arXiv subjects

Bernard Deconinck

Publications and source records attributed to Bernard Deconinck.

At least 19 recordsLinked to original sources

arXiv2026

The Benjamin-Feir instability in KdV-like equations with general dispersion and monomial nonlinearity

Nonlinear waves in dispersive media can be succeptible to modulational instabilities. We examine a category of scalar equations, with general dispersion and monomial nonlinearity, including a large variety of KdV-like equations. For small-amplitude traveling wave solutions, we provide a complete characterization of the spectrum near the origin of the linear operator obtained from linearizing about periodic traveling waves. We prove rigorously that, when the modulational instability is present, the spectrum connected to the origin consists of curves that invariably form a closed figure-eight pattern.

math.AP↗

arXiv2025

Stokes Waves in Water of Finite Depth

Periodic water waves of permanent form traveling at constant speed, the so-called Stokes waves, are studied in water of fixed finite depth using methods previously used in water of infinite depth. We apply our methods to waves of varying steepness over a range of fixed depths in order to determine how a number of physical quantities related to the waves change as the steepness of the waves increases. Finally, we examine the Taylor sign condition for these waves, as well as the complex singularities outside of their domain of definition when the waves are considered as a function of a conformal variable.

nlin.PS↗

arXiv2024

The explicit solution of linear, dissipative, second-order initial-boundary value problems with variable coefficients

We derive explicit solution representations for linear, dissipative, second-order Initial-Boundary Value Problems (IBVPs) with coefficients that are spatially varying, with linear, constant-coefficient, two-point boundary conditions. We accomplish this by considering the variable-coefficient problem as the limit of a constant-coefficient interface problem, previously solved using the Unified Transform Method of Fokas. Our method produces an explicit representation of the solution, allowing us to determine properties of the solution directly. As explicit examples, we demonstrate the solution procedure for different IBVPs of variations of the heat equation, and the linearized complex Ginzburg-Landau (CGL) equation (periodic boundary conditions). We can use this to find the eigenvalues of dissipative second-order linear operators (including non-self-adjoint ones) as roots of a transcendental function, and we can write their eigenfunctions explicitly in terms of the eigenvalues.

math.AP↗

arXiv2024

Self-similarity and recurrence in stability spectra of near-extreme Stokes waves

We consider steady surface waves in an infinitely deep two--dimensional ideal fluid with potential flow, focusing on high-amplitude waves near the steepest wave with a 120 degree corner at the crest. The stability of these solutions with respect to coperiodic and subharmonic perturbations is studied, using new matrix-free numerical methods. We provide evidence for a plethora of conjectures on the nature of the instabilities as the steepest wave is approached, especially with regards to the self-similar recurrence of the stability spectrum near the origin of the spectral plane.

physics.flu-dyn↗

arXiv2022

The instability of near-extreme Stokes waves

We study the stability of Stokes waves on a free surface of an ideal fluid of infinite depth. For small steepness the modulational instability dominates the dynamics, but its growth rate is vastly surpassed for steeper waves by an instability due to disturbances localized at the wave crest, explaining why long propagating ocean swell consists of small-amplitude waves. The dominant localized disturbances are either co-periodic with the Stokes wave, or have twice its period. The nonlinear stage of instability for steep wave evolution reveals the formation of a plunging breaker.

physics.flu-dyn↗

arXiv2022

Solving the heat equation with variable thermal conductivity

We consider the heat equation with spatially variable thermal conductivity and homogeneous Dirichlet boundary conditions. Using the Method of Fokas or Unified Transform Method, we derive solution representations as the limit of solutions of constant-coefficient interface problems where the number of subdomains and interfaces becomes unbounded. This produces an explicit representation of the solution, from which we can compute the solution and determine its properties. Using this solution expression, we can find the eigenvalues of the corresponding variable-coefficient eigenvalue problem as roots of a transcendental function. We can write the eigenfunctions explicitly in terms of the eigenvalues. The heat equation is the first example of more general variable-coefficient second-order initial-boundary value problems that can be solved using this approach.

math.AP↗

arXiv2022

The analytic extension of solutions to initial-boundary value problems outside their domain of definition

We examine the analytic extension of solutions of linear, constant-coefficient initial-boundary value problems outside their spatial domain of definition. We use the Unified Transform Method or Method of Fokas, which gives a representation for solutions to half-line and finite-interval initial-boundary value problems as integrals of kernels with explicit spatial and temporal dependence. These solution representations are defined within the spatial domain of the problem. We obtain the extension of these representation formulae via Taylor series outside these spatial domains and find the extension of the initial condition that gives rise to a whole-line initial-value problem solved by the extended solution. In general, the extended initial condition is not differentiable or continuous unless the boundary and initial conditions satisfy compatibility conditions. We analyze dissipative and dispersive problems, and problems with continuous and discrete spatial variables.

math.AP↗

arXiv2022

A High-Order Asymptotic Analysis of the Benjamin-Feir Instability Spectrum in Arbitrary Depth

We investigate the Benjamin-Feir (or modulational) instability of Stokes waves, i.e., small-amplitude, one-dimensional periodic gravity waves of permanent form and constant velocity, in water of finite and infinite depth. We develop a perturbation method to describe to high-order accuracy the unstable spectral elements associated with this instability, obtained by linearizing Euler's equations about the small-amplitude Stokes waves. These unstable elements form a figure-eight curve centered at the origin of the complex spectral plane, which is parameterized by a Floquet exponent. Our asymptotic expansions of this figure-eight are in excellent agreement with numerical computations as well as recent rigorous results by Berti, Maspero, and Ventura. From our expansions, we derive high-order estimates for the growth rates of the Benjamin-Feir instability and for the parameterization of the Benjamin-Feir figure-eight curve with respect to the Floquet exponent. We are also able to compare the Benjamin-Feir and high-frequency instability spectra analytically for the first time, revealing three different regimes of the Stokes waves, depending on the predominant instability.

physics.flu-dyn↗

arXiv2021

The numerical solution of semidiscrete linear evolution problems on the finite interval using the Unified Transform Method

We study a semidiscrete analogue of the Unified Transform Method introduced by A. S. Fokas, to solve initial-boundary-value problems for linear evolution partial differential equations with constant coefficients on the finite interval $x \in (0,L)$. The semidiscrete method is applied to various spatial discretizations of several first and second-order linear equations, producing the exact solution for the semidiscrete problem, given appropriate initial and boundary data. From these solutions, we derive alternative series representations that are better suited for numerical computations. In addition, we show how the Unified Transform Method treats derivative boundary conditions and ghost points introduced by the choice of discretization stencil and we propose the notion of "natural" discretizations. We consider the continuum limit of the semidiscrete solutions and compare with standard finite-difference schemes.

math.NA↗

arXiv2021

High-Frequency Instabilities of Stokes Waves

Euler's equations govern the behavior of gravity waves on the surface of an incompressible, inviscid, and irrotational fluid of arbitrary depth. We investigate the spectral stability of sufficiently small-amplitude, one-dimensional Stokes waves, i.e., periodic gravity waves of permanent form and constant velocity, in both finite and infinite depth. Using a nonlocal formulation of Euler's equations developed by Ablowitz et al. (2006), we develop a perturbation method to describe the first few high-frequency instabilities away from the origin, present in the spectrum of the linearization about the small-amplitude Stokes waves. Asymptotic and numerical computations of these instabilities are compared for the first time to excellent agreement.

physics.flu-dyn↗

arXiv2021

The Numerical Unified Transform Method for the Nonlinear Schrödinger equation on the half-line

We implement the Numerical Unified Transform Method to solve the Nonlinear Schrödinger equation on the half-line. For so-called linearizable boundary conditions, the method solves the half-line problems with comparable complexity as the Numerical Inverse Scattering Transform solves whole-line problems. In particular, the method computes the solution at any $x$ and $t$ without spatial discretization or time stepping. Contour deformations based on the method of nonlinear steepest descent are used so that the method's computational cost does not increase for large $x,t$ and the method is more accurate as $x,t$ increase. Our ideas also apply to some cases where the boundary conditions are not linearizable.

math.NA↗

arXiv2021

The numerical solutions of linear semi-discrete evolution problems on the half-line using the Unified Transform Method

We discuss a semi-discrete analogue of the Unified Transform Method, introduced by A. S. Fokas, to solve initial-boundary-value problems for linear evolution partial differential equations of constant coefficients. The semi-discrete method is applied to various spacial discretizations of several first and second-order linear equations on the half-line $x \geq 0$, producing the exact solution for the semi-discrete problem, given appropriate initial and boundary data. We additionally show how the Unified Transform Method treats derivative boundary conditions and ghost points introduced by the choice of discretization stencil. We consider the continuum limit of the semi-discrete solutions and provide several numerical examples.

math.NA↗

arXiv2021

High-Frequency Instabilities of a Boussinesq-Whitham System: A Perturbative Approach

We analyze the spectral stability of small-amplitude, periodic, traveling-wave solutions of a Boussinesq-Whitham system. These solutions are shown numerically to exhibit high-frequency instabilities when subject to bounded perturbations on the real line. We use a formal perturbation method to estimate the asymptotic behavior of these instabilities in the small-amplitude regime. We compare these asymptotic results with direct numerical computations. This is the second paper in a series of three that investigates high-frequency instabilities of Stokes waves.

math.AP↗

arXiv2021

High-Frequency Instabilities of the Kawahara Equation: A Perturbative Approach

We analyze the spectral stability of small-amplitude, periodic, traveling-wave solutions of the Kawahara equation. These solutions exhibit high-frequency instabilities when subject to bounded perturbations on the whole real line. We introduce a formal perturbation method to determine the asymptotic growth rates of these instabilities, among other properties. Explicit numerical computations are used to verify our asymptotic results.

math.AP↗

arXiv2020

Real Lax spectrum implies spectral stability

We consider the dynamical stability of periodic solutions of integrable equations with 2x2 Lax pairs. We construct the eigenfunctions and hence the Floquet discriminant for such Lax pairs. The boundedness of the eigenfunctions determines the Lax spectrum. We use a connection between the Lax spectrum and the stability spectrum to show that the subset of the real line which gives rise to stable eigenvalues is contained in the Lax spectrum. This subset is the full spectrum for self-adjoint members of the AKNS hierarchy. For non-self-adjoint members of the AKNS hierarchy admitting a common reduction, the real line is always part of the Lax spectrum and maps to stable eigenvalues of the stability problem. We demonstrate our methods work for a variety of examples.

nlin.SI↗

arXiv2020

Stability of elliptic solutions to the sinh-Gordon equation

Using the integrability of the sinh-Gordon equation, we demonstrate the spectral stability of its elliptic solutions. By constructing a Lyapunov functional using higher-order conserved quantities of the sinh-Gordon equation, we show that these elliptic solutions are orbitally stable with respect to subharmonic perturbations of arbitrary period.

nlin.SI↗

arXiv2020

The Numerical Unified Transform Method for Initial-boundary Value Problems on the Half-line

We implement the Unified Transform Method of Fokas as a numerical method to solve linear partial differential equations on the half-line. The method computes the solution at any x and t without spatial discretization or time stepping. With the help of contour deformations and oscillatory integration techniques, the method's complexity does not increase for large x,t and the method is more accurate as x,t increase. Our goal is to make no assumptions on the functional form of the initial or boundary functions while maintaining high accuracy in a large region of the (x,t) plane.

math.NA↗

arXiv2019

The orbital stability of elliptic solutions of the Focusing Nonlinear Schrödinger Equation

We examine the stability of the elliptic solutions of the focusing nonlinear Schrödinger equation (NLS) with respect to subharmonic perturbations. Using the integrability of NLS, we discuss the spectral stability of the elliptic solutions, establishing that solutions of smaller amplitude are stable with respect to larger classes of perturbations. We show that spectrally stable solutions are orbitally stable by constructing a Lyapunov functional using higher-order conserved quantities of NLS.

nlin.SI↗