An analytical approximation to measure the extinction cross-section using: Localized Waves

We present a general expression for the optical theorem in terms of Localized Waves. This representation is well-known and commonly used to generate Frozen waves, Xwaves, and other propagation invariant beams. We analyze several examples using different input beam sources on a circular detector to measure the extinction cross-section.

A Novel Dynamic of Localized Solitary Waves for the Hirota-Maccari System

This paper investigates a particular family of semi-rational solutions in determinant form by using the KP hierarchy reduction method, which describe resonant collisions among lumps or resemble line rogue waves and dark solitons in the Hirota-Maccari system. Due to the resonant collisions, the line resemble rogue waves are generated and attenuated in the background of dark solitons with line profiles of finite length, it takes a short time for the lumps to appear from and disappear into the dark solitons background. These novel dynamic of localized solitary waves may be help to understand some physical phenomena of nonlinear localized waves propagation in many physical settings.

Y-shaped soliton solutions for the (2+1)-dimensional bidirectional Sawada–Kotera equation

On the basis of the Hirota bilinear method, resonance Y-shaped soliton and its interaction with other localized waves of (2+1)-dimensional bidirectional Sawada–Kotera equation are derived by introducing the constraint conditions. These types of mixed soliton solutions exhibit complex interaction phenomenon between the resonance Y-shaped solitons and line waves, breather waves, and high-order lump waves. The dynamic behaviors of the interaction solutions are analyzed and illustrated.

Resonant Collisions Among Two-Dimensional Localized Waves in the Mel'nikov Equation

We study the resonant collisions among different types of localized solitary waves in the Mel'nikov equation, which are described by exact solutions constructed using Hirota's direct method. The elastic collisions among different solitary waves can be transformed into resonant collisions when the phase shifts of these solitary waves tend to infinity . First, we study the resonant collision among a breather and a dark line soliton. We obtain two collision scenarios: (i) the breather is semi-localized in space and is not localized in time when it obliquely intersects with the dark line soliton, and (ii) the breather is semi-localized in time and is not localized in space when it parallelly intersects with the dark line soliton. The resonant collision of a lump and a dark line soliton, as the limit case of resonant collision of a breather and a dark line soliton, shows the fusing process of the lump into the dark line soliton. Then we investigate the resonant collision among a breather and two dark line solitons. In this evolution process we also obtain two dynamical behaviors: (iii) when the breather and the two dark line solitons obliquely intersect each other we get that the breather is completely localized in space and is not localized in time, and (iv) when the breather and the two dark line solitons are parallel to each other, we get that the breather is completely localized in time and is not localized in space. The resonant collision of a lump and two dark line solitons is obtained as the limit case of the resonant collision among a breather and two dark line solitons. In this special case the lump first detaches from a dark line soliton and then disappears into the other dark line soliton. Eventually, we also investigate the intriguing phenomenon that when a resonant collision among a breather and four dark line solitons occurs, we get the interesting situation that two of the four dark line solitons are degenerate and the corresponding solution displays the same shape as that of the resonant collision among a breather and two dark line solitons, except for the phase shifts of the solitons, which are not only dependent of the parameters controlling the waveforms of the solitons and the breather, but also dependent of some parameters irrelevant to the waveforms.

Novel Localized Waves and their Interaction Solutions for a Dimensionally Reduced (2+1)-dimensional Boussinesq Equation from N-soliton Solutions

The Boussinesq equation (BqE) 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 over-topping, inundation, and near-shore wave process in which nonlinearity and dispersion effects are taken into consideration. The study deals with the dynamics of localized waves and their interaction solutions to a dimensionally reduced (2 + 1)-dimensional BqE from N-soliton solutions with the use of Hirota’s bilinear method (HBM). Taking the long-wave limit approach in coordination with some constraint parameters in the N-soliton solutions, the localized waves (i.e., soliton, breather, lump, and rogue waves) and their interaction solutions are constructed. The interaction solutions can be obtained among localized waves, such as (i) one breather or one lump from the two solitons, (ii) one stripe and one breather, and one stripe and one lump from the three solitons, and (iii) 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. It is to be found that all interactions among the solitons are elastic. The energy, phase shift, shape, and propagation direction of these localized waves and their interaction solutions can be influenced and controlled by the involved constraint parameters. The dynamical characteristics of these localized waves and their 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.