The formation of rogue waves in shallow water is presented in this Rapid Communication by providing the three lowest-order exact rational solutions to the Korteweg-de Vries (KdV) equation. They have been obtained from the modified KdV equation by using the complex Miura transformation. It is found that the amplitude amplification factor of such waves formed in shallow water is much larger than the amplitude amplification factor of those occurring in deep water. These solutions clearly demonstrate a potential hazard for coastal areas. They can also provide a solid mathematical basis for the existence of abnormally large-amplitude waves in other branches of nonlinear physics such as optics, unidirectional crystal growth, and in quantum mechanics.
We present an infinitely-extended KdV equation that contains an infinite number of arbitrary real coefficients controlling higher-order terms in the extended evolution equation. The higher-order terms are chosen in a way that maintains the integrability of the whole equation. Another significant step in this work is that this extended equation admits complex-valued solutions. This generalization allows us to consider both solitons and rogue waves in the form of rational solutions of this equation. Special choices of the arbitrary coefficients lead to particular cases—the basic KdV and its higher-order versions. Using the extended KdV, instead of the basic one, may improve the accuracy of the description of rogue waves in shallow water.
Rogue waves can appear in optical fibers and other optical systems, as well as in natural events like water waves. Their mathematical description is based on partial differential equations that have solutions that are localized both in time and space. One example is the Peregrine solution of the nonlinear Schrödinger equation (NLSE). When higher-order terms in the equation are involved, the solution becomes distorted, but its main features remain localized in space and time. Although exact solutions are not obtained in all cases, approximations which describe the solutions with reasonable accuracy do exist. Here, we consider approximate rogue wave solutions of the NLSE with an optically-relevant Raman delay term.
A close analysis of dust charging process in the presence of radio frequency (RF) discharge on low pressure and fully ionized plasma for both weak and strong discharge's electric field is considered. When the electromagnetic waves pass throughout fully ionized plasma, the collision frequency of the plasma is derived. Moreover, the disturbed distribution function of plasma particles in the presence of the RF discharge is obtained. In this article, by using the Krook model, we separate the distribution function in two parts, the Maxwellian part and the perturbed part. The perturbed part of distribution can make an extra current, so-called the accretion rate of electron (or ion) current, towards a dust particle as a function of the average electron-ion collision frequency. It is proven that when the potential of dust grains increases, the accretion rate of electron current experiences an exponential reduction. Furthermore, the accretion rate of electron current for a strong electric field is relatively smaller than that for a weak electric field. The reasons are elaborated. V C 2015 AIP Publishing LLC. [http://dx.
Propagation of dust lattice waves (DLWs) in a two-dimensional bi-crystal lattice in an arbitrary direction in zero gravity is studied. Interaction of the nearest neighbor in comparison with the second neighbor under the Yukawa and overlapping Debye sphere (ODS) potential is illustrated. In the last part of this article, DLW modes for mono-crystals, for more simplification, are found to be hybrid and have both longitudinal and transverse components; however, they become purely longitudinal or transverse modes when the angle of propagation θ becomes equal to zero or π /2. Dispersion relations for the Yukawa and ODS potentials are compared with experimental data and good agreement is found.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.