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Jonatan Lenells

Publications and source records attributed to Jonatan Lenells.

88 records · Page 5Linked to original sources
arXiv2011

The geometry of the two-component Camassa-Holm and Degasperis-Procesi equations

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.

math.AP↗
arXiv2011

Geometry of diffeomorphism groups, complete integrability and optimal transport

We study the geometry of the space of densities $\VolM$, which is the quotient space $\Diff(M)/\Diff_μ(M)$ of the diffeomorphism group of a compact manifold $M$ by the subgroup of volume-preserving diffemorphisms, endowed with a right-invariant homogeneous Sobolev $\dot{H}^1$-metric. We construct an explicit isometry from this space to (a subset of) an infinite-dimensional sphere and show that the associated Euler-Arnold equation is a completely integrable system in any space dimension. We also prove that its smooth solutions break down in finite time. Furthermore, we show that the $\dot{H}^1$-metric induces the Fisher-Rao (information) metric on the space of probability distributions, and thus its Riemannian distance is the spherical version of Hellinger distance. We compare it to the Wasserstein distance in optimal transport which is induced by an $L^2$-metric on $\Diff(M)$. The $\dot{H}^1$ geometry we introduce in this paper can be seen as an infinite-dimensional version of the geometric theory of statistical manifolds.

math.DG↗
arXiv2010

Stability of the $μ$-Camassa-Holm Peakons

The $μ$-Camassa-Holm ($μ$CH) equation is a nonlinear integrable partial differential equation closely related to the Camassa-Holm equation. We prove that the periodic peaked traveling wave solutions (peakons) of the $μ$CH equation are orbitally stable.

math.AP↗
arXiv2010

An integrable generalization of the sine-Gordon equation on the half-line

We analyze a generalization of the sine-Gordon equation in laboratory coordinates on the half-line. Using the Fokas transform method for the analysis of initial-boundary value problems for integrable PDEs, we show that the solution $u(x,t)$ can be constructed from the initial and boundary values via the solution of a $2\times 2$-matrix Riemann-Hilbert problem.

nlin.SI↗
arXiv2010

The solution of the global relation for the derivative nonlinear Schrödinger equation on the half-line

We consider initial-boundary value problems for the derivative nonlinear Schrödinger (DNLS) equation on the half-line $x > 0$. In a previous work, we showed that the solution $q(x,t)$ can be expressed in terms of the solution of a Riemann-Hilbert problem with jump condition specified by the initial and boundary values of $q(x,t)$. However, for a well-posed problem, only part of the boundary values can be prescribed; the remaining boundary data cannot be independently specified, but are determined by the so-called global relation. In general, an effective solution of the problem therefore requires solving the global relation. Here, we present the solution of the global relation in terms of the solution of a system of nonlinear integral equations. This also provides a construction of the Dirichlet-to-Neumann map for the DNLS equation on the half-line.

nlin.SI↗
arXiv2010

Dressing for a novel integrable generalization of the nonlinear Schrödinger equation

We implement the dressing method for a novel integrable generalization of the nonlinear Schrödinger equation. As an application, explicit formulas for the $N$-soliton solutions are derived. As a by-product of the analysis, we find a simplification of the formulas for the $N$-solitons of the derivative nonlinear Schrödinger equation given by Huang and Chen.

nlin.SI↗
arXiv2010

Boundary value problems for the stationary axisymmetric Einstein equations: a disk rotating around a black hole

We solve a class of boundary value problems for the stationary axisymmetric Einstein equations corresponding to a disk of dust rotating uniformly around a central black hole. The solutions are given explicitly in terms of theta functions on a family of hyperelliptic Riemann surfaces of genus 4. In the absence of a disk, they reduce to the Kerr black hole. In the absence of a black hole, they reduce to the Neugebauer-Meinel disk.

nlin.SI↗
arXiv2009

Integrable evolution equations on spaces of tensor densities and their peakon solutions

We study a family of equations defined on the space of tensor densities of weight $λ$ on the circle and introduce two integrable PDE. One of the equations turns out to be closely related to the inviscid Burgers equation while the other has not been identified in any form before. We present their Lax pair formulations and describe their bihamiltonian structures. We prove local wellposedness of the corresponding Cauchy problem and include results on blow-up as well as global existence of solutions. Moreover, we construct "peakon" and "multi-peakon" solutions for all $λ\neq 0,1$, and "shock-peakons" for $λ= 3$. We argue that there is a natural geometric framework for these equations that includes other well-known integrable equations and which is based on V. Arnold's approach to Euler equations on Lie groups.

math.AP↗
arXiv2009

Exact Results for Perturbative Chern-Simons Theory with Complex Gauge Group

We develop several methods that allow us to compute all-loop partition functions in perturbative Chern-Simons theory with complex gauge group G_C, sometimes in multiple ways. In the background of a non-abelian irreducible flat connection, perturbative G_C invariants turn out to be interesting topological invariants, which are very different from finite type (Vassiliev) invariants obtained in a theory with compact gauge group G. We explore various aspects of these invariants and present an example where we compute them explicitly to high loop order. We also introduce a notion of "arithmetic TQFT" and conjecture (with supporting numerical evidence) that SL(2,C) Chern-Simons theory is an example of such a theory.

hep-th↗
arXiv2009

On the N=2 Supersymmetric Camassa-Holm and Hunter-Saxton Equations

We consider N=2 supersymmetric extensions of the Camassa-Holm and Hunter-Saxton equations. We show that they admit geometric interpretations as Euler equations on the superconformal algebra of contact vector fields on the 1|2-dimensional supercircle. We use the bi-Hamiltonian formulation to derive Lax pairs. Moreover, we present some simple examples of explicit solutions. As a by-product of our analysis we obtain a description of the bounded traveling-wave solutions for the two-component Hunter-Saxton equation.

nlin.SI↗
arXiv2008

Exactly solvable model for nonlinear pulse propagation in optical fibers

The nonlinear Schrödinger (NLS) equation is a fundamental model for the nonlinear propagation of light pulses in optical fibers. We consider an integrable generalization of the NLS equation which was first derived by means of bi-Hamiltonian methods in [A. S. Fokas, {\it Phys. D} {\bf 87} (1995), 145--150]. The purpose of the present paper is threefold: (a) We show how this generalized NLS equation arises as a model for nonlinear pulse propagation in monomode optical fibers when certain higher-order nonlinear effects are taken into account; (b) We show that the equation is equivalent, up to a simple change of variables, to the first negative member of the integrable hierarchy associated with the derivative nonlinear Schrödinger equation; (c) We analyze traveling-wave solutions.

physics.optics↗
arXiv2008

The derivative nonlinear Schrödinger equation on the half-line

We analyze the derivative nonlinear Schrödinger equation $iq_t + q_{xx} = i(|q|^2q)_x$ on the half-line using the Fokas method. Assuming that the solution $q(x,t)$ exists, we show that it can be represented in terms of the solution of a matrix Riemann-Hilbert problem formulated in the plane of the complex spectral parameter $ζ$. The jump matrix has explicit $x,t$ dependence and is given in terms of the spectral functions $a(ζ)$, $b(ζ)$ (obtained from the initial data $q_0(x) = q(x,0)$) as well as $A(ζ)$, $B(ζ)$ (obtained from the boundary values $g_0(t) = q(0,t)$ and $g_1(t) = q_x(0,t)$). The spectral functions are not independent, but related by a compatibility condition, the so-called global relation. Given initial and boundary values $\{q_0(x), g_0(t), g_1(t)\}$ such that there exist spectral function satisfying the global relation, we show that the function $q(x,t)$ defined by the above Riemann-Hilbert problem exists globally and solves the derivative nonlinear Schrödinger equation with the prescribed initial and boundary values.

nlin.SI↗
arXiv2008

A bi-Hamiltonian supersymmetric geodesic equation

A supersymmetric extension of the Hunter-Saxton equation is constructed. We present its bi-Hamiltonian structure and show that it arises geometrically as a geodesic equation on the space of superdiffeomorphisms of the circle that leave a point fixed endowed with a right-invariant metric.

math-ph↗
arXiv2008

Generalized Hunter-Saxton equation and the geometry of the group of circle diffeomorphisms

We study an equation lying `mid-way' between the periodic Hunter-Saxton and Camassa-Holm equations, and which describes evolution of rotators in liquid crystals with external magnetic field and self-interaction. We prove that it is an Euler equation on the diffeomorphism group of the circle corresponding to a natural right-invariant Sobolev metric. We show that the equation is bihamiltonian and admits both cusped, as well as smooth, traveling-wave solutions which are natural candidates for solitons. We also prove that it is locally well-posed and establish results on the lifespan of its solutions. Throughout the paper we argue that despite similarities to the KdV, CH and HS equations, the new equation manifests several distinctive features that set it apart from the other three.

math.DG↗