Time-evolution-proof Scattering Data for the Focusing and Defocusing Zakharov-Shabat Systems
نویسنده
چکیده
Nonlinear Schrödinger (NLS) equations have attracted the attention of the physical and mathematical community for over four decades. NLS equations arise in such diverse fields as deep water waves [4, 25], plasma physics [24], fiber optics [11], and Bose-Einstein condensation [17]. The basic method for solving the NLS initial-value problem is the inverse scattering transform (IST) method [2–4, 6, 10, 15, 21, 25], where the NLS time evolution is transcribed into the time evolution of the scattering data of the so-called Zakharov-Shabat system. In this article we characterize the scattering data for the Zakharov-Shabat (ZS) system
منابع مشابه
Scattering Operators for Matrix Zakharov-Shabat Systems
In this article the scattering matrix pertaining to the defocusing matrix Zakharov-Shabat system on the line is related to the scattering operator arising from time-dependent scattering theory. Further, the scattering data allowing for a unique retrieval of the potential in the defocusing matrix Zakharov-Shabat system are characterized. Mathematics Subject Classification (2000). Primary 34A55, ...
متن کاملWave Operators for Defocusing Matrix Zakharov-shabat Systems with Potentials Nonvanishing at Infinity
In this article we prove that the wave operators describing the direct scattering of the defocusing matrix Zakharov-Shabat system with potentials having distinct nonzero values with the same modulus at ±∞ exist, are asymptotically complete, and lead to a unitary scattering operator. We also prove that the free Hamiltonian operator is absolutely continuous.
متن کاملOn the Location of the Discrete Eigenvalues for Defocusing Zakharov-Shabat Systems having Potentials with Nonvanishing Boundary Conditions
In this article we prove that the discrete eigenvalues of the Zakharov-Shabat system belong to certain neighborhoods of the endpoints of the spectral gap and the discrete eigenvalue of the free Hamiltonian.
متن کاملMarchenko Equations and Norming Constants of the Matrix Zakharov–shabat System
In this article we derive the Marchenko integral equations for solving the inverse scattering problem for the matrix Zakharov-Shabat system with a potential without symmetry properties and having L1 entries under a technical hypothesis preventing the accumulation of discrete eigenvalues on the continuous spectrum. Wederive additional symmetry properties in the focusing case. The norming constan...
متن کاملOn N-wave Type Systems and Their Gauge Equivalent
Abstract. The class of nonlinear evolution equations (NLEE) – gauge equivalent to the N-wave equations related to the simple Lie algebra g are derived and analyzed. They are written in terms of S(x, t) ∈ g satisfying r = rank g nonlinear constraints. The corresponding Lax pairs and the time evolution of the scattering data are found. The Zakharov–Shabat dressing method is appropriately modified...
متن کامل