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Propagation and Energy Transfer
Published in James D. Taylor, Introduction to Ultra-Wideband Radar Systems, 2020
Robert Roussel-Dupré, Terence W. Barrett
A definition of a soliton is that it is a solitary wave which preserves its shape and speed in a collision with another solitary wave.141-151 What is a solitary wave? A solitary wave is one which travels without change in shape or profile. They are both mathematical objects which can describe physical phenomena. How is a soliton obtained? It is obtained by matching either the radiation pulse area to the dispersion or to the intensity-dependent nonlinearity in the medium (Figure 7.42). Here we will be concerned only with matching to the dispersion.
Extreme Faraday Waves
Published in Shamil U. Galiev, Evolution of Extreme Waves and Resonances, 2020
Solitons. A soliton is a solitary traveling wave, i.e., a spatially localized wave with spectacular stability properties. It was first observed in 1834 on the surface of a canal by John Scott Russell [101]. Solitons have stimulated large interest and debates since that time. In particular, it was found that a few very popular mathematical equations do have soliton solutions. This is why solitons have interested mathematicians very much.
Nonlinear Wave and Solitons
Published in K.T. Chau, Applications of Differential Equations in Engineering and Mechanics, 2019
This kind of soliton was first observed in nature by Scottish engineer John Scott Russell in 1834. When Russell conducted experiments on the efficiency of canal boats at the Union Canal in Scotland, he discovered that if the boat was pulled at the “right” speed (corresponding to a Ursell number of about one) by a pair of horses, a wave of “translation” was generated in front of the boat. This wave maintained its form (i.e., did not disperse at all) and traveled with a constant speed upstream. Russell rode on horse to chase the wave for miles before it vanished. Russell gave a very detailed account of the observation of this wave. This wave is now known as a “soliton.” It was known that dispersion causes a wave to attenuate along the travel distance, but nonlinearity effects on the other hand lead to shock wave formation or the eventual breaking wave. The formation of a soliton requires a delicate balance between dispersion and nonlinearity such that the shape of the soliton remains constant. Dutch professor Korteweg and his PhD student de Vries were the first to derive a nonlinear PDE that can successfully depict the wave observed by Russell in Scotland. Naturally, their PDE was eventually known as the KdV (short form from their last names) equation, which will be the main focus of this chapter. However, it was later discovered that the same PDE and soliton solution was published in a book by Boussinesq.
Two efficient methods for solving the generalized regularized long wave equation
Published in Applicable Analysis, 2022
Seydi Battal Gazi Karakoç, Liquan Mei, Khalid K. Ali
In mathematics and physics, a soliton or solitary wave is a self-reinforcing wave packet that maintains its shape while it propagates at a constant velocity. Solitons are caused by a cancellation of nonlinear and dispersive effects in the medium. (Dispersive effects are a property of certain systems where the speed of a wave depends on its frequency.) Solitons are the solutions of a widespread class of weakly nonlinear dispersive partial differential equations describing physical systems. Many researchers have obtained several soliton solutions for different equations see [1–10].
Bright, dark and dark-singular soliton solutions of nonlinear Schrödinger's equation with spatio-temporal dispersion
Published in Journal of Modern Optics, 2018
Solitons are self-reinforcing wave packets that hold their shapes while moving at a constant velocity. Solitons are caused by the cancellation of nonlinear and dispersive influences as they propagate. Currently, the study of solitons is quite active. The nonlinear Schrödinger (NLS) equation is a generic model that governs the evolution of the wave in a wide range of physical environments, including water waves and nonlinear optics (1–17). The NLS equation is a well-known model for transmission in fibre optics (18–30) and describes the propagation of pulses of bright and dark solitons in anomalous and normal dispersion regimes (4,8) In Equation (1), is a complex-valued function. The independent variables are the spatial variable x and the temporal variable t. The coefficients of and are group velocity dispersion and spatio-temporal dispersion, respectively. The coefficients of and are quadratic and cubic nonlinear terms. The first term is the linear evolution term of the soliton pulse. In this article, we consider an extended NLS equation with spatio-temporal dispersion effect and quadratic–cubic nonlinearity. We analytically investigate a wide class of solutions by using the principle of undetermined coefficients as well as the travelling wave hypothesis. In particular, we derive optical bright, dark and dark-singular soliton solutions under certain parametric conditions. By using Jacobi elliptic functions (JEFs) as an ansatz, dark and bright soliton solutions are obtained. An extended direct algebraic method provides us dark and dark-singular soliton solutions.
Interaction of ion acoustic solitons for Zakharov Kuznetsov equation in relativistically degenerate quantum magnetoplasmas
Published in Waves in Random and Complex Media, 2021
M. Yousaf Khattak, W. Masood, R. Jahangir, M. Siddiq
In real systems, waves propagate as single as well as train of solitons and it is, therefore, important to study the multi-soliton solutions. In physical systems, multi-soliton solutions are important with regard to the transport of energy and momentum in the system as solitons retain their shape after interaction. Multi-soliton solutions also give us new solutions that can shed light on the ways waves can propagate and interact in a system equipping us with a fascinating insight into the interaction of nonlinear waves in physical systems. Multi-soliton solutions can be obtained by using various methods such as Darboux transformation [41,42], inverse scattering transformation [43], Bäcklund transformation [44–46], Painlevé test [47,48], and Hirota method [49,50]. The benefit of Hirota's method over the remaining methods is that it is algebraic rather than analytic. In Hirota's bilinear method, the soliton solutions are obtained by truncating the perturbation expansion at different levels in an easier and elegant manner by comparison with other methods. Although multi-soliton solutions of KdV, KP equations and their variants have been investigated many times in unmagnetized plasmas using different methods [51–56], multi-soliton solutions of ZK equation have received relatively less attention in plasma physics [57,58]. It is in the fitness of the situation to mention here that rational multi-soliton solutions of ZK equation have recently been presented by Osman [59] in the context of quantum plasmas where it has been summarized that the solutions are very useful to apprehend the nature of multidimensional ion acoustic waves in extreme conditions. In this paper, our primary focus is to study the multi-soliton solutions of ZK equation in a quantum plasma with relativistically degenerate electrons. We have explored in detail the interaction of nonlinear electrostatic ion acoustic ZK solitons in both nonrelativistic and ultrarelativistic limits.