Functional equation of inverse scattering for the schrödinger equation with continuous and discrete spectra (short note)

2008 ◽  
pp. 327-331
Author(s):  
Saeyoung Ahn
Keyword(s):  
Functional Equation ◽  
Schrödinger Equation ◽  
Inverse Scattering ◽  
Schrodinger Equation ◽  
Short Note ◽  
Continuous And Discrete Spectra ◽  
Discrete Spectra

scholarly journals USING MIXED DATA IN THE INVERSE SCATTERING PROBLEM

2008 ◽  
Vol 22 (23) ◽  
pp. 2181-2189 ◽  
Author(s):  
M. LASSAUT ◽  
S. Y. LARSEN ◽  
S. A. SOFIANOS ◽  
J. C. WALLET
Keyword(s):  
Schrödinger Equation ◽  
Inverse Scattering ◽  
Regular Solution ◽  
Schrodinger Equation ◽  
Scattering Problem ◽  
Inverse Scattering Problem ◽  
Phase Shifts ◽  
Mixed Data ◽  
Monotonic Functions ◽  
Unique Potential

Consider the fixed-ℓ inverse scattering problem. We show that the zeros of the regular solution of the Schrödinger equation, rn(E), which are monotonic functions of the energy, determine a unique potential when the domain of the energy is such that the rn(E) range from zero to infinity. This suggests that the use of the mixed data of phase-shifts {δ(ℓ0, k), k ≥ k0} ∪ {δ(ℓ, k0), ℓ ≥ ℓ0}, for which the zeros of the regular solution are monotonic in both domains, and range from zero to infinity, offers the possibility of determining the potential in a unique way.

Riemann–Hilbert approach and multi-soliton solutions of a variable-coefficient fifth-order nonlinear Schrödinger equation with N distinct arbitrary-order poles

2021 ◽  
pp. 2150194
Author(s):  
Zhi-Qiang Li ◽  
Shou-Fu Tian ◽  
Tian-Tian Zhang ◽  
Jin-Jie Yang
Keyword(s):  
Schrödinger Equation ◽  
Inverse Scattering ◽  
Nonlinear Schrödinger Equation ◽  
Variable Coefficient ◽  
Schrodinger Equation ◽  
Arbitrary Order ◽  
Soliton Solutions ◽  
Nonlinear Schrodinger Equation ◽  
Nonlinear Schrödinger ◽  
Fifth Order

Based on inverse scattering transformation, a variable-coefficient fifth-order nonlinear Schrödinger equation is studied through the Riemann–Hilbert (RH) approach with zero boundary conditions at infinity, and its multi-soliton solutions with [Formula: see text] distinct arbitrary-order poles are successfully derived. By deriving the eigenfunction and scattering matrix, and revealing their properties, a RH problem (RHP) is constructed based on inverse scattering transformation. Via solving the RHP, the formulae of multi-soliton solutions are displayed when the reflection coefficient possesses [Formula: see text] distinct arbitrary-order poles. Finally, some appropriate parameters are selected to analyze the interaction of multi-soliton solutions graphically.

Painlevé analysis of a nonlinear Schrödinger equation discussing dynamics of solitons in optical fiber

2020 ◽  
pp. 2150005
Author(s):  
Syed T. R. Rizvi ◽  
Aly R. Seadawy ◽  
Ijaz Ali ◽  
Muhammad Younis
Keyword(s):  
Schrödinger Equation ◽  
Inverse Scattering ◽  
Nonlinear Schrödinger Equation ◽  
Schrodinger Equation ◽  
Nonlinear Schrodinger Equation ◽  
Complete Integrability ◽  
Branch Points ◽  
Painlevé Test ◽  
Nonlinear Schrödinger ◽  
New Form

In this paper, we investigated a new form of nonlinear Schrödinger equation (NLSE), namely the Biswas–Arshed model (BAM) for the analysis of complete integrability with the help of Painlevé test ([Formula: see text]-test). By applying this test, we analyze the singularity structure of the solutions of BAM, knowing the fact that the absence of specific sort of singularities like moveable branch points is a patent signal for the complete integrability of the discussed model. Passing the [Formula: see text]-test is a powerful indicator that the studied model is resolvable by means of inverse scattering transformation (IST).

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