Electric double layer in porous media is studied with direct numerical simulations of the Nernst-Planck-Poisson equation. The time evolution of the charging process of the electric double-layer along a straight pore is first studied, and confirm that the time evolution obeys a power law of the exponent 1/2. We find that the diffusion constant increases effectively by the effect of the width of the pore. Next it is found that the time evolution of the charging process in fractal porous media obeys a power law, and the exponent $\alpha$ is related to the fracton dimension. Finally, we propose a coupled map lattice model for the creation of pore structures by gas activation processes, and perform numerical simulation of the charging dynamics of the electric double layer.

This article provides a focused review of recent findings which demonstrate, in some cases quite counter-intuitively, the existence of bound states with a singularity of the density pattern at the center; the states are physically meaningful because their total norm converges. One model of this type is based on the 2D Gross–Pitaevskii equation (GPE), which combines the attractive potential $\sim r^{-2}$ and the quartic self-repulsive nonlinearity, induced by the Lee–Huang–Yang effect (quantum fluctuations around the mean-field state). The GPE demonstrates suppression of the 2D quantum collapse, driven by the attractive potential, and emergence of a stable ground state (GS), whose density features an integrable singularity $\sim r^{-4/3}$ at $r\to 0$ . Modes with embedded angular momentum exist too, but they are unstable. A counter-intuitive peculiarity of the model is that the GS exists even if the sign of the potential is reversed from attraction to repulsion, provided that its strength is small enough. This peculiarity finds a relevant explanation. The other model outlined in the review includes 1D, 2D, and 3D GPEs, with the septimal (seventh-order), quintic, and cubic self-repulsive terms, respectively. These equations give rise to stable singular solitons, which represent the GS for each dimension D, with the density singularity $\sim r^{-2/(4-D)}$ . Such states may be considered the results of screening a “bare” delta-functional attractive potential by the respective nonlinearities.

We construct soliton solutions for a self-attractive Bose-Einstein condensate trapped in a rotating optical lattice. The rotation pivot is set at a local maximum of the lattice potential. We demonstrate that fully localized stable solitons, containing $N\sim \mathrm{10,000}$ atoms, may be supported by the lattice, rotating along with it, if the angular velocity $\omega$ is taken below a critical value ${\omega }_c$ (which is $\lesssim 10\mathrm{kHz}$). A monotonously increasing dependence of ${\omega }_c$ on $N$ is found. In the regime of rapid rotation, the lattice potential is nearly tantamount to the axisymmetric Bessel potential, with a maximum at the center. The latter potential supports fundamental ring-shaped solitons, and also dipole states, which have a bipolar form of two adjacent ring solitons with opposite signs. Stability regions are found for both species (which is the first example of stable azimuthally uniform solitons in a self-focusing model with a radial potential). Unstable solitons of these types evolve into strongly localized nonrotating states. At smaller $\omega$, they start to drift slowly, following the rotating lattice. The second critical value, $\omega ={\omega }_c^{\left(2\right)}$, is found, below which the drifting solitons are destroyed. The rotating lattice supports no stable states in the interval of ${\omega }_c<\omega <{\omega }_c^{\left(2\right)}$.

We address effects of spin–orbit coupling (SOC), phenomenologically added to a two-component Bose–Einstein condensate composed of particles moving by Lévy flights, in one- and two-dimensional (1D) and (2D) settings. The corresponding system of coupled Gross–Pitaevskii equations includes fractional kinetic-energy operators, characterized by the Lévy index, α < 2 (the normal kinetic energy corresponds to α = 2). The SOC terms, with strength λ, produce strong effects in the 2D case: they create families of stable solitons of the semi-vortex and mixed-mode types in the interval of 1 < α < 2, where the supercritical collapse does not admit the existence of stable solitons in the absence of the SOC. At λ → 0, amplitudes of these solitons vanish ∼λ ^1/(α−1).

We consider possibilities to control dynamics of solitons of two types, maintained by the combination of cubic attraction and spin-orbit coupling (SOC) in a two-component system, namely, semi-dipoles (SDs) and mixed modes (MMs), by making the relative strength of the cross-attraction, $\gamma$ , a function of time periodically oscillating around the critical value, $\gamma =1$ , which is an SD/MM stability boundary in the static system. The structure of SDs is represented by the combination of a fundamental soliton in one component and localized dipole mode in the other, while MMs combine fundamental and dipole terms in each component. Systematic numerical analysis reveals a finite bistability region for the SDs and MMs around $\gamma =1$ , which does not exist in the absence of the periodic temporal modulation (“management”), as well as emergence of specific instability troughs and stability tongues for the solitons of both types, which may be explained as manifestations of resonances between the time-periodic modulation and intrinsic modes of the solitons. The system can be implemented in Bose-Einstein condensates (BECs), and emulated in nonlinear optical waveguides.