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We prove existence of small-amplitude modulated solitary waves for the full-dispersion Kadomtsev--Petviashvilii (FDKP) equation with weak surface tension. The resulting waves are small-order perturbations of scaled, translated and frequency-shifted solutions of a Davey--Stewartson (DS) type equation. The construction is variational and relies upon a series of reductive steps which transform the FDKP functional to a perturbed scaling of the DS functional, for which least-energy ground states are found. We also establish a convergence result showing that scalings of FDKP solitary waves converge to ground states of the DS functional as the scaling parameter tends to zero. Our method is robust and applies to nonlinear dispersive equations with the properties that (i) their dispersion relation has a global minimum (or maximum) at a non-zero wave number, and (ii) the associated formal weakly nonlinear analysis leads to a DS equation of elliptic-elliptic focussing type. We present full details for the FDKP equation., Comment: 35 pages

The paper considers the existence of one- or two-hump solutions of a singularly perturbed nonlinear Schrodinger (NLS) equation, which is the standard NLS equation with a third order perturbation. In particular, this equation appears in the field of nonlinear optics, where it is used to describe pulses in optical fibers near a zero dispersion wavelength. It has been shown formally and numerically that the perturbed NLS equation has one- or multi-hump solutions with small oscillations at infinity, called generalized one- or multi-hump solutions. The main purpose of the paper is to provide the first rigorous proof of the existence of generalized one- or two-hump solutions of the singularly perturbed NLS equation. The several invariant properties of the equation, i.e., the translational invariance, the gauge invariance and the reversibility property, are essential to obtain enough free constants to prove the existence. The ideas and methods presented here may be applicable to show existence of generalized 2 k -hump solutions of the equation.

Background & Aims: Golexanolone is a novel small molecule GABA-A receptor-modulating steroid antagonist under development for the treatment of cognitive and vigilance disorders caused by allosteric over-activation of GABA-A receptors by neurosteroids. It restored spatial learning and motor coordination in animal models of hepatic encephalopathy (HE) and mitigated the effects of intravenous allopregnanolone in healthy adults in a dose-dependent fashion. Herein, we report data on the safety, pharmacokinetics (PK) and efficacy of golexanolone in adult patients with cirrhosis. Methods: Following single/multiple ascending dose studies, adults with Child-Pugh A/B cirrhosis and abnormal continuous reaction time (CRT) on screening were randomized to 3 weeks dosing with golexanolone (10, 40 or 80 mg BID) or placebo. CRT, psychometric hepatic encephalopathy score (PHES), animal naming test (ANT), Epworth sleepiness scale (ESS) and electroencephalogram (mean dominant frequency [MDF]; delta+theta/alpha+beta ratio [DT/AB]) were obtained at baseline, 10, and 21 days. Results: Golexanolone exhibited satisfactory safety and PK. Baseline characteristics were similar between the 12 and 33 patients randomized to placebo or golexanolone, respectively. By prespecified analyses, golexanolone was associated with directionally favourable changes vs. placebo in ESS (p = 0.047), MDF (p = 0.142) and DT/AB (p = 0.021). All patients also showed directionally favourable changes in CRT, PHES and ANT, but with no statistical difference between golexanolone and placebo. Post hoc analyses taking into account the variability and improvement in CRT, PHES and ANT observed between screening and baseline suggested an efficacy signal by cognitive measures as well. Conclusion: Golexanolone was well tolerated and associated with improvement in cognitive performance. These results implicate GABA-A receptor-modulating neurosteroids in the pathogenesis of HE and support the therapeutic potential of golexanolone. Lay summary: Many patients with cirrhosis experience subtle but disabling cognitive problems, including sleepiness and poor attention span, that impair their ability to be gainfully employed or carry out activities of daily living. This pilot study tested the hypothesis that these problems with cognition, for which there is no approved treatment, might be improved by an experimental drug, golexanolone, designed to normalize the function of receptors which inhibit brain function. The results of this study suggest that golexanolone is well tolerated and may improve cognition, as reflected by measures of sleepiness, attention span and brain wave activity, paving the way for future larger studies of this promising experimental drug. (C) 2021 The Author(s). Published by Elsevier B.V. on behalf of European Association for the Study of the Liver. Funding Agencies|Umecrine Cognition AB

The travelling wave problem for a particular bidirectional Whitham system modelling surface water waves is under consideration. This system firstly appeared in Dinvay et al. (2019), where it was numerically shown to be stable and a good approximation to the incompressible Euler equations. In subsequent papers (Dinvay, 2019; Dinvay et al., 2019) the initial-value problem was studied and well-posedness in classical Sobolev spaces was proved. Here we prove existence of solitary wave solutions and provide their asymptotic description. Our proof relies on a variational approach and a concentration-compactness argument. The main difficulties stem from the fact that in the considered Euler–Lagrange equation we have a non-local operator of positive order appearing both in the linear and non-linear parts. Our approach allows us to obtain solitary waves for a particular Boussinesq system as well.

OBJECTIVES The prevalence of idiopathic normal pressure hydrocephalus (iNPH) and vascular comorbidity increases with age. It has not been clarified if high age and vascular disease are negative predictors of shunt surgery outcome in patients with iNPH. The aim of this study was to investigate the impact of high age and vascular comorbidity on outcome after shunt surgery in patients with iNPH. METHODS All 332 patients with iNPH who were treated with shunts between 2011 and 2015 at a single centre were consecutively included. Hellstrom iNPH scale, without the neuropsychological tests, was calculated preoperatively and at follow-up 12 months after shunt surgery. Outcome was defined as the difference between the post-operative and preoperative iNPH scale scores. A multivariable model was used to investigate the predictive effects of age and vascular comorbidity on shunt surgery outcome. RESULTS In a multivariable analysis of covariance (ANCOVA) with post-operative outcome as the dependent variable, increasing age (years, B = -0.63, P

In this note, we study solitary wave solutions of a class of Whitham–Boussinesq systems which include the bidirectional Whitham system as a special example. The travelling wave version of the evolution system can be reduced to a single evolution equation, similar to a class of equations studied by Ehrnstrom et al. (Nonlinearity 25:2903–2936, 2012). In that paper, the authors prove the existence of solitary wave solutions using a constrained minimization argument adapted to noncoercive functionals, developed by Buffoni (Arch Ration Mech Anal 173:25–68, 2004), Groves and Wahlen (J Math Fluid Mech 13:593–627, 2011), together with the concentration–compactness principle.

We consider a two-dimensional inviscid irrotational flow in a two layer fluid under the effects of gravity and interfacial tension. The upper fluid is bounded above by a rigid lid, and the lower fluid is bounded below by a rigid bottom. We use a spatial dynamics approach and formulate the steady Euler equations as a Hamiltonian system, where we consider the unbounded horizontal coordinate x as a time-like coordinate. The linearization of the Hamiltonian system is studied, and bifurcation curves in the (β,α)-plane are obtained, where α and β are two parameters. The curves depend on two additional parameters ρ and h, where ρ is the ratio of the densities and h is the ratio of the fluid depths. However, the bifurcation diagram is found to be qualitatively the same as for surface waves. In particular, we find that a Hamiltonian-Hopf bifurcation, Hamiltonian real 1:1 resonance, and a Hamiltonian 02-resonance occur for certain values of (β,α). Of particular interest are solitary wave solutions of the Euler equations. Such solutions correspond to homoclinic solutions of the Hamiltonian system. We investigate the parameter regimes where the Hamiltonian-Hopf bifurcation and the Hamiltonian real 1:1 resonance occur. In both these cases, we perform a center manifold reduction of the Hamiltonian system and show that homoclinic solutions of the reduced system exist. In contrast to the case of surface waves, we find parameter values ρ and h for which the leading order nonlinear term in the reduced system vanishes. We make a detailed analysis of this phenomenon in the case of the real 1:1 resonance. We also briefly consider the Hamiltonian 02-resonance and recover the results found by Kirrmann. (Less)

We consider three-dimensional inviscid irrotational flow in a two layer fluid under the effects of gravity and surface tension, where the upper fluid is bounded above by a rigid lid and the lower fluid is bounded below by a flat bottom. We use a spatial dynamics approach and formulate the steady Euler equations as an infinite-dimensional Hamiltonian system, where an unbounded spatial direction $x$ is considered as a time-like coordinate. In addition we consider wave motions that are periodic in another direction $z$. By analyzing the dispersion relation we detect several bifurcation scenarios, two of which we study further: a type of $00(\mathrm{i}s)(\mathrm{i}\kappa_0)$ resonance and a Hamiltonian-Hopf bifurcation. The bifurcations are investigated by performing a center-manifold reduction, which yields a finite-dimensional Hamiltonian system. For this finite-dimensional system we establish the existence of periodic and homoclinic orbits, which correspond to, respectively, doubly periodic travelling waves and oblique travelling waves with a dark or bright solitary wave profile in the $x$-direction. The former are obtained using a variational Lyapunov-Schmidt reduction and the latter by first applying a normal form transformation and then studying the resulting canonical system of equations., Comment: 34 pages

This paper presents existence theories for several families of axisymmetric solitary waves on the surface of an otherwise cylindrical ferrofluid jet surrounding a stationary metal rod. The ferrofluid, which is governed by a general (nonlinear) magnetisation law, is subject to an azimuthal magnetic field generated by an electric current flowing along the rod. The ferrohydrodynamic problem for axisymmetric travelling waves is formulated as an infinite-dimensional Hamiltonian system in which the axial direction is the time-like variable. A centre-manifold reduction technique is employed to reduce the system to a locally equivalent Hamiltonian system with a finite number of degrees of freedom, and homoclinic solutions to the reduced system, which correspond to solitary waves, are detected by dynamical-systems methods.