Finite-gap Solutions of the Vortex Filament Equation: Isoperiodic Deformations

We study the topology of quasiperiodic solutions of the vortex filament equation in a neighborhood of multiply covered circles. We construct these solutions by means of a sequence of isoperiodic deformations, at each step of which a real double point is "unpinched" to produce a new pair of branch points and therefore a solution of higher genus. We prove that every step in this process corresponds to a cabling operation on the previous curve, and we provide a labelling scheme that matches the deformation data with the knot type of the resulting filament.Comment: 33 pages, 5 figures; submitted to Journal of Nonlinear Scienc

The Effects of Viscosity on the Linear Stability of Damped Stokes Waves, Downshifting, and Rogue Wave Generation

We investigate a higher order nonlinear Schrodinger equation with linear damping and weak viscosity, recently proposed as a model for deep water waves exhibiting frequency downshifting. Through analysis and numerical simulations, we discuss how the viscosity affects the linear stability of the Stokes wave solution, enhances rogue wave formation, and leads to permanent downshift in the spectral peak. In particular, we study the wave evolution over short-to-moderate time scales, when most rogue wave activity occurs, and explain the transition of the perturbed solution from the initial Benjamin-Feir instability to a predominantly oscillatory behavior. Finally, we determine the mechanism and timing of permanent downshift in the spectral peak and its relation to the location of the global minimum of the momentum and the magnitude of its second derivative

Topological Aspect of Knotted Vortex Filaments in Excitable Media

Scroll waves exist ubiquitously in three-dimensional excitable media. It's rotation center can be regarded as a topological object called vortex filament. In three-dimensional space, the vortex filaments usually form closed loops, and even linked and knotted. In this letter, we give a rigorous topological description of knotted vortex filaments. By using the $\phi$-mapping topological current theory, we rewrite the topological current form of the charge density of vortex filaments and use this topological current we reveal that the Hopf invariant of vortex filaments is just the sum of the linking and self-linking numbers of the knotted vortex filaments. We think that the precise expression of the Hopf invariant may imply a new topological constraint on knotted vortex filaments.Comment: 4 pages, no figures, Accepted by Chin. Phys. Let

Numerical investigation of stability of breather-type solutions of the nonlinear Schrödinger equation

In this article we conduct a broad numerical investigation of stability of breather-type solutions of the nonlinear Schrödinger (NLS) equation, a widely used model of rogue wave generation and dynamics in deep water. NLS breathers rising over an unstable background state are frequently used to model rogue waves. However, the issue of whether these solutions are robust with respect to the kind of random perturbations occurring in physical settings and laboratory experiments has just recently begun to be addressed. Numerical experiments for spatially periodic breathers with one or two modes involving large ensembles of perturbed initial data for six typical random perturbations suggest interesting conclusions. Breathers over an unstable background with N unstable modes are generally unstable to small perturbations in the initial data unless they are "maximal breathers" (i.e., they have N spatial modes). Additionally, among the maximal breathers with two spatial modes, the one of highest amplitude due to coalescence of the modes appears to be the most robust. The numerical observations support and extend to more realistic settings the results of our previous stability analysis, which we hope will provide a useful tool for identifying physically realizable wave forms in experimental and observational studies of rogue waves

Symptomatic Uncomplicated Diverticular Disease (SUDD): Practical Guidance and Challenges for Clinical Management

Symptomatic Uncomplicated Diverticular Disease (SUDD) is a syndrome within the diverticular disease spectrum, characterized by local abdominal pain with bowel movement changes but without systemic inflammation. This narrative review reports current knowledge, delivers practical guidance, and reveals challenges for the clinical management of SUDD. A broad and common consensus on the definition of SUDD is still needed. However, it is mainly considered a chronic condition that impairs quality of life (QoL) and is characterized by persistent left lower quadrant abdominal pain with bowel movement changes (eg, diarrhea) and low-grade inflammation (eg, elevated calprotectin) but without systemic inflammation. Age, genetic predisposition, obesity, physical inactivity, low-fiber diet, and smoking are considered risk factors. The pathogenesis of SUDD is not entirely clarified. It seems to result from an interaction between fecal microbiota alterations, neuro-immune enteric interactions, and muscular system dysfunction associated with a low-grade and local inflammatory state. At diagnosis, it is essential to assess baseline clinical and Quality of Life (QoL) scores to evaluate treatment efficacy and, ideally, to enroll patients in cohort studies, clinical trials, or registries. SUDD treatments aim to improve symptoms and QoL, prevent recurrence, and avoid disease progression and complications. An overall healthy lifestyle – physical activity and a high-fiber diet, with a focus on whole grains, fruits, and vegetables – is encouraged. Probiotics could effectively reduce symptoms in patients with SUDD, but their utility is missing adequate evidence. Using Rifaximin plus fiber and Mesalazine offers potential in controlling symptoms in patients with SUDD and might prevent acute diverticulitis. Surgery could be considered in patients with medical treatment failure and persistently impaired QoL. Still, studies with well-defined diagnostic criteria for SUDD that evaluate the safety, QoL, effectiveness, and cost-effectiveness of these interventions using standard scores and comparable outcomes are needed

Numerical instability of the Akhmediev breather and a finite-gap model of it

In this paper we study the numerical instabilities of the NLS Akhmediev breather, the simplest space periodic, one-mode perturbation of the unstable background, limiting our considerations to the simplest case of one unstable mode. In agreement with recent theoretical findings of the authors, in the situation in which the round-off errors are negligible with respect to the perturbations due to the discrete scheme used in the numerical experiments, the split-step Fourier method (SSFM), the numerical output is well-described by a suitable genus 2 finite-gap solution of NLS. This solution can be written in terms of different elementary functions in different time regions and, ultimately, it shows an exact recurrence of rogue waves described, at each appearance, by the Akhmediev breather. We discover a remarkable empirical formula connecting the recurrence time with the number of time steps used in the SSFM and, via our recent theoretical findings, we establish that the SSFM opens up a vertical unstable gap whose length can be computed with high accuracy, and is proportional to the inverse of the square of the number of time steps used in the SSFM. This neat picture essentially changes when the round-off error is sufficiently large. Indeed experiments in standard double precision show serious instabilities in both the periods and phases of the recurrence. In contrast with it, as predicted by the theory, replacing the exact Akhmediev Cauchy datum by its first harmonic approximation, we only slightly modify the numerical output. Let us also remark, that the first rogue wave appearance is completely stable in all experiments and is in perfect agreement with the Akhmediev formula and with the theoretical prediction in terms of the Cauchy data.Comment: 27 pages, 8 figures, Formula (30) at page 11 was corrected, arXiv admin note: text overlap with arXiv:1707.0565

The branch processes of vortex filaments and Hopf Invariant Constraint on Scroll Wave

In this paper, by making use of Duan's topological current theory, the evolution of the vortex filaments in excitable media is discussed in detail. The vortex filaments are found generating or annihilating at the limit points and encountering, splitting, or merging at the bifurcation points of a complex function $Z(\vec{x},t)$. It is also shown that the Hopf invariant of knotted scroll wave filaments is preserved in the branch processes (splitting, merging, or encountering) during the evolution of these knotted scroll wave filaments. Furthermore, it also revealed that the "exclusion principle" in some chemical media is just the special case of the Hopf invariant constraint, and during the branch processes the "exclusion principle" is also protected by topology.Comment: 9 pages, 5 figure

Giant Magnons and Singular Curves

We obtain the giant magnon of Hofman-Maldacena and its dyonic generalisation on R x S^3 < AdS_5 x S^5 from the general elliptic finite-gap solution by degenerating its elliptic spectral curve into a singular curve. This alternate description of giant magnons as finite-gap solutions associated to singular curves is related through a symplectic transformation to their already established description in terms of condensate cuts developed in hep-th/0606145.Comment: 34 pages, 17 figures, minor change in abstrac

Gross-Neveu Models, Nonlinear Dirac Equations, Surfaces and Strings

Recent studies of the thermodynamic phase diagrams of the Gross-Neveu model (GN2), and its chiral cousin, the NJL2 model, have shown that there are phases with inhomogeneous crystalline condensates. These (static) condensates can be found analytically because the relevant Hartree-Fock and gap equations can be reduced to the nonlinear Schr\"odinger equation, whose deformations are governed by the mKdV and AKNS integrable hierarchies, respectively. Recently, Thies et al have shown that time-dependent Hartree-Fock solutions describing baryon scattering in the massless GN2 model satisfy the Sinh-Gordon equation, and can be mapped directly to classical string solutions in AdS3. Here we propose a geometric perspective for this result, based on the generalized Weierstrass spinor representation for the embedding of 2d surfaces into 3d spaces, which explains why these well-known integrable systems underlie these various Gross-Neveu gap equations, and why there should be a connection to classical string theory solutions. This geometric viewpoint may be useful for higher dimensional models, where the relevant integrable hierarchies include the Davey-Stewartson and Novikov-Veselov systems.Comment: 27 pages, 1 figur