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## Publications

Publications (152)

We present a detailed numerical study of the transverse stability of line solitons of two-dimensional, generalized Zakharov-Kusnetsov equations with various power nonlinearities. In the $L^{2}$-subcritical case, in accordance with a theorem due to Yamazaki we find a critical velocity, below which the line soliton is stable. For higher velocities, t...

The study of complex geometric optics solutions to a system of d-bar equations appearing in the context of electrical impedance tomography and the scattering theory of the integrable Davey-Stewartson II equations for large values of the spectral parameter $k$ in \cite{KlSjSt20} is extended to the reflection coefficient. For the case of potentials $...

We consider in this paper modified fractional Korteweg–de Vries and related equations (modified Burgers–Hilbert and Whitham). They have the advantage with respect to the usual fractional KdV equation to have a defocusing case with a different dynamics. We will distinguish the weakly dispersive case where the phase velocity is unbounded for low freq...

An efficient high precision hybrid numerical approach for integrable Davey–Stewartson (DS) I equations for trivial boundary conditions at infinity is presented for Schwartz class initial data. The code is used for a detailed numerical study of DS I solutions in this class. Localized stationary solutions are constructed and shown to be unstable agai...

We present a numerical approach for generalised Korteweg-de Vries (KdV) equations on the real line. In the spatial dimension we compactify the real line and apply a Chebyshev collocation method. The time integration is performed with an implicit Runge-Kutta method of fourth order. Several examples are discussed: initial data bounded but not vanishi...

Complex geometric optics solutions to a system of d‐bar equations appearing in the context of electrical impedance tomography and the scattering theory of the integrable Davey‐Stewartson II equations are studied for large values of the spectral parameter k. For potentials q∈⋅−2Hsℂ for some s∈1,2, it is shown that the solution converges as the geome...

This introductory chapter contains some well-known facts and a few more recent results. It is mainly addressed to non-specialists of either nonlinear dispersive PDEs or Inverse Scattering techniques and vocabulary. We first recall some basic elementary notions on (dispersive) wave propagation. Then, taking as a guideline the Korteweg–de Vries equat...

This chapter focuses on two asymptotic models for internalwaves, the Benjamin–Ono (BO) and Intermediate LongWave (ILW) equations, which are integrable by inverse scattering techniques (IST). After briefly recalling their (rigorous) derivations, we will review old and recent results on the Cauchy problem, comparing those obtained by IST and PDE tech...

This chapter aims to survey the known results on the Kadomtsevs–Petviashvili equations and their variants from the point of viewof modeling, PDEs and IST.Numerical simulations will illustrate the results and provide useful hints for open problems and conjectures.

We have tried in this book to develop in some detail rigorous results on the Cauchy problem for relevant dispersive integrable equations (and some of their variants) by methods of inverse scattering or PDEs (in the large).

We present a numerical approach for generalised Korteweg-de Vries (KdV) equations on the real line. In the spatial dimension we compactify the real line and apply a Chebyshev collocation method. The time integration is performed with an implicit Runge-Kutta method of fourth order. Several examples are discussed: initial data bounded but not vanishi...

To appear in Applied Mathematical Sciences (Springer)

We present a numerical approach to study solutions to the dispersionless Kadomtsev–Petviashvili (dKP) equation on \({\mathbb {R}}\times {\mathbb {T}}\). The dependence on the coordinate x is considered on the compactified real line, and the dependence on the coordinate y is assumed to be periodic. Critical behavior, the formation of a shock in the...

An efficient high precision hybrid numerical approach for integrable Davey-Stewartson (DS) I equations for trivial boundary conditions at infinity is presented for Schwartz class initial data. The code is used for a detailed numerical study of DS I solutions in this class. Localized stationary solutions are constructed and shown to be unstable agai...

A multi-domain spectral method is presented to compute the Hilbert transform on the whole compactified real line, with a special focus on piece-wise analytic functions and functions with algebraic decay towards infinity. Several examples of these and other types of functions are discussed. As an application solitons to generalized Benjamin–Ono equa...

We present a detailed numerical study of solutions to the (generalized) Zakharov–Kuznetsov equation in two spatial dimensions with various power nonlinearities. In the L2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \s...

We present a detailed numerical study of solutions to the Zakharov-Kuznetsov equation in three spatial dimensions. The equation is a three-dimensional generalization of the Korteweg–de Vries equation, though, not completely integrable. This equation is L2-subcritical, and thus, solutions exist globally, for example, in the H1 energy space.
We first...

This paper is concerned with the efficient numerical computation of solutions to the 1D stationary Schrödinger equation in the semiclassical limit in the highly oscillatory regime. A previous approach to this problem based on explicitly incorporating the leading terms of the WKB approximation is enhanced in two ways: first a refined error analysis...

We study a 1D nonlinear Schrödinger equation appearing in the description of a particle inside an atomic nucleus. For various nonlinearities, the ground states are discussed and given in explicit form. Their stability is studied numerically via the time evolution of perturbed ground states. In the time evolution of general localized initial data, t...

We present a multi-domain spectral approach with an exterior compactified domain for the Maxwell equations for monochromatic fields. The Sommerfeld radiation condition is imposed exactly at infinity being a finite point on the numerical grid. As an example, axisymmetric situations in spherical and prolate spheroidal coordinates are discussed, as we...

A multi-domain spectral method is presented to compute the Hilbert transform on the whole compactified real line, with a special focus on piece-wise analytic functions and functions with algebraic decay towards infinity. Several examples of these and other types of functions are discussed. As an application solitons to generalized Benjamin-Ono equa...

We perform numerical experiments on the Serre-Green-Naghdi (SGN) equations and a fully dispersive "Whitham-Green-Naghdi" (WGN) counterpart in dimension 1. In particular, solitary wave solutions of the WGN equations are constructed and their stability, along with the explicit ones of the SGN equations, is studied. Additionally, the emergence of modu...

We present a detailed numerical study of solutions to the Zakharov-Kuznetsov equation in three spatial dimensions. The equation is a three-dimensional generalization of the Korteweg-de Vries equation, though, not completely integrable. This equation is $L^2$-subcritical, and thus, solutions exist globally, for example, in the $H^1$ energy space. We...

We consider in this paper modified fractional Korteweg-de Vries and related equations (modified Burgers-Hilbert and Whitham). They have the advantage with respect to the usual fractional KdV equation to have a defocusing case with a different dynamics. We will distinguish the weakly dispersive case where the phase velocity is unbounded for low freq...

In a previous work on the large $|k|$ behavior of complex geometric optics solutions to a system of d-bar equations, we treated in detail the situation when a certain potential is the characteristic function of a strictly convex set with real-analytic boundary. We here extend the results to the case of sets with smooth boundary, by using almost hol...

We present a multidomain spectral approach with an exterior compactified domain for the Maxwell equations for monochromatic fields. The Sommerfeld radiation condition is imposed exactly at infinity being a finite point on the numerical grid. As an example, axisymmetric situations in spherical and prolate spheroidal coordinates are discussed.

Complex geometric optics solutions to a system of d-bar equations appearing in the context of electrical impedance tomography and the scattering theory of the integrable Davey-Stewartson II equations are studied for large values of the spectral parameter $k$. For potentials \( q\in \langle \cdot \rangle^{-2} H^{s}(\mathbb{C}) \) for some $s \in]1,2...

We present an efficient high-precision numerical approach for Davey-Stewartson (DS) II type equations , treating initial data from the Schwartz class of smooth, rapidly decreasing functions. As with previous approaches, the presented code uses discrete Fourier transforms for the spatial dependence and Driscoll's composite Runge-Kutta method for the...

We perform numerical experiments on the Serre-Green-Nagdi (SGN) equationsand a fully dispersive "Whitham-Green-Naghdi" (WGN) counterpart in dimension 1. In particular, solitary wave solutions of the WGN equations are constructed and their stability, along with the explicit ones of the SGN equations, is studied. Additionally, the onset of dispersive...

We present a multidomain spectral approach for Fuchsian ordinary differential equations in the particular case of the hypergeometric equation. Our hybrid approach uses Frobenius’ method and Moebius transformations in the vicinity of each of the singular points of the hypergeometric equation, which leads to a natural decomposition of the real axis i...

We numerically study the transverse stability of the Peregrine solution, an exact solution to the one dimensional nonlinear Schrödinger (NLS) equation and thus a y-independent solution to the 2D NLS. To this end we generalise a previously published approach based on a multi-domain spectral method on the whole real line. We do this in two ways: firs...

We present a detailed numerical study of solutions to the (generalized) Zakharov-Kuznetsov equation in two spatial dimensions with various power nonlinearities. In the $L^{2}$-subcritical case, numerical evidence is presented for the stability of solitons and the soliton resolution for generic initial data. In the $L^2$-critical and supercritical c...

We study a 1D nonlinear Schr{\"o}dinger equation appearing in the description of a particle inside an atomic nucleus. For various nonlinearities, the ground states are discussed and given in explicit form. Their stability is studied numerically via the time evolution of perturbed ground states. In the time evolution of general localized initial dat...

We present an efficient high-precision numerical approach for the Davey-Stewartson (DS) II equation, treating initial data from the Schwartz class of smooth, rapidly decreasing functions. As with previous approaches, the presented code uses discrete Fourier transforms for the spatial dependence and Driscoll's composite Runge-Kutta method for the ti...

We numerically study the transverse stability of the Peregrine solution, an exact solution to the one dimensional nonlinear Schr\"odinger (NLS) equation and thus a $y$-independent solution to the 2D NLS. To this end we generalise a previously published approach based on a multi-domain spectral method on the whole real line. We do this in two ways:...

The inverse scattering approach for the defocusing Davey–Stewartson II equation is given by a system of D-bar equations. We present a numerical approach to semi-classical D-bar problems for real analytic rapidly decreasing potentials. We treat the D-bar problem as a complex linear second order integral equation which is solved with discrete Fourier...

We present analytical results and numerical simulations for a class of nonlinear dispersive equations in two spatial dimensions. These equations are of (derivative) nonlinear Schrödinger type and have recently been obtained in [11] in the context of nonlinear optics. In contrast to the usual nonlinear Schrödinger equation, this new model incorporat...

We present a multidomain spectral approach for Fuchsian ordinary differential equations in the particular case of the hypergeometric equation. Our hybrid approach uses Frobenius' method and Moebius transformations in the vicinity of each of the singular points of the hypergeometric equation, which leads to a natural decomposition of the real axis i...

In this work we present spectral approaches for d-bar problems with potentials having compact support on a disk. Our algorithms use polar coordinates and implement a Chebychev collocation scheme for the radial dependence and a Fourier spectral method for the azimuthal dependence. The focus is placed on the construction of complex geometric optics (...

The inverse scattering approach for the defocusing Davey-Stewartson II equation is given by a system of D-bar equations. We present a numerical approach to semi-classical D-bar problems for real analytic rapidly decreasing potentials. We treat the D-bar problem as a complex linear second order integral equation which is solved with discrete Fourier...

A multidomain spectral approach for Painlevé transcendents on unbounded domains is presented. This method is designed to study solutions determined uniquely by a, possibly divergent, asymptotic series valid near infinity in a sector and approximates the solution on straight lines lying entirely within said sector without the need of evaluating trun...

We present analytical results and numerical simulations for a class of nonlinear dispersive equations in two spatial dimensions. These equations are of (derivative) nonlinear Schr\"odinger type and have recently been obtained in \cite{DLS} in the context of nonlinear fiber optics. In contrast to the usual nonlinear Schr\"odinger equation, this new...

We present a detailed numerical study of various blow‐up issues in the context of the focusing Davey–Stewartson II equation. To this end, we study Gaussian initial data and perturbations of the lump and the explicit blow‐up solution due to Ozawa. Based on the numerical results it is conjectured that the blow‐up in all cases is self‐similar, and tha...

We present a detailed numerical study of various blow-up issues in the context of the focusing Davey-Stewartson II equation. To this end we study Gaussian initial data and perturbations of the lump and the explicit blow-up solution due to Ozawa. Based on the numerical results it is conjectured that the blow-up in all cases is self similar, and that...

The defocusing Davey‐Stewartson II equation has been shown in numerical experiments to exhibit behavior in the semiclassical limit that qualitatively resembles that of its one‐dimensional reduction, the defocusing nonlinear Schrödinger equation, namely the generation from smooth initial data of regular rapid oscillations occupying domains of space‐...

The aim of this paper is to study, via theoretical analysis and numerical simulations, the dynamics of Whitham and related equations. In particular we establish rigorous bounds between solutions of the Whitham and KdV equations and provide some insights into the dynamics of the Whitham equation in different regimes, some of them being outside the r...

A detailed numerical study of the long time behaviour of dispersive shock waves in solutions to the Kadomtsev-Petviashvili (KP) I equation is presented. It is shown that modulated lump solutions emerge from the dispersive shock waves. For the description of dispersive shock waves, Whitham modulation equations for KP are obtained. It is shown that t...

An important step in the efficient computation of multi-dimensional theta functions is the construction of appropriate symplectic transformations for a given Riemann matrix assuring a rapid convergence of the theta series. An algorithm is presented to approximately map the Riemann matrix to the Siegel fundamental domain. The shortest vector of the...

We study numerically the evolution of perturbed Korteweg-de Vries solitons and of well localized initial data by the Novikov-Veselov (NV) equation at different levels of the "energy" parameter $ E $. We show that as $ |E| \to \infty $, NV behaves, as expected, similarly to its formal limit, the Kadomtsev-Petviashvili equation. However at intermedia...

A multidomain spectral method with compactified exterior domains combined
with stable second and fourth order time integrators is presented for
Schr\"odinger equations. The numerical approach allows high precision numerical
studies of solutions on the whole real line. At examples for the linear and
cubic nonlinear Schr\"odinger equation, this code...

The Kerr solution in coordinates corotating with the horizon is studied as a testbed for a spacetime with a helical Killing vector in the Ernst picture. The solution is numerically constructed by solving the Ernst equation with a spectral method and a Newton iteration. We discuss convergence of the iteration for several initial iterates and differe...

A purely numerical approach to compact Riemann surfaces starting from plane
algebraic curves is presented. The critical points of the algebraic curve are
computed via a two-dimensional Newton iteration. The starting values for this
iteration are obtained from the resultants with respect to both coordinates of
the algebraic curve and a suitable pair...

An asymptotic description of the formation of dispersive shock waves in solutions to the generalized Kadomtsev–Petviashvili (KP) equation is conjectured. The asymptotic description based on a multiscales expansion is given in terms of a special solution to an ordinary differential equation of the Painlevé I hierarchy. Several examples are discussed...

We present the first numerical approach to D-bar problems having spectral
convergence for real analytic rapidly decreasing potentials. The proposed
method starts from a formulation of the problem in terms of an integral
equation which is solved with Fourier techniques. The singular integrand is
regularized analytically. The resulting integral equat...

The Peregrine breather is widely discussed as a model for rogue waves in deep
water. We present here a detailed numerical study of perturbations of the
Peregrine breather as a solution to the nonlinear Schr\"odinger (NLS)
equations. We first address the modulational instability of the constant
modulus solution to NLS. Then we study numerically loca...

For equation , the second member in the PI hierarchy, we prove existence of various degenerate solutions depending on the complex parameter and evaluate the asymptotics in the complex plane for and . Using this result, we identify the most degenerate solutions , , , called tritronqu,e; describe the quasi-linear Stokes phenomenon; and find the large...

The dispersionless Kadomtsev-Petviashvili (dKP) equation
$(u_t+uu_x)_x=u_{yy}$ is one of the simplest nonlinear wave equations
describing two-dimensional shocks. To solve the dKP equation we use a
coordinate transformation inspired by the method of characteristics for the
one-dimensional Hopf equation $u_t+uu_x=0$. We show numerically that the
solu...

We present a detailed numerical study of solutions to general Korteweg-de Vries equations with critical and supercritical nonlinearity, both in the context of dispersive shocks and blow-up. We study the stability of solitons and show that they are unstable against being radiated away and blow-up. In the critical case, the blow-up mechanism by Marte...

Using a Fourier spectral method, we provide a detailed numerical investigation of dispersive Schrödinger-type equations involving a fractional Laplacian in an one-dimensional case. By an appropriate choice of the dispersive exponent, both mass and energy sub- and supercritical regimes can be identified. This allows us to study the possibility of fi...

We survey and compare, mainly in the two-dimensional case, various results
obtained by IST and PDE techniques for integrable equations. We also comment on
what can be predicted from integrable equations on non integrable ones.

We present a computational approach to general hyperelliptic Riemann surfaces
in Weierstrass normal form. The surface is either given by a list of the branch
points, the coefficients of the defining polynomial or a system of cuts for the
curve. A canonical basis of the homology is introduced algorithmically for this
curve. The periods of the holomo...

We provide a numerical study of various issues pertaining to the dynamics of
the Davey-Stewartson systems of the DS II type. In particular we investigate
whether or not the properties (blow-up, radiation,...) displayed by the
focusing and defocusing DS II integrable systems persist in the non integrable
case.

Using a Fourier spectral method, we provide a detailed numerically
investigation of dispersive Schr\"odinger type equations involving a fractional
Laplacian. By an appropriate choice of the dispersive exponent, both mass and
energy sub- and supercritical regimes can be computed in one spatial dimension,
only. This allows us to study the possibility...

We present the first detailed numerical study of the Toda equations in 2 + 1 dimensions in the limit of long wavelengths, both for the hyperbolic and elliptic case. We first study the continuum limit of the Toda equations and solve initial value problems for the resulting system up to the point of gradient catastrophe. It is shown that the break-up...

We consider a semiclassically scaled Schrodinger equation with WKB initial data. We prove that in the classical limit the corresponding Bohmian trajectories converge (locally in measure) to the classical trajectories before the appearance of the first caustic. In a second step we show that after caustic onset this convergence in general no longer h...

We present the first detailed numerical study of the semiclassical limit of
the Davey-Stewartson II equations both for the focusing and the defocusing
variant. We concentrate on rapidly decreasing initial data with a single hump.
The formal limit of these equations for vanishing semiclassical parameter
$\epsilon$, the semiclassical equations, are n...

We provide a detailed numerical study of various issues pertaining to the
dynamics of the Burgers equation perturbed by a weak dispersive term: blow-up
in finite time versus global existence, nature of the blow-up, existence for
"long" times, and the decomposition of the initial data into solitary waves
plus radiation. We numerically construct soli...

We study the critical behaviour of solutions to weakly dispersive Hamiltonian
systems considered as perturbations of elliptic and hyperbolic systems of
hydrodynamic type with two components. We argue that near the critical point of
gradient catastrophe of the dispersionless system, the solutions to a suitable
initial value problem for the perturbed...

We present a numerical study of solutions to the