for Lagrangian systems § 6. The Whitham equations of hydrodynamics of weakly deformed soliton lattices for Hamiltonian field-theoretic systems. The principle of conservation of the Hamiltonian structure under averaging § 7. Modulations of soliton lattices of completely integrable evolutionary systems. Krichever's method. The analytic solution of the Gurevich-Pitaevskii problem on the dispersive analogue of a shock wave § 8. Evolution of the oscillatory zone in the KdV theory. Multi-valued functions in the hydrodynamics of soliton lattices. Numerical studies § 9. Influence of small viscosity on the evolution of the oscillatory zone References
521 citations