arXiv subjects

Patrik V. Nabelek

Publications and source records attributed to Patrik V. Nabelek.

5 recordsLinked to original sources

arXiv2021

Algebro-Geometric Finite Gap Solutions to the Korteweg--de Vries Equation as Primitive Solutions

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 (see remark 1 for the precise meaning of this statement). This is done using the the primitive solution framework initiated by [5,28,31]. One implication of this result is that the N-soliton solutions can approximate any bounded periodic solution to the Korteweg--de Vries equation arbitrarily well in the limit as N diverges to infinity. We also study primitive solutions numerically that have the same spectral properties as the algebro-geometric finite gap solutions but are not algebro-geometric solutions.

nlin.SI↗

arXiv2021

On Solutions to the Nonlocal $\overline{\partial}$ Problem and (2+1) Dimensional Completely Integrable Systems

In this short note we discuss a new formula for solving the nonlocal $\overline{\partial}$-problem, and discuss application to the Manakov--Zakharov dressing method. We then explicitly apply this formula to solving the complex (2+1)D Kadomtsev--Petviashvili equation and complex (2+1)D completely integrable generalization of the (2+1)D Kaup--Broer (or Kaup--Boussinesq) system. We will also discuss how real (1+1)D solutions are expressed using this formalism. It is simple to express the formalism for finite gap primitive solutions from [10], [8] using the formalism of this note. We also discuss recent results on the infinite soliton limit for the (1+1)D Korteweg--de Vries equation and the (2+1)D Kaup--Broer system. In an appendix, the classical solutions to the 3D Laplace equation (2+1)D d'Alembert wave equation by Whittaker are described. This appendix is included to elucidate an analogy between the dressing method and the Whittaker solutions.

nlin.SI↗

arXiv2021

Distributions Supported on Fractal Sets and Solutions to the Kadomtsev--Petviashvili Equation

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 discuss a family distributions that are holomorphic on the Riemann sphere except on the singular sets homeomorphic to a Cantor set or Sierpinski gasket. These distributions allow us to produce solutions to the Kadomtsev--Petviashvili equation. These distributions are limits of families of rational functions that can also be associated with holomorphic line bundles on surfaces with a finite number of doubly degenerate singular points. We conjecture that a subset of these distributions can be used to formulate a definition of a holomorphic line bundle on some surfaces that are homeomorphic to spheres except where they become doubly degenerate on singular sets homeomorphic to a Cantor set or Sierpinski gasket.

math-ph↗

arXiv2019

A Riemann--Hilbert Problem Approach to Periodic Infinite Gap Hill's Operators and the Korteweg--de Vries Equation

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 spectral bands and gaps. We establish a uniqueness theorem for this Riemann--Hilbert problem, which provides a new route to establishing unique determination of periodic potentials from spectral data. As the potential evolves according to the KdV equation, we use integrability to derive an associated Riemann--Hilbert problem with explicit time dependence. Basic principles from the theory of Riemann--Hilbert problems yield a new characterization of spectra for periodic potentials in terms of the existence of a solution to a scalar Riemann--Hilbert problem, and we derive a similar condition on the spectrum for the temporal periodicity for an evolution under the KdV equation.

math.SP↗

arXiv2019

Solutions to the Kaup--Broer System and Its 2+1 Dimensional Integrable Generalization via the Dressing Method

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]. The method for the 1+1 dimensional Kaup--Broer systems are reductions of a method for a complex valued 2+1 dimensional completely integrable partial differential equation first introduced in [23]. This method allows computation of solutions to all cases of the Kaup--Broer system. We then consider the case of non-capillary waves with usual gravitational forcing, and use the dressing method to compute N-soliton solutions and more general solutions in the closure of the N-soliton solutions in the topology of uniform convergence in compact sets called primitive solutions. These more general solutions are an analogue of the solutions derived in [11, 30, 31] for the KdV equation. We derive dressing functions for finite gap solutions. We compute counter propagating dispersive shockwave type solutions numerically.

nlin.SI↗