A new nonlinear equation governing asymptotic dynamics of ripples is derived by using a short wave perturbative expansion on a generalized version of the Green-Naghdi system. It admits peakon solutions with amplitude, velocity and width in interrelation and static compacton solutions with amplitude and width in interrelation. Short wave pattern formation is shown to result from a balance betwee...

We present the first experimental demonstration of a new type of localized state in the continuum, namely, compacton-like linear states in flat-band lattices. To this end, we employ photonic Lieb lattices, which exhibit three tight-binding bands, with one being perfectly flat. Discrete predictions are confirmed by realistic continuous numerical simulations as well as by direct experiments. Our ...

We propose and investigate several complex versions of extensions restrictions the Skyrme model with a well-defined Bogomolny-Prasad-Sommerfield (BPS) limit. The models studied possess kink, anti-kink, semi-kink, massless purely imaginary compacton BPS solutions that all have real energies. reality energies for particular solution is guaranteed when modified antilinear CPT-symmetry maps Hamilto...

We consider fifth-order nonlinear dispersive K(m,n, p) type equations to study the effect of nonlinear dispersion. Using simple scaling arguments we show, how, instead of the conventional solitary waves like solitons, the interaction of the nonlinear dispersion with nonlinear convection generates compactons the compact solitary waves free of exponential tails. This interaction also generates ma...

The numerical simulation of compactons, solitary waves with compact support, is characterized by the presence of spurious phenomena, as numerically-induced radiation, which is illustrated here using four numerical methods applied to the RosenauHyman K(p, p) equation. Both forward and backward radiations are emitted from the compacton presenting a self-similar shape which has been illustrated gr...

A numerical study of the nonlinear wave solutions of the Rosenau-Pikovsky K(cos) equation is presented. This equation supports at least two kind of solitary waves with compact support: compactons of varying amplitude and speed, both bounded, and kovatons which have the maximum compacton amplitude, but arbitrary width. A new Padé numerical method is used to simulate the propagation and, with sma...

The two-dimensional (2D) version of ferromagnet XXZ spin chain with DzyaloshinskyMoriya (DM) interaction, recently introduced by F. Kenmogne and coworkers is reexplored. Firstly using the Dyson-Maleev transformation, 2-D discrete nonlinear Schrödinger (DNLS) equation, governing quantum states behaviors found. Next semidiscrete multiple-scale method, DNLS equation reduced to 2D extended (ENLS) w...

In this lecture we outline main results of our investigations of certain field-theoretic systems which have V-shaped field potential. After presenting physical examples of such systems we show that in static problems the exact ground state value of the field is achieved on a finite distance – there are no exponential tails. This applies in particular to soliton-like object called the topologica...