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Sökning: WFRF:(Tovbis Alexander)

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1.
  • Bertola, Marco, et al. (författare)
  • Diagonalization of the finite Hilbert transform on two adjacent intervals : the Riemann-Hilbert approach
  • 2020
  • Ingår i: Analysis and Mathematical Physics. - : Springer Nature. - 1664-2368 .- 1664-235X. ; 10:3
  • Tidskriftsartikel (refereegranskat)abstract
    • In this paper we study the spectra of bounded self-adjoint linear operators that are related to finite Hilbert transforms H-L : L-2([b(L), 0]) -> L-2([0, b(R)]) and H-R : L-2([0, b(R)]) -> L-2([b(L), 0]). These operators arise when one studies the interior problem of tomography. The diagonalization of H-R, H-L has been previously obtained, but only asymptotically when b(L) not equal -b(R). We implement a novel approach based on the method of matrix Riemann-Hilbert problems (RHP) which diagonalizes H-R, H-L explicitly. We also find the asymptotics of the solution to a related RHP and obtain error estimates.
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2.
  • Fromm, Samuel, 1991- (författare)
  • The defocusing nonlinear Schrödinger equation with step-like oscillatory data
  • 2021
  • Doktorsavhandling (övrigt vetenskapligt/konstnärligt)abstract
    • The thesis at hand consists of three papers as well as an introductory chapter and a summary of results. The topic of the thesis is the study of the defocusing nonlinear Schrödinger equation with step-like oscillatory data.Paper A studies the Cauchy problem for the defocusing nonlinear Schrödinger equation on the line with step-like oscillatory boundary conditions. More precisely, the solution is required to approach a single exponential as x → -∞ and to decay to zero as x → +∞. We prove existence of a global solution and show that the solution can be expressed in terms of the solution of a Riemann-Hilbert problem. We also compute the long-time asymptotics of the solution and apply the results to a related initial-boundary value problem on the half-line.Paper B studies an initial-boundary value problem for the defocusing nonlinear Schrödinger equation on the half-line with asymptotically oscillatory boundary conditions. More precisely, the solution is required to approach a single exponential on the boundary as t → +∞ and to decay to zero as x → +∞. We construct a solution of the problem in a sector close to the boundary and compute its long-time behaviour.Paper C studies a similar problem as Paper B but instead of the nonlinear Schrödinger equation we study the Gerdjikov-Ivanov equation. We give necessary conditions for the existence of a solution of the associated initial-boundary value problem under asymptotically oscillatory boundary conditions.
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