arXiv subjects

Dag Nilsson

Publications and source records attributed to Dag Nilsson.

14 recordsLinked to original sources

arXiv2025

An existence theory for solitary waves on a ferrofluid jet

We discuss axisymmetric solitary waves on the surface of an otherwise cylindrical ferrofluid jet surrounding a stationary metal rod. The ferrofluid, which is governed by a general (nonlinear) magnetisation law, is subject to an azimuthal magnetic field generated by an electric current flowing along the rod. We treat the governing equations using a modification of the Zakharov-Craig-Sulem formulation for water waves, reducing the problem to a single nonlocal equation for the free-surface elevation variable $η$. The nonlocality in the equation takes the form of a Dirichlet-Neumann operator whose analyticity (in standard function spaces) is demonstrated by studying its defining boundary-value problem in newly introduced Sobolev spaces for radial functions.\ Using rudimentary fixed-point arguments and Fourier analysis we rigorously reduce the equation for $η$ to a perturbation of a Korteweg-de Vries equation (for strong surface tension) or a nonlinear Schrödinger equation (for weak surface tension), both of which have nondegenerate explicit solitary-wave solutions. The existence theory is completed using an appropriate version of the implicit-function theorem.

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arXiv2024

Analytical Study of a generalised Dirichlet-Neumann operator and application to three-dimensional water waves on Beltrami flows

In this paper we consider three-dimensional steady water waves with vorticity, under the action of gravity and surface tension; in particular we consider so-called Beltrami flows, for which the velocity field and the vorticity are collinear. We discuss a recent variational formulation of the problem which involves a generalisation of the classical Dirichlet-Neumann operator. We study this operator in detail, extending some well-known results for the classical Dirichlet-Neumann operator, such as the Taylor expansion in homogeneous powers of the wave profile, the computation of its differential and the asymptotic expansion of its associated symbol. A new formulation of the problem as a single equation for the wave profile is also presented and discussed in a similar vein. As an application of these results we rigorously prove existence of doubly periodic gravity-capillary steady waves and construct approximate doubly periodic gravity steady waves.

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arXiv2024

Periodically modulated solitary waves of the CH-KP-I equation

We consider the CH-KP-I equation. For this equation we prove the existence of steady solutions, which are solitary in one horizontal direction and periodic in the other. We show that such waves bifurcate from the line solitary wave solutions, i.e. solitary wave solutions to the Camassa-Holm equation, in a dimension-breaking bifurcation. This is achieved through reformulating the problem as a dynamical system for a perturbation of the line solitary wave solutions, where the periodic direction takes the role of time, then applying the Lyapunov-Iooss theorem.

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arXiv2023

A resonant Lyapunov centre theorem with an application to doubly periodic travelling hydroelastic waves

We present a Lyapunov centre theorem for an antisymplectically reversible Hamiltonian system exhibiting a nondegenerate $1:1$ or $1:-1$ semisimple resonance as a detuning parameter is varied. The system can be finite- or infinite dimensional (and quasilinear) and have a non-constant symplectic structure. We allow the origin to be a "trivial" eigenvalue arising from a translational symmetry or, in an infinite-dimensional setting, to lie in the continuous spectrum of the linearised Hamiltonian vector field provided a compatibility condition on its range is satisfied. As an application we show how Kirchgässner's spatial dynamics approach can be used to construct doubly periodic travelling waves on the surface of a three-dimensional body of water (of finite or infinite depth) beneath a thin ice sheet (hydroelastic waves). The hydrodynamic problem is formulated as a reversible Hamiltonian system in which an arbitrary horizontal spatial direction is the time-like variable and the infinite-dimensional phase space consists of wave profiles which are periodic (with fixed period) in a second, different horizontal direction. Applying our Lyapunov centre theorem at a point in parameter space associated with a $1:1$ or $1:-1$ semisimple resonance yields a periodic solution of the spatial Hamiltonian system corresponding to a doubly periodic hydroelastic wave.

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arXiv2023

Lump solutions of the fractional Kadomtsev--Petviashvili equation

Of concern are lump solutions for the fractional Kadomtsev--Petviashvili (fKP) equation. As in the classical Kadomtsev--Petviashvili equation, the fKP equation comes in two versions: fKP-I (strong surface tension case) and fKP-II (weak surface tension case). We prove the existence of nontrivial lump solutions for the fKP-I equation in the energy subcritical case $α>\frac{4}{5}$ by means of variational methods. It is already known that there exist neither nontrivial lump solutions belonging to the energy space for the fKP-II equation nor for the fKP-I when $α\leq \frac{4}{5}$. Furthermore, we show that for any $α>\frac{4}{5}$ lump solutions for the fKP-I equation are smooth and decay quadratically at infinity. Numerical experiments are performed for the existence of lump solutions and their decay. Moreover, numerically, we observe cross-sectional symmetry of lump solutions for the fKP-I equation.

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arXiv2022

Traveling waves and transverse instability for the fractional Kadomtsev-Petviashvili equation

Of concern are traveling wave solutions for the fractional Kadomtsev--Petviashvili (fKP) equation. The existence of periodically modulated solitary wave solutions is proved by dimension breaking bifurcation. Moreover, the line solitary wave solutions and their transverse (in)stability are discussed. Analogous to the classical Kadmomtsev--Petviashvili (KP) equation, the fKP equation comes in two versions: fKP-I and fKP-II. We show that the line solitary waves of fKP-I equation are transversely linearly instable. We also perform numerical experiments to observe the (in)stability dynamics of line solitary waves for both fKP-I and fKP-II equations.

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arXiv2021

Existence of Davey--Stewartson type solitary waves for the fully dispersive Kadomtsev--Petviashvilii equation

We prove existence of small-amplitude modulated solitary waves for the full-dispersion Kadomtsev--Petviashvilii (FDKP) equation with weak surface tension. The resulting waves are small-order perturbations of scaled, translated and frequency-shifted solutions of a Davey--Stewartson (DS) type equation. The construction is variational and relies upon a series of reductive steps which transform the FDKP functional to a perturbed scaling of the DS functional, for which least-energy ground states are found. We also establish a convergence result showing that scalings of FDKP solitary waves converge to ground states of the DS functional as the scaling parameter tends to zero. Our method is robust and applies to nonlinear dispersive equations with the properties that (i) their dispersion relation has a global minimum (or maximum) at a non-zero wave number, and (ii) the associated formal weakly nonlinear analysis leads to a DS equation of elliptic-elliptic focussing type. We present full details for the FDKP equation.

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arXiv2021

Solitary wave solutions of a Whitham-Boussinesq system

The travelling wave problem for a particular bidirectional Whitham system modelling surface water waves is under consideration. This system firstly appeared in [Dinvay, Dutykh, Kalisch 2018], where it was numerically shown to be stable and a good approximation to the incompressible Euler equations. In subsequent papers [Dinvay 2018], [Dinvay, Selberg, Tesfahun 2019] the initial-value problem was studied and well-posedness in classical Sobolev spaces was proved. Here we prove existence of solitary wave solutions and provide their asymptotic description. Our proof relies on a variational approach and a concentration-compactness argument. The main difficulties stem from the fact that in the considered Euler-Lagrange equation we have a non-local operator of positive order appearing both in the linear and non-linear parts. Our approach allows us to obtain solitary waves for a particular Boussinesq system as well.

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arXiv2019

Extended lifespan of the fractional BBM equation

For $0<α<1$ and with initial data $\vert\vert u_0\vert\vert_{H^{N+\fracα{2}}}=\varepsilon$, sufficently small, we show that the existence time for solutions of the fractional BBM equation $\partial_tu+\partial_xu+u\partial_xu+\vert\mathrm{D}\vert^α\partial_tu=0$, can be extended beyond the hyperbolic existence time $\frac{1}{\varepsilon}$, to $\frac{1}{\varepsilon^2}$. For the proof we use a modified energy, based on a normal form transformation as in [Hunter, Ifrim, Tataru, Wong, 2015]. In addition we employ ideas and techniques from [Ehrnström, Wang, 2018], in which the authors obtain an enhanced existence time for the fractional KdV equation.

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arXiv2018

Solitary wave solutions to a class of Whitham-Boussinesq systems

In this note we study solitary wave solutions of a class of Whitham-Boussinesq systems which includes the bi-directional Whitham system as a special example. The travelling wave version of the evolution system can be reduced to a single evolution equation, similar to a class of equations studied by Ehrnström, Groves and Wahlén. In that paper the authors prove the existence of solitary wave solutions using a constrained minimization argument adapted to noncoercive functionals, developed by Buffoni, Groves and Wahlén, together with the concentration-compactness principle.

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arXiv2018

Three-dimensional internal gravity-capillary waves in finite depth

We consider three-dimensional inviscid irrotational flow in a two layer fluid under the effects of gravity and surface tension, where the upper fluid is bounded above by a rigid lid and the lower fluid is bounded below by a flat bottom. We use a spatial dynamics approach and formulate the steady Euler equations as an infinite-dimensional Hamiltonian system, where an unbounded spatial direction $x$ is considered as a time-like coordinate. In addition we consider wave motions that are periodic in another direction $z$. By analyzing the dispersion relation we detect several bifurcation scenarios, two of which we study further: a type of $00(\mathrm{i}s)(\mathrm{i}κ_0)$ resonance and a Hamiltonian-Hopf bifurcation. The bifurcations are investigated by performing a center-manifold reduction, which yields a finite-dimensional Hamiltonian system. For this finite-dimensional system we establish the existence of periodic and homoclinic orbits, which correspond to, respectively, doubly periodic travelling waves and oblique travelling waves with a dark or bright solitary wave profile in the $x$-direction. The former are obtained using a variational Lyapunov-Schmidt reduction and the latter by first applying a normal form transformation and then studying the resulting canonical system of equations.

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arXiv2017

Solitary wave solutions to a class of modified Green-Naghdi systems

We provide the existence and asymptotic description of solitary wave solutions to a class of modified Green-Naghdi systems, modeling the propagation of long surface or internal waves. This class was recently proposed by Duch{ê}ne, Israwi and Talhouk (Stud. Appl. Math.,137 (2016)) in order to improve the frequency dispersion of the original Green-Naghdi system while maintaining the same precision. The solitary waves are constructed from the solutions of a constrained minimization problem. The main difficulties stem from the fact that the functional at stake involves low order non-local operators, intertwining multiplications and convolutions through Fourier multipliers.

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arXiv2017

Spatial dynamics methods for solitary waves on a ferrofluid jet

This paper presents existence theories for several families of axisymmetric solitary waves on the surface of an otherwise cylindrical ferrofluid jet surrounding a stationary metal rod. The ferrofluid, which is governed by a general (nonlinear) magnetisation law, is subject to an azimuthal magnetic field generated by an electric current flowing along the rod. The ferrohydrodynamic problem for axisymmetric travelling waves is formulated as an infinite-dimensional Hamiltonian system in which the axial direction is the time-like variable. A centre-manifold reduction technique is employed to reduce the system to a locally equivalent Hamiltonian system with a finite number of degrees of freedom, and homoclinic solutions to the reduced system, which correspond to solitary waves, are detected by dynamical-systems methods.

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arXiv2016

Internal gravity-capillary solitary waves in finite depth

We consider two-dimensional inviscid irrotational flow in a two layer fluid under the effects of gravity and surface tension. The upper fluid is bounded above by a rigid lid, and the lower fluid is bounded below by a rigid bottom. We use a spatial dynamics approach and formulate the steady Euler equations as a Hamiltonian system, where we consider the unbounded horizontal coordinate $x$ as a time-like coordinate. The linearization of the Hamiltonian system is studied, and bifurcation curves in the $(β,α)$-plane are obtained, where $α$ and $β$ are two parameters. The curves depend on two additional parameters $ρ$ and $h$, where $ρ$ is the ratio of the densities and $h$ is the ratio of the fluid depths. However, the bifurcation diagram is found to be qualitatively the same as for surface waves. In particular we find that a Hamiltonian-Hopf bifurcation and a Hamiltonian real 1:1 resonance occur for certain values of $(β,α)$. Of particular interest are solitary wave solutions of the Euler equations. Such solutions correspond to homoclinic solutions of the Hamiltonian system. We investigate the parameter regimes where the Hamiltonian-Hopf bifurcation and the Hamiltonian real 1:1 resonance occur. In both these cases we perform a center manifold reduction of the Hamiltonian system and show that homoclinic solutions of the reduced system exist. In contrast to the case of surface waves we find parameter values $ρ$ and $h$ for which the leading order nonlinear term in the reduced system vanishes. We make a detailed analysis of this phenomenon in the case of the real 1:1 resonance.

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