arXiv subjects
Publications and source records attributed to Martin Kohlmann.
In this paper, we study a weakly dissipative variant of the periodic Degasperis-Procesi equation. We show the local well-posedness of the associated Cauchy problem in $H^s(§)$, $s>3/2$, and discuss the precise blow-up scenario for $s=3$. We also present explicit examples for globally existing solutions and blow-up.
We consider the periodic $\muDP$ equation (a modified version of the Degasperis-Procesi equation) as the geodesic flow of a right-invariant affine connection $\nabla$ on the Fréchet Lie group $\Diff^{\infty}(§^1)$ of all smooth and orientation-preserving diffeomorphisms of the circle $§^1=\R/\Z$. On the Lie algebra $\C^{\infty}(§^1)$ of $\Diff^{\infty}(§^1)$, this connection is canonically given by the sum of the Lie bracket and a bilinear operator. For smooth initial data, we show the short time existence of a smooth solution of $\muDP$ which depends smoothly on time and on the initial data. Furthermore, we prove that the exponential map defined by $\nabla$ is a smooth local diffeomorphism of a neighbourhood of zero in $\C^{\infty}(§^1)$ onto a neighbourhood of the unit element in $\Diff^{\infty}(§^1)$. Our results follow from a general approach on non-metric Euler equations on Lie groups, a Banach space approximation of the Fréchet space $\C^{\infty}(§^1)$, and a sharp spatial regularity result for the geodesic flow.
We use geometric methods to study two natural two-component generalizations of the periodic Camassa-Holm and Degasperis-Procesi equations. We show that these generalizations can be regarded as geodesic equations on the semidirect product of the diffeomorphism group of the circle $\Diff(S^1)$ with some space of sufficiently smooth functions on the circle. Our goals are to understand the geometric properties of these two-component systems and to prove local well-posedness in various function spaces. Furthermore, we perform some explicit curvature calculations for the two-component Camassa-Holm equation, giving explicit examples of large subspaces of positive curvature.
In this paper, we study two-component versions of the periodic Hunter-Saxton equation and its $μ$-variant. Considering both equations as a geodesic flow on the semidirect product of the circle diffeomorphism group $\Diff(§)$ with a space of scalar functions on $§$ we show that both equations are locally well-posed. The main result of the paper is that the sectional curvature associated with the 2HS is constant and positive and that 2$μ$HS allows for a large subspace of positive sectional curvature. The issues of this paper are related to some of the results for 2CH and 2DP presented in [J. Escher, M. Kohlmann, and J. Lenells, J. Geom. Phys. 61 (2011), 436-452].
In this paper, we study the $μ$-variant of the periodic $b$-equation and show that this equation can be realized as a metric Euler equation on the Lie group $\Diff^{\infty}(§)$ if and only if $b=2$ (for which it becomes the $μ$-Camassa-Holm equation). In this case, the inertia operator generating the metric on $\Diff^{\infty}(§)$ is given by $L=μ-\partial_x^2$. In contrast, the $μ$-Degasperis-Procesi equation (obtained for $b=3$) is not a metric Euler equation on $\Diff^{\infty}(§)$ for any regular inertia operator $A\in\mathcal L_{\text{is}}^{\text{sym}}(C^{\infty}(§))$. The paper generalizes some recent results of [J. Escher and B. Kolev, DOI 10.1007/s00209-010-0778-2], [J. Escher and J. Seiler, J. Math. Phys. 51 (2010), 053101.1-053101.6] and [B. Kolev, Wave Motion 46 (2009), 412-419].
As a simple model for lattice defects like grain boundaries in solid state physics we consider potentials which are obtained from a periodic potential $V = V(x,y)$ on $\R^2$ with period lattice $\Z^2$ by setting $W_t(x,y) = V(x+t,y)$ for $x < 0$ and $W_t(x,y) = V(x,y)$ for $x \ge 0$, for $t \in [0,1]$. For Lipschitz-continuous $V$ it is shown that the Schrödinger operators $H_t = -Δ+ W_t$ have spectrum (surface states) in the spectral gaps of $H_0$, for suitable $t \in (0,1)$. We also discuss the density of these surface states as compared to the density of the bulk. Our approach is variational and it is first applied to the well-known dislocation problem [E. Korotyaev, Commun. Math. Phys. 213 (2000), 471-489], [E. Korotyaev, Asymptotic Anal. 45 (2005), 73-97] on the real line. We then proceed to the dislocation problem for an infinite strip and for the plane. In an appendix, we discuss regularity properties of the eigenvalue branches in the one-dimensional dislocation problem for suitable classes of potentials.