arXiv subjects
Publications and source records attributed to Thomas Kappeler.
We investigate the spectrum of the Lax operator $L_u$ of the Benjamin-Ono equation on the torus for complex valued potentials $u$ in the Sobolev space $H^{-s}(\mathbb{T},\mathbb{C})$, $0 \le s < 1/2$, with small imaginary part and prove analytic properties of the moment map, defined in terms of spectral data of $L_u$.
Near an arbitrary finite gap potential we construct real analytic, canonical coordinates for the Benjamin-Ono equation on the torus having the following two main properties: (1) up to a remainder term, which is smoothing to any given order, the coordinate transformation is a pseudo-differential operator of order 0 with principal part given by a modified Fourier transform (modification by a phase factor) and (2) the pullback of the Hamiltonian of the Benjamin-Ono is in normal form up to order three and the corresponding Hamiltonian vector field admits an expansion in terms of para-differential operators. Such coordinates are a key ingredient for studying the stability of finite gap solutions of the Benjamin-Ono equation under small, quasi-linear perturbations.
We prove smoothing properties of the solutions of the Benjamin-Ono equation in the Sobolev space $H^{s}(\mathbb{T},\mathbb{R})$ for any $s\ge 0$. To this end we show that Tao's gauge transform is a high frequency approximation of the nonlinear Fourier transform $Φ$ for the Benjamin-Ono equation, constructed in our previous work. The results of this paper are manifestations of the quasi-linear character of the Benjamin-Ono equation.
In this paper we obtain the following stability result for periodic multi-solitons of the KdV equation: We prove that under any given semilinear Hamiltonian perturbation of small size $\varepsilon > 0$, a large class of periodic multi-solitons of the KdV equation, including ones of large amplitude, are orbitally stable for a time interval of length at least $O(\varepsilon^{-2})$. To the best of our knowledge, this is the first stability result of such type for periodic multi-solitons of large size of an integrable PDE.
In this paper, we survey our recent results on the Benjamin-Ono equation on the torus. As an application of the methods developed we construct large families of periodic or quasiperiodic solutions, which are not $C^\infty$-smooth.
Two-sided bounds for the efficiency of the torsion function are obtained in terms of the square of the distance to the boundary function under the hypothesis that the Dirichlet Laplacian satisfies a strong Hardy inequality. Localisation properties of the torsion function are obtained under that hypothesis. An example is analysed in detail.
The spectral curve associated with the sinh-Gordon equation on the torus is defined interms of the spectrum of the Lax operator appearing in the Lax pair formulation of the equation. If thespectrum is simple, it is an open Riemann surface of infinite genus. In this paper we construct normalizeddifferentials and derive estimates for the location of their zeroes.
We prove an infinite dimensional version of the Arnold-Liouville theorem for integrable non-linear PDEs: In a case study we consider the {\em focusing} NLS equation with periodic boundary conditions.
We prove that for any $0 < s < 1/2$, the Benjamin--Ono equation on the torus is globally in time $C^0-$well-posed on the Sobolev space $H^{-s}(\T, \R)$,in the sense that the solution map, which is known to be defined for smooth data, continuously extends to $H^{-s}(\T,\R)$. The solution map does not extend continuously to $H^{-s}(\T, \R)$ with $s > 1/2$. Hence the critical Sobolev exponent $s_c=-1/2$ of the Benjamin--Ono equation is the threshold for well-posedness on the torus. The obtained solutions are almost periodic in time. Furthermore, we prove that the traveling wave solutions of the Benjamin--Ono equation on the torus are orbitally stable in $H^{-s}(\T,\R)$ for any $0\le s<1/2$.
In this paper we prove that the Benjamin-Ono equation, when considered on the torus, is an integrable (pseudo)differential equation in the strongest possible sense: it admits global Birkhoff coordinates on the space $L^2(\T)$. These are coordinates which allow to integrate it by quadrature and hence are also referred to as nonlinear Fourier coefficients. As a consequence, all the $L^2(\T)$ solutions of the Benjamin--Ono equation are almost periodic functions of the time variable. The construction of such coordinates relies on the spectral study of the Lax operator in the Lax pair formulation of the Benjamin--Ono equation and on the use of a generating functional, which encodes the entire Benjamin--Ono hierarchy.
In this paper we prove the persistence of space periodic multi-solitons of arbitrary size under any quasi-linear Hamiltonian perturbation, which is smooth and sufficiently small. This answers positively a longstanding question whether KAM techniques can be further developed to prove the existence of quasi-periodic solutions of arbitrary size of strongly nonlinear perturbations of integrable PDEs.
Near an arbitrary finite gap potential we construct real analytic, canonical coordinates for the KdV equation on the torus having the following two main properties: (1) up to a remainder term, which is smoothing to any given order, the coordinate transformation is a pseudodifferential operator of order 0 with principal part given by the Fourier transform and (2) the pullback of the KdV Hamiltonian is in normal form up to order three and the corresponding Hamiltonian vector field admits an expansion in terms of a paradifferential operator. Such coordinates are a key ingredient for studying the stability of finite gap solutions of the KdV equation under small, quasi-linear perturbations.
Bounds are obtained for the $L^p$ norm of the torsion function $v_Ω$, i.e. the solution of $-Δv=1,\, v\in H_0^1(Ω),$ in terms of the Lebesgue measure of $Ω$ and the principal eigenvalue $λ_1(Ω)$ of the Dirichlet Laplacian acting in $L^2(Ω)$. We show that these bounds are sharp for $1\le p\le 2$.
We study the maps induced on link Floer homology by elementary decorated link cobordisms. We compute these for births, deaths, stabilizations, and destabilizations, and show that saddle cobordisms can be computed in terms of maps in a decorated skein exact triangle that extends the oriented skein exact triangle in knot Floer homology. In particular, we completely determine the Alexander and Maslov grading shifts. As a corollary, we compute the maps induced by elementary cobordisms between unlinks. We show that these give rise to a $(1+1)$-dimensional TQFT that coincides with the reduced Khovanov TQFT. Hence, when applied to the cube of resolutions of a marked link diagram, it gives the complex defining the reduced Khovanov homology of the knot. Finally, we define a spectral sequence from (reduced) Khovanov homology using these cobordism maps, and we prove that it is an invariant of the (marked) link.
In a case study for integrable PDEs, we construct real analytic, canonical coordinates for the defocusing NLS equation on the circle, specifically taylored towards the needs in perturbation theory. They are defined in neighbourhoods of families of finite dimensional invariant tori and are shown to satisfy together with their derivatives tame estimates. When expressed in these coordinates, the dNLS Hamiltonian is in normal form up to order three.
We prove that the renormalized defocusing mKdV equation on the circle is locally in time $C^{0}$-wellposed on the Fourier Lebesgue space ${\mathcal{F}\ell}^p$ for any $2 < p < \infty$. The result implies that the defocusing mKdV equation itself is illposed on these spaces since the renormalizing phase factor becomes infinite. The proof is based on the fact that the mKdV equation is an integrable PDE whose Hamiltonian is in the NLS hierarchy. A key ingredient is a novel way of representing the bi-infinite sequence of frequencies of the renormalized defocusing mKdV equation, allowing to analytically extend them to ${\mathcal{F}\ell}^p$ for any $2 \le p < \infty$ and to deduce asymptotics for $n \to \pm \infty$.
In form of a case study for the KdV and the KdV2 equations, we present a novel approach of representing the frequencies of integrable PDEs which allows to extend them analytically to spaces of low regularity and to study their asymptotics. Applications include convexity properties of the Hamiltonians and wellposedness results in spaces of low regularity. In particular, it is proved that on $H^{s}$ the KdV2 equation is $C^{0}$-wellposed if $s\ge 0$ and illposed (in a strong sense) if $s < 0$.
We prove that small, semi-linear Hamiltonian perturbations of the defocusing nonlinear Schrödinger (dNLS) equation on the circle have an abundance of invariant tori of any size and (finite) dimension which support quasi-periodic solutions. When compared with previous results the novelty consists in considering perturbations which do not satisfy any symmetry condition (they may depend on $x$ in an arbitrary way) and need not be analytic. The main difficulty is posed by pairs of almost resonant dNLS frequencies. The proof is based on the integrability of the dNLS equation, in particular the fact that the nonlinear part of the Birkhoff coordinates is one smoothing. We implement a Newton-Nash-Moser iteration scheme to construct the invariant tori. The key point is the reduction of linearized operators, coming up in the iteration scheme, to $ 2 \times 2 $ block diagonal ones with constant coefficients together with sharp asymptotic estimates of their eigenvalues.