arXiv subjects

Ramazan Ercan

Publications and source records attributed to Ramazan Ercan.

4 recordsLinked to original sources

arXiv2026

The Marchenko method for soliton solutions to the Sawada--Kotera equation

Associated with the third-order linear differential operator, we present the Marchenko integral equation using as input the bound-state poles of a transmission coefficient and the time-evolved bound-state dependency constants. We derive the $\mathbf N$-soliton solution to the Sawada--Kotera equation, for an arbitrary positive integer $\mathbf N,$ by recovering that soliton solution from the solution to our Marchenko integral equation. Our method explains the origin of the $2\mathbf N$ real parameters appearing in the $\mathbf N$-soliton solution formula obtained by the ad-hoc method of Hirota. We show that $\mathbf N$ of those parameters are related to the $\mathbf N$ bound-state poles of the left transmission coefficient and the remaining $\mathbf N$ parameters are related to the bound-state dependency constants. Our Marchenko integral equation corresponds to the ``GLM (Gel'fand--Levitan--Marchenko) integral equation'' Kaup relentlessly but unsuccessfully tried to obtain.

nlin.SI↗

arXiv2026

Inverse scattering for the linear system associated with the coupled Gerdjikov--Ivanov equations

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 Schr\"odinger) 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 Schr\"odinger) 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.

math-ph↗

arXiv2022

The Marchenko method to solve the general system of derivative nonlinear Schr\"odinger equations

A system of linear integral equations is presented, which is the analog of the system of Marchenko integral equations, to solve the inverse scattering problem for the linear system associated with the derivative NLS equations. The corresponding direct and inverse scattering problems are analyzed, and the recovery of the potentials and the Jost solutions from the solution to the Marchenko system is described. When the reflection coefficients are zero, some explicit solution formulas are provided for the potentials and the Jost solutions in terms of a pair of constant matrix triplets representing the bound-state information for any number of bound states and any multiplicities. In the reduced case, when the two potentials in the linear system are related to each other through complex conjugation, the corresponding reduced Marchenko integral equation is obtained. The solution to the derivative NLS equation is obtained from the solution to the reduced Marchenko integral equation. The theory presented is illustrated with some explicit examples.

nlin.SI↗

arXiv2018

Direct and inverse scattering problems for a first-order system with energy-dependent potentials

The direct and inverse scattering problems on the full line are analyzed for a first-order system of ordinary linear differential equations associated with the derivative nonlinear Schrödinger equation and related equations. The system contains a spectral parameter and two potentials, where the potentials are proportional to the spectral parameter and hence are called energy-dependent potentials. Using the two potentials as input, the direct problem is solved by determining the scattering coefficients and the bound-state information consisting of bound-state energies, their multiplicities, and the corresponding norming constants. By using two different methods, the corresponding inverse problem is solved by determining the two potentials when the scattering data set is used as input. The first method involves the transformation of the energy-dependent system into two distinct energy-independent systems. The second method involves the establishment of the so-called alternate Marchenko system of linear integral equations and the recovery of the energy-dependent potentials from the solution to the alternate Marchenko system.

math-ph↗