arXiv subjects
Publications and source records attributed to Halis Yilmaz.
The Hirota equation is an integrable higher order nonlinear Schrödinger type equation which describes the propagation of ultrashort light pulses in optical fibers. We present a standard Darboux transformation for the Hirota equation and then construct its quasideterminant solutions. The multisoliton and breather solutions of the Hirota equation are given explicitly.
We present a construction of a large class of Laplace invariants for linear hyperbolic partial differential operators of fairly general form and arbitrary order.
The Sasa-Satsuma equation is an integrable higher-order nonlinear Schrödinger equation. Higher-order and multicomponent generalisations of the nonlinear Schrödinger equation are important in various applications, e.g., in optics. One of these equations is the Sasa-Satsuma equation. We present the binary Darboux transformations for the Sasa-Satsuma equation and then construct its quasigrammians solutions by iterating its binary Darboux transformations. Periodic, one-soliton, two-solitons and breather solutions are given as explicit examples.
We study the Gerdjikov-Ivanov (GI) equation and present a standard Darboux transformation for it. The solution is given in terms of quasideterminants. Further, the parabolic, soliton and breather solutions of the GI equation are given as explicit examples.
We consider Darboux transformations for the derivative nonlinear Schrödinger equation. A new theorem for Darboux transformations of operators with no derivative term are presented and proved. The solution is expressed in quasideterminant forms. Additionally, the parabolic and soliton solutions of the derivative nonlinear Schrödinger equation are given as explicit examples.
We consider the generalization of Laplace invariants to linear differential systems of arbitrary rank and dimension. We discuss completeness of certain subsets of invariants.