Department of Mathematics
California State University San Marcos
San Marcos, CA 92096, USA
email: ramercan44@gmail.com
Ramazan Ercan
Abstract:
We consider a certain first-order linear system of ordinary differential equations,
and we analyze the direct and inverse scattering problems for that linear system.
The linear system involves two potentials in the Schwartz class, and those potentials
linearly depend on the spectral parameter.
This linear system is related to the integrable system of nonlinear partial differential
equations known as the DNLS (derivative nonlinear Schrodinger) system III, which is also
known as the Gerdjikov-Ivanov system.
When analyzing the direct problem, we describe the pertinent properties of the Jost
solutions and the scattering coefficients.
The bound states poles and the associated normalization constants are represented via
a matrix triplet pair, enabling us to deal with
any number of bound states and any multiplicities. The inverse scattering problem
comprises the determination of
the two potentials when the reflection coefficients and the bound-state information
are available. To solve the inverse problem, we establish a linear system of integral
equations where the kernel and
nonhomogeneous term are determined by the Fourier transforms of the reflection
coefficients and the matrix triplet pair representing the bound-state information.
This system of linear integral equations is the counterpart of the system of Marchenko
integral equations available for the AKNS system associated with the integrable NLS
(nonlinear Schrodinger) system. We recover the potentials from the solution of our
established Marchenko integral system.
When we use the time-evolved reflection coefficients and the time-evolved matrix
triplets, the corresponding time-evolved
potential pair yields a solution of the Gerdjikov-Ivanov system.
Submitted March 26, 2026. Published May 13, 2026.
Math Subject Classifications: 34A55, 34L40, 37K15.
Key Words: Scattering for first-order linear systems; energy dependent potentials;
inverse scattering with energy-dependent potentials; Marchenko method;
derivative nonlinear Schrodinger equations; Gerdjikov-Ivanov system; DNLS system III
DOI: 10.58997/ejde.2026.37
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Ramazan Ercan Department of Mathematics California State University San Marcos San Marcos, CA 92096, USA email: ramercan44@gmail.com |
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