Interactions of solitary waves in the Adlam-Allen model.

We study the interactions of two or more solitary waves in the Adlam-Allen model describing the evolution of a (cold) plasma of positive and negative charges, in the presence of electric and transverse magnetic fields. In order to show that the interactions feature an exponentially repulsive nature, we elaborate two distinct approaches: (a) using energetic considerations and the Hamiltonian structure of the model, and (b) using the so-called Manton method. We compare these findings with results of direct simulations, and we identify adjustments necessary to achieve a quantitative match between them. Additional connections are made, such as with solitons of the Korteweg-de Vries equation. New challenges are identified in connection to this model and its solitary waves.

Soliton pairs in two-dimensional nonlocal media.

We study the interaction of optical beams of different wavelengths, described by a two-component, two-dimensional (2D) nonlocal nonlinear Schrödinger (NLS) model. Using a multiscale expansion method the NLS model is asymptotically reduced to the completely integrable 2D Mel'nikov system, the soliton solutions of which are used to construct approximate dark-bright and antidark-bright soliton solutions of the original NLS model; the latter being unique to the nonlocal NLS system with no relevant counterparts in the local case. Direct numerical simulations show that, for sufficiently small amplitudes, both these types of soliton stripes do exist and propagate undistorted, in excellent agreement with the analytical predictions. Larger amplitude of these soliton stripes, when perturbed along the transverse direction, disintegrate either to filled vortex structures (the dark-bright solitons) or to radiation (the antidark-bright solitons).

Bright and gap solitons in membrane-type acoustic metamaterials.

We study analytically and numerically envelope solitons (bright and gap solitons) in a one-dimensional, nonlinear acoustic metamaterial, composed of an air-filled waveguide periodically loaded by clamped elastic plates. Based on the transmission line approach, we derive a nonlinear dynamical lattice model which, in the continuum approximation, leads to a nonlinear, dispersive, and dissipative wave equation. Applying the multiple scales perturbation method, we derive an effective lossy nonlinear Schrödinger equation and obtain analytical expressions for bright and gap solitons. We also perform direct numerical simulations to study the dissipation-induced dynamics of the bright and gap solitons. Numerical and analytical results, relying on the analytical approximations and perturbation theory for solions, are found to be in good agreement.

Light Meets Water in Nonlocal Media: Surface Tension Analogue in Optics.

Shallow water wave phenomena find their analogue in optics through a nonlocal nonlinear Schrödinger (NLS) model in 2+1 dimensions. We identify an analogue of surface tension in optics, namely, a single parameter depending on the degree of nonlocality, which changes the sign of dispersion, much like surface tension does in the shallow water wave problem. Using multiscale expansions, we reduce the NLS model to a Kadomtsev-Petviashvili (KP) equation, which is of the KPII (KPI) type, for strong (weak) nonlocality. We demonstrate the emergence of robust optical antidark solitons forming Y-, X-, and H-shaped wave patterns, which are approximated by colliding KPII line solitons, similar to those observed in shallow waters.

Ring dark and antidark solitons in nonlocal media.

Ring dark and antidark solitons in nonlocal media are found. These structures have, respectively, the form of annular dips or humps on top of a stable CW background, and exist in a weak or strong nonlocality regime, defined by the sign of a characteristic parameter. It is demonstrated analytically that these solitons satisfy an effective cylindrical Kadomtsev-Petviashvili (aka Johnson's) equation and, as such, can be written explicitly in closed form. Numerical simulations show that they propagate undistorted and undergo quasi-elastic collisions, attesting to their stability properties.

Dark solitons in the presence of higher-order effects.

Dark soliton propagation is studied in the presence of higher-order effects, including third-order dispersion, self-steepening, linear/nonlinear gain/loss, and Raman scattering. It is found that for certain values of the parameters a stable evolution can exist for both the soliton and the relative continuous-wave background. Using a newly developed perturbation theory we show that the perturbing effects give rise to a shelf that accompanies the soliton in its propagation. Although, the stable solitons are not affected by the shelf it remains an integral part of the dynamics otherwise not considered so far in studies of higher-order nonlinear Schrödinger models.