Dynamics Of Localized Waves Research Articles

Dynamics of localized waves and interactions in a [formula omitted]-dimensional equation from combined bilinear forms of Kadomtsev–Petviashvili and extended shallow water wave equations

This work investigates new exact solutions within a unique (2+1)-dimensional problem by combining the Hirota bilinear forms of the Kadomtsev–Petviashvili and the Shallow Water Wave equations. We obtain resonant Y-type and X-type soliton, breather, and lump waves by introducing novel limitations on the N-soliton. Additionally, it generates various types of solutions (bulk-soliton, fissionable soliton-lump, breather-lump, and soliton-breather-lump) based on velocity and module resonance conditions. Furthermore, we investigate the lump wave’s paths interacting with the waves prior to and following collision using the long-wave limit approach. We show that the lump wave either avoids collision with other waves or maintains a persistent state of collision with them by applying additional limitations. Specifically, we provide a series of figures depicting all solutions, comprehensively capturing their dynamical behavior and interactions.

Novel localized waves and interaction solutions for a dimensionally reduced (2 + 1)-dimensional Boussinesq equation from N-soliton solutions

The Boussinesq equation has been of considerable interest in coastal and ocean engineering models for simulating surface water waves in shallow seas and harbors, tsunami wave propagation, wave overtopping, inundation, and near-shore wave process. Consequently, the study deals with the dynamics of localized waves and interaction solutions to a dimensionally reduced (2 + 1)-dimensional Boussinesq equation from N-soliton solutions by Hirota’s bilinear method. The integrability of the equation of interest was confirmed via the Painlevé test. Taking the long-wave limit approach in coordination with some constraint parameters available in the N-soliton solutions, the localized waves and interaction solutions are constructed. The interaction solutions can be obtained among the localized waves, such as (1) one breather or one lump from the two solitons, (2) one stripe and one breather, and one stripe and one lump from the three solitons, and (3) two stripes and one breather, one lump and one periodic breather, two stripes and one lump, two breathers, and two lumps from the four solitons. The interactions among the solitons are found to be elastic. The energy, phase shift, shape, period, amplitude, and propagation direction of each of the localized waves and interaction solutions are found to be influenced and controlled by the parameters involved. The dynamical characteristics of these localized waves and interaction solutions are demonstrated through some 3D and density graphs. The outcomes achieved in this study can be used to illustrate the wave interaction phenomena in shallow water.

3D Physical Modeling of an Artificial Beach Nourishment: Laboratory Procedures and Nourishment Performance

Beach nourishment represents a type of coastal defense intervention, keeping the beach as a natural coastal defense system. Altering the cross-shore profile geometry, due to the introduction of new sediments, induces a non-equilibrium situation regarding the local wave dynamics. This work aims to increase our knowledge concerning 3D movable bed physical modeling and beach nourishment impacts on the hydrodynamics, sediment transport, and morphodynamics. A set of experiments with an artificial beach nourishment movable bed model was prepared. Hydrodynamic, sediment transport, and morphological variations and impacts due to the presence of the nourishment were monitored with specific equipment. Special attention was given to the number and positioning of the monitoring equipment and the inherent constraints of 3D movable beds laboratory tests. The nourishment induced changes in the beach dynamics, leading to an increase in the flow velocities range and suspended sediment concentration, and effectively increasing the emerged beach width. Predicting and anticipating the morphological evolution of the modeled beach has a major impact on data accuracy, since it might influence the monitoring equipment’s correct position. Laboratory results and constraints were characterized to help better define future laboratory procedures and strategies for increasing movable bed models’ accuracy and performance.

Modeling Microstructural Effects on Heterogeneous Temperature Fields within Polycrystalline Explosives

AbstractThe paper addresses the role of crystal anisotropy on the evolution of heterogeneous temperature fields in plastic‐bonded explosives (PBXs) under conditions of weak shock. The modeling approach is based on simplified idealizations of PBX microstructure including RDX grains and estane binder regions subjected to velocity boundary conditions representative of impact conditions in situ. The constitutive description of the microstructure constituents includes a dislocation‐based, anisotropic, single crystal plasticity model for the explosive grains and a linear viscoelastic model to represent the estane binder. Large suites of simulations were used to systematically study the correlation in heating of the grains with local wave dynamics, crystal orientation, and the microstructural neighborhood. These correlations were identified through the selection of characteristics of crystal anisotropy including oriented wave speeds and Taylor factor. A key observation is that the wave dispersion within a certain distance from the impact interface plays a dominant role in the temperature field. Beyond this distance, individual crystal orientations play a more dominant role, but cannot entirely account for the observed heterogeneity without consideration of the local microstructure neighborhood.

Dynamics of localized waves in a (3+1)-dimensional nonlinear evolution equation

In this paper, a (3[Formula: see text]+[Formula: see text]1)-dimensional nonlinear evolution equation is studied via the Hirota method. Soliton, lump, breather and rogue wave, as four types of localized waves, are derived. The obtained N-soliton solutions are dark solitons with some constrained parameters. General breathers, line breathers, two-order breathers, interaction solutions between the dark soliton and general breather or line breather are constructed by choosing suitable parameters on the soliton solution. By the long wave limit method on the soliton solution, some new lump and rogue wave solutions are obtained. In particular, dark lumps, interaction solutions between dark soliton and dark lump, two dark lumps are exhibited. In addition, three types of solutions related with rogue waves are also exhibited including line rogue wave, two-order line rogue waves, interaction solutions between dark soliton and dark lump or line rogue wave.

Dynamics of localized waves in one-dimensional random potentials: Statistical theory of the coherent forward scattering peak

As recently discovered [PRL ${\bf 109}$ 190601(2012)], Anderson localization in a bulk disordered system triggers the emergence of a coherent forward scattering (CFS) peak in momentum space, which twins the well-known coherent backscattering (CBS) peak observed in weak localization experiments. Going beyond the perturbative regime, we address here the long-time dynamics of the CFS peak in a 1D random system and we relate this novel interference effect to the statistical properties of the eigenfunctions and eigenspectrum of the corresponding random Hamiltonian. Our numerical results show that the dynamics of the CFS peak is governed by the logarithmic level repulsion between localized states, with a time scale that is, with good accuracy, twice the Heisenberg time. This is in perfect agreement with recent findings based on the nonlinear $\sigma$-model. In the stationary regime, the width of the CFS peak in momentum space is inversely proportional to the localization length, reflecting the exponential decay of the eigenfunctions in real space, while its height is exactly twice the background, reflecting the Poisson statistical properties of the eigenfunctions. Our results should be easily extended to higher dimensional systems and other symmetry classes.

Wave analysis of Airy beams

The Airy beams are analyzed in order to provide a cogent physical explanation to their intriguing features which include weak diffraction, curved propagation trajectories in free-space, and self healing. The asymptotically exact analysis utilizes the method of uniform geometrical optics (UGO), and it is also verified via a uniform asymptotic evaluation of the Kirchhoff-Huygens integral. Both formulations are shown to fully agree with the exact Airy beam solution in the paraxial zone where the latter is valid, but they are also valid outside this zone. Specifically it is shown that the beam along the curved propagation trajectory is not generated by contributions from the main lobe in the aperture, i.e., it is not described by a local wave-dynamics along this trajectory. Actually, this beam is identified as a caustic of rays that emerge sideways from points in the initial aperture that are located far away from the main lobe. The field of these focusing rays, described h e by the UGO, fully agrees with the Airy beam solution. These observations explain that the "weak-diffraction" and the "self healing" properties are generated, in fact, by a continuum of sideways contributions to the field, and not by local self-curving dynamics. The uniform ray representation provides a systematic framework to synthesize aperture sources for other beam solutions with similar properties in uniform or in non-uniform media.

Dynamics of localized waves: Pulsed microwave transmissions in quasi-one-dimensional media

We have measured pulsed microwave transmission through quasi-one-dimensional quasi-1D samples with lengths up to three times the localization length as determined from measurements of the variance of intensity fluctuations. Measurements are analyzed using four complementary approaches, each appropriate in a specific time range: i diffusion theory; ii self-consistent localization theory SCLT with a renormalized diffusion coefficient in space and frequency, Dz,; iii a dynamic single parameter scaling DSPS model, which reflects the decay of localized modes which do not overlap in space and frequency; and iv simulations of 1D random media. For times up to twice the diffusion time D, diffusion theory gives an excellent fit to the data. For times up to 4D, the slowing of the decay rate of transmission is in accord with SCLT. For longer times, transmission decays more slowly than the predictions of the SCLT, indicating the inability of this modified diffusion theory to capture the decay of long-lived localized states. Beyond the Heisenberg time, the decay rate approaches the predictions of the DSPS model, reflecting the increasing proportion of wave energy in longlived localized states. The decay rates obtained from 1D simulations are then in good agreement with measurements in quasi-1D samples and coincide with decay rates given by the DSPS model.

Linear planetary wave dynamics in a 2.5-layer ventilated thermocline model

Linear planetary wave dynamics in a 2.5-layer ventilated thermocline model is investigated by a local eigenvalue analysis and simple numerical computations. It is known that there are two types of waves in this system; we refer to one of them as the N-mode (Non-Doppler shift mode), which propagates almost westward, and the other as the A-mode (Advection mode), which propagates almost along the second layer basic current. First, we study the local longwave dynamics, assuming that the wavelength is much longer than the deformation radius and shorter than the gyre scale. It is shown that the N-mode and the A-mode cannot neatly be separated in the ventilated zone, and even in the Rhines and Young pool, a compact A-mode disturbance cannot exist by itself. When anomalous Ekman pumping is applied in the ventilated zone, the N-mode is generated in the forcing region, and the A-mode is generated at the wave front of the N-mode. For diabatic forcing, a similar phenomenon occurs. It is also found that the shadow zone is unstable to longwave disturbances. Secondly, the wave behavior in the linear planetary geostrophic model is numerically investigated. The main features can be interpreted by the local wave dynamics, and the disturbances around the maximum amplitudes are dominated by the waves whose wavenumber vectors are perpendicular to the second layer potential vorticity contours. The amplitude changes during the wave propagation are also discussed. Finally, the effects of the finite wavelength are studied. The N-mode is strongly dispersive at the scale of the Rossby deformation radius, while the A-mode is weakly dispersive. It is also shown that the ventilated zone is unstable to shortwave disturbances, although it is stable to longwave disturbances.

Dynamics of localized waves with large amplitude in a weakly dispersive medium with a quadratic and positive cubic nonlinearity

The dynamics of localized waves is analyzed in the framework of a model described by the Korteweg-de Vries (KdV) equation with account made for the cubic positive nonlinearity (the Gardner equation). In particular, the interaction process of two solitons is considered, and the dynamics of a “breathing” wave packet (a breather) is discussed. It is shown that solitons of the same polarity interact as in the case of the Korteweg-de Vries equation or modified Korteweg-de Vries equation, whereas the interaction of solitons of different polarity is qualitatively different from the classical case. An example of “unpredictable” behavior of the breather of the Gardner equation is discussed.