Infinite propagation speed of the Camassa-Holm equation

We use the inverse scattering transform to show that a solution of the Camassa-Holm equation is identically zero whenever it vanishes on two horizontal half-lines in the x-t space. In particular, a solution that has compact support at two different times vanishes everywhere, proving that the Camassa-Holm equation has infinite propagation speed.     

A two-component geodesic equation on a space of constant positive curvature

We propose a new two-component geodesic equation with the unusual property that the underlying space has constant positive curvature. In the special case of one space dimension, the equation reduces to the two-component Hunter–Saxton equation.     

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.     

Stable hypersurfaces via the first eigenvalue of the anisotropic Laplacian operator

In this paper we provide a characterization for stable hypersurfaces with constant anisotropic mean curvature immersed in the Euclidean space \(\mathbb {R}^{n+1}\) through the analysis of the first eigenvalue of the anisotropic Laplacian operator.     

The derivative nonlinear Schrödinger equation on the half-line

We analyze the derivative nonlinear Schrödinger equation 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 q₀(x)=q(x,0)) as well as A(ζ), B(ζ) (obtained from the boundary values g₀(t)=q(0,t) and g₁(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 {q₀(x),g₀(t),g₁(t)} such that there exist spectral functions 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.     

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.      

Riemannian geometry on the diffeomorphism group of the circle

The topological group of diffeomorphisms of the unit circle of Sobolev class H ^k , for k large enough, is a Banach manifold modeled on the Hilbert space . In this paper we show that the H ¹ right-invariant metric obtained by right-translation of the H ¹ inner product on defines a smooth Riemannian metric on , and we explicitly construct a compatible smooth affine connection. Once this framework has been established results from the general theory of affine connections on Banach manifolds can be applied to study the exponential map, geodesic flow, parallel translation, curvature etc. The diffeomorphism group of the circle provides the natural geometric setting for the Camassa–Holm equation – a nonlinear wave equation that has attracted much attention in recent years – and in this context it has been remarked in various papers how to construct a smooth Riemannian structure compatible with the H ¹ right-invariant metric. We give a self-contained presentation that can serve as a detailed mathematical foundation for the future study of geometric aspects of the Camassa–Holm equation.     

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 ≠ 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.     

Toward the computer-aided discovery of FabH inhibitors. Do predictive QSAR models ensure high quality virtual screening performance?

Antibiotic resistance has increased over the past two decades. New approaches for the discovery of novel antibacterials are required and innovative strategies will be necessary to identify novel and effective candidates. Related to this problem, the exploration of bacterial targets that remain unexploited by the current antibiotics in clinical use is required. One of such targets is the \(\beta \) -ketoacyl-acyl carrier protein synthase III (FabH). Here, we report a ligand-based modeling methodology for the virtual-screening of large collections of chemical compounds in the search of potential FabH inhibitors. QSAR models are developed for a diverse dataset of 296 FabH inhibitors using an in-house modeling framework. All models showed high fitting, robustness, and generalization capabilities. We further investigated the performance of the developed models in a virtual screening scenario. To carry out this investigation, we implemented a desirability-based algorithm for decoys selection that was shown effective in the selection of high quality decoys sets. Once the QSAR models were validated in the context of a virtual screening experiment their limitations arise. For this reason, we explored the potential of ensemble modeling to overcome the limitations associated to the use of single classifiers. Through a detailed evaluation of the virtual screening performance of ensemble models it was evidenced, for the first time to our knowledge, the benefits of this approach in a virtual screening scenario. From all the obtained results, we could arrive to a significant main conclusion: at least for FabH inhibitors, virtual screening performance is not guaranteed by predictive QSAR models.