5 papers
Distributions Supported on Fractal Sets and Solutions to the Kadomtsev--Petviashvili Equation
Patrik V. Nabelek
In this note we will discuss a potentially interesting extension of some recent results on primitive solutions to completely integrable partial differential equations. We will disc…
Solutions to the Kaup--Broer System and Its 2+1 Dimensional Integrable Generalization via the Dressing Method
Patrik V. Nabelek, Vladimir E. Zakharov
In this paper we formulate the nonlocal dbar problem dressing method of Manakov and Zakharov [28, 29, 27] for the 4 scaling classes of the 1+1 dimensional Kaup--Broer system [7, 13…
Algebro-Geometric Finite Gap Solutions to the Korteweg--de Vries Equation as Primitive Solutions
Patrik V. Nabelek
In this paper we show that all algebro-geometric finite gap solutions to the Korteweg--de Vries equation can be realized as a limit of N-soliton solutions as N diverges to infinity…
On symmetric primitive potentials
Patrik Nabelek, Dmitry Zakharov, Vladimir Zakharov
The concept of a primitive potential for the Schroedinger operator on the line was introduced in [2,3,4]. Such a potential is determined by a pair of positive functions on a finite…
A Riemann--Hilbert Problem Approach to Periodic Infinite Gap Hill's Operators and the Korteweg--de Vries Equation
Kenneth T-R. McLaughlin, Patrik V. Nabelek
We formulate the inverse spectral theory of infinite gap Hill's operators with bounded periodic potential as a Riemann--Hilbert problem on a typically infinite collection of spectr…