Stabilization of 1D solitons by fractional derivatives in systems with quintic nonlinearity

We study theoretically the properties of a soliton solution of the fractional Schrödinger equation with quintic nonlinearity. Under “fractional” we understand the Schrödinger equation, where ordinary Laplacian (second spatial derivative in 1D) is substituted by its fractional counterpart with Lévy index $$\alpha$$ α . We speculate that the latter substitution corresponds to phenomenological account for disorder in a system. Using analytical (variational and perturbative) and numerical arguments, we have shown that while in the case of Schrödinger equation with the ordinary Laplacian (corresponding to Lévy index $$\alpha =2$$ α = 2 ), the soliton is unstable, even infinitesimal difference $$\alpha$$ α from 2 immediately stabilizes the soliton texture. Our analytical and numerical investigations of $$\omega (N)$$ ω ( N ) dependence ($$\omega$$ ω is soliton frequency and N its mass) show (within the famous Vakhitov–Kolokolov criterion) the stability of our soliton texture in the fractional $$\alpha <2$$ α < 2 case. Direct numerical analysis of the linear stability problem of soliton texture also confirms this point. We show analytically and numerically that fractional Schrödinger equation with quintic nonlinearity admits the existence of (stable) soliton textures at $$2/3<\alpha <2$$ 2 / 3 < α < 2 , which is in accord with existing literature data. These results may be relevant to both Bose–Einstein condensates in cold atomic gases and optical solitons in the disordered media.

Stabilization of 1D Solitons By Fractional Derivatives In Systems With Quintic Nonlinearity

We study theoretically the properties of a soliton solution of the fractional Schrödinger equation with quintic nonlinearity. Under ”fractional” we understand the Schrödinger equation, where ordinary Laplacian (second spatial derivative in 1D) is substituted by its fractional counterpart with Lévy index α. We speculate that the latter substitution corresponds to phenomenological account for disorder in a system. Using analytical (variational and perturbative) and numerical arguments, we have shown that while in the case of Schrödinger equation with the ordinary Lapla-cian (corresponding to Lévy index α = 2), the soliton is unstable, even infinitesimal difference α from 2 immediately stabilizes the soliton texture. Our analytical and numerical investigations of ω(N) dependence (ω is soliton frequency and N its mass) show (within the famous Vakhitov-Kolokolov criterion) the stability of our soliton texture in the fractional α < 2 case. Direct numerical analysis of the linear stability problem of soliton texture also confirms this point. We show analytically and numerically that fractional Schrödinger equation with quintic nonlinearity admits the existence of (stable) soliton textures at 2/3 < α < 2, which is in accord with existing literature data. These results may be relevant to both Bose-Einstein condensates in cold atomic gases and optical solitons in the disordered media.

Polydisperse streaming instability – II. Methods for solving the linear stability problem

ABSTRACT Occurring in protoplanetary discs composed of dust and gas, streaming instabilities are a favoured mechanism to drive the formation of planetesimals. The polydispserse streaming instability is a generalization of the streaming instability to a continuum of dust sizes. This second paper in the series provides a more in-depth derivation of the governing equations and presents novel numerical methods for solving the associated linear stability problem. In addition to the direct discretization of the eigenproblem at second order introduced in the previous paper, a new technique based on numerically reducing the system of integral equations to a complex polynomial combined with root finding is found to yield accurate results at much lower computational cost. A related method for counting roots of the dispersion relation inside a contour without locating those roots is also demonstrated. Applications of these methods show they can reproduce and exceed the accuracy of previous results in the literature, and new benchmark results are provided. Implementations of the methods described are made available in an accompanying python package psitools.

Stability Analysis for an Interface with a Continuous Internal Structure

A general method for solving a linear stability problem of an interface with a continuous internal structure is described. Such interfaces or fronts are commonly found in various branches of physics, such as combustion and plasma physics. It extends simplified analysis of an infinitely thin discontinuous front by means of numerical integration along the steady-state solution. Two examples are presented to demonstrate the application of the method for 1D pulsating instability in magnetic deflagration and 2D Darrieus–Landau instability in a laser ablation wave.

Weakly nonlinear mode interactions in spherical Rayleigh–Bénard convection

In an annular spherical domain with separation $d$, the onset of convective motion occurs at a critical Rayleigh number $Ra=Ra_{c}$. Solving the axisymmetric linear stability problem shows that degenerate points $(d=d_{c},Ra_{c})$ exist where two modes simultaneously become unstable. Considering the weakly nonlinear evolution of these two modes, it is found that spatial resonances play a crucial role in determining the preferred convection pattern for neighbouring modes $(\ell ,\ell \pm 1)$ and non-neighbouring even modes $(\ell ,\ell \pm 2)$. Deriving coupled amplitude equations relevant to all degeneracies, we outline the possible solutions and the influence of changes in $d,Ra$ and Prandtl number $Pr$. Using direct numerical simulation (DNS) to verify all results, time periodic solutions are also outlined for small $Pr$. The $2:1$ periodic signature observed to be general for oscillations in a spherical annulus is explained using the structure of the equations. The relevance of all solutions presented is determined by computing their stability with respect to non-axisymmetric perturbations at large $Pr$.

TheN-flagella problem: elastohydrodynamic motility transition of multi-flagellated bacteria

Peritrichous bacteria such asEscherichia coliswim in viscous fluids by forming a helical bundle of flagellar filaments. The filaments are spatially distributed around the cell body to which they are connected via a flexible hook. To understand how the swimming direction of the cell is determined, we theoretically investigate the elastohydrodynamic motility problem of a multi-flagellated bacterium. Specifically, we consider a spherical cell body with a numberNof flagella which are initially symmetrically arranged in a plane in order to provide an equilibrium state. We solve the linear stability problem analytically and find that at most six modes can be unstable and that these correspond to the degrees of freedom for the rigid-body motion of the cell body. Although there exists a rotation-dominated mode that generates negligible locomotion, we show that for the typical morphological parameters of bacteria the most unstable mode results in linear swimming in one direction accompanied by rotation around the same axis, as observed experimentally.

Settling-driven instability in two-component stably stratified Hele-Shaw flows

We investigate the onset of instability in a stably stratified two-component fluid in a vertical Hele-Shaw cell when the unstably stratified scalar has a settling velocity. This linear stability problem is analysed on the basis of Darcy’s law, for constant-gradient base states. The settling velocity is found to trigger a novel instability mode characterized by pairs of inclined waves. For unequal diffusivities, this new settling-driven mode competes with the traditional double-diffusive mode. Below a critical value of the settling velocity, the double-diffusive elevator mode dominates, while, above this threshold, the inclined waves associated with the settling-driven instability exhibit faster growth. The analysis yields neutral stability curves and allows for the discussion of various asymptotic limits.

Linear and Weakly Nonlinear Instability of Shallow Mixing Layers with Variable Friction

Linear and weakly nonlinear instability of shallow mixing layers is analysed in the present paper. It is assumed that the resistance force varies in the transverse direction. Linear stability problem is solved numerically using collocation method. It is shown that the increase in the ratio of the friction coefficients in the main channel to that in the floodplain has a stabilizing influence on the flow. The amplitude evolution equation for the most unstable mode (the complex Ginzburg–Landau equation) is derived from the shallow water equations under the rigid-lid assumption. Results of numerical calculations are presented.