Tuncay Aktosun

The uniqueness in the inverse problem for transmission eigenvalues for the spherically symmetric variable-speed wave equation

The recovery of a spherically symmetric wave speed v is considered in a bounded spherical region of radius b from the set of the corresponding transmission eigenvalues for which the corresponding eigenfunctions are also spherically symmetric. If the integral of 1/v on the interval [0, b] is less than b, assuming that there exists at least one v corresponding to the data, it is shown that v is uniquely determined by the data consisting of such transmission eigenvalues and their ‘multiplicities’, where the ‘multiplicity’ is defined as the multiplicity of the transmission eigenvalue as a zero of a key quantity. When that integral is equal to b, the unique recovery is obtained when the data contain one additional piece of information. Some similar results are presented for the unique determination of the potential from the transmission eigenvalues with ‘multiplicities’ for a related Schrodinger equation.

Exact solutions to the focusing nonlinear Schrödinger equation

A method is given to construct globally analytic (in space and time) exact solutions to the focusing cubic nonlinear Schrodinger equation on the line. An explicit formula and its equivalents are presented to express such exact solutions in a compact form in terms of matrix exponentials. Such exact solutions can alternatively be written explicitly as algebraic combinations of exponential, trigonometric and polynomial functions of the spatial and temporal coordinates.

Inverse problem on the line without phase information

The one-dimensional Schrodinger equation is considered for real potentials that are integrable, have finite first moment, and contain no bound states. The recovery of a potential with support in a right half-line is studied in terms of the scattering data consisting of the magnitude of the reflection coefficient, a known potential placed to the left of the unknown potential, and the magnitude of the reflection coefficient of the combined potential. Several kinds of methods are described for retrieval of the reflection coefficient corresponding to the unknown potential. Some illustrative examples are provided.

On the Riemann-Hilbert problem for the one-dimensional Schrödinger equation

A matrix Riemann–Hilbert problem associated with the one‐dimensional Schrodinger equation is considered, and the existence and uniqueness of its solutions are studied. The solution of this Riemann–Hilbert problem yields the solution of the inverse scattering problem for a larger class of potentials than the usual Faddeev class. Some examples of explicit solutions of the Riemann–Hilbert problem are given, and the connection with ambiguities in the inverse scattering problem is established.

Inverse spectral-scattering problem with two sets of discrete spectra for the radial Schrödinger equation

The Schrodinger equation on the half-line is considered with a real-valued, integrable potential having a finite first moment. It is shown that the potential and the boundary conditions are uniquely determined by the data containing the discrete eigenvalues for a boundary condition at the origin, the continuous part of the spectral measure for that boundary condition and a subset of the discrete eigenvalues for a different boundary condition. This result extends the celebrated two-spectrum uniqueness theorem of Borg and Marchenko to the case where there is also a continuous spectrum.

Explicit solutions to the Korteweg–de Vries equation on the half line

Certain explicit solutions to the Korteweg–de Vries equation in the first quadrant of the xt-plane are presented. Such solutions involve algebraic combinations of truly elementary functions, and their initial values correspond to rational reflection coefficients in the associated Schrodinger equation. In the reflectionless case such solutions reduce to pure N-soliton solutions. An illustrative example is provided.

Direct and inverse scattering for selfadjoint Hamiltonian systems on the line

A direct and inverse scattering theory on the full line is developed for a class of first-order selfadjoint 2n×2n systems of differential equations with integrable potential matrices. Various properties of the corresponding scattering matrices including unitarity and canonical Wiener-Hopf factorization are established. The Marchenko integral equations are derived and their unique solvability is proved. The unique recovery of the potential from the solutions of the Marchenko equations is shown. In the case of rational scattering matrices, state space methods are employed to construct the scattering matrix from a reflection coefficient and to recover the potential explicitly.

Wave scattering in one dimension with absorption

Wave scattering is analyzed in a one-dimensional nonconservative medium governed by the generalized Schrodinger equation d2ψ/dx2+k2ψ=[ikP(x)+Q(x)]ψ, where P(x) and Q(x) are real, integrable potentials with finite first moments. Various properties of the scattering solutions are obtained. The corresponding scattering matrix is analyzed, and its small-k and large-k asymptotics are established. The bound states, which correspond to the poles of the transmission coefficient in the upper-half complex plane, are studied in detail. When the medium is not purely absorptive, i.e., unless P(x)⩽0, it is shown that there may be bound states at complex energies, degenerate bound states, and singularities of the transmission coefficient imbedded in the continuous spectrum. Some explicit examples are provided illustrating the theory.

Exact solutions to the sine-Gordon equation

A systematic method is presented to provide various equivalent solution formulas for exact solutions to the sine-Gordon equation. Such solutions are analytic in the spatial variable

Inverse Theory: Problem on the Line1

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