Collision Of Solitary Waves Research Articles

Error analysis of the exponential wave integrator sine pseudo-spectral method for the higher-order Boussinesq equation

AbstractIn this paper, we derive a new exponential wave integrator sine pseudo-spectral (EWI-SP) method for the higher-order Boussinesq equation involving the higher-order effects of dispersion. The method is fully-explicit and it has fourth order accuracy in time and spectral accuracy in space. We rigorously carry out error analysis and establish error bounds in the Sobolev spaces. The performance of the EWI-SP method is illustrated by examining the long-time evolution of the single solitary wave, single wave splitting, and head-on collision of solitary waves. Numerical experiments confirm the theoretical results.

Multi-bounce resonances in the interaction of walking droplets

Discrete dynamical models of walking droplets (“walkers”) have allowed swift numerical experiments revealing heretofore unobserved quantum statistics and related behaviors in a classical hydrodynamic system. We present evidence that one such model of walking droplets exhibits the empirically elusive n-bounce resonances that are traditionally seen in the scattering of solitary waves governed by covariant nonlinear field theories with polynomial self-interaction. A numerical investigation of the chosen model of interacting walking droplets reveals a fractal structure of resonances in the velocity in–velocity out diagram, much like the usual maps constructed for collisions of solitary waves. We suggest avenues for further theoretical analysis of walker collisions, which may connect this discrete model to the field-theoretic setting, as well as directions towards new experimental realizations n-bounce resonances.

Collisions of ion-acoustic solitary waves in a plasma with negative ions

Head-on collisions of rarefactive ion-acoustic solitary waves and head-on collisions of compressive ion-acoustic solitary waves in a plasma with negative ions are considered. It is shown that such collisions at large amplitudes of the colliding waves lead to a decrease in their amplitudes after the collision. Physical phenomena leading to a loss of energy and a corresponding decrease in the amplitude of the wave after a collision are considered. The phenomenon of wave breaking is studied and the breaking amplitude is determined. The formation of ion-acoustic solitary waves with trapped ions is described. Such a wave, unlike ordinary ion-acoustic solitary waves, transfers not only momentum and energy, but also matter. It is shown that ion-acoustic solitary waves of large amplitudes in a plasma with negative ions cannot be classified as ion-acoustic solitons.

Interaction of Schamel type solitary waves with superthermal trapped electrons

The expanded Poincar'e-Lighthill-Kuo approach is used to explore the collision of Schamel type ion acoustic solitary waves. The phase shift, the most crucial characteristic, is calculated using specific solutions on the basis of the Schamel type Korteweg-de Vries equations. It has been revealed that, in given regime of physically acceptable conditions, the effects of electron confinement and superthermality have a significant influence in altering the phase shift. The results will be a valuable contribution to ongoing research on plasma waves interactions as well as to inquiries into a number of laboratory plasma experiments.

Solitary-Wave Solutions of Benjamin–Ono and Other Systems for Internal Waves: II. Dynamics

Considered here are two systems of equations modeling the two-way propagation of long-crested, long-wavelength internal waves along the interface of a two-layer system of fluids in the Benjamin–Ono and the Intermediate Long-Wave regime, respectively. These systems were previously shown to have solitary-wave solutions, decaying to zero algebraically for the Benjamin–Ono system, and exponentially in the Intermediate Long-Wave regime. Several methods to approximate solitary-wave profiles were introduced and analyzed by the authors in Part I of this project. A natural continuation of this previous work, pursued here, is to study the dynamics of the solitary-wave solutions of these systems. This will be done by computational means using a discretization of the periodic initial-value problem. The numerical method used here is a Fourier spectral method for the spatial approximation coupled with a fourth-order, explicit Runge–Kutta time stepping. The resulting, fully discrete scheme is used to study computationally the stability of the solitary waves under small and large perturbations, the collisions of solitary waves, the resolution of initial data into trains of solitary waves, and the formation of dispersive shock waves. Comparisons with related unidirectional models are also undertaken.

Investigating overtaking collisions of solitary waves in the Schamel equation

This article presents a numerical investigation of overtaking collisions between two solitary waves in the context of the Schamel equation. Our study reveals different regimes characterized by the behavior of the wave interactions. In certain regimes, the collisions maintain two well-separated crests consistently over time, while in other regimes, the number of local maxima undergoes variations following the patterns of 2→1→2→1→2 or 2→1→2. These findings demonstrate that the geometric Lax-categorization observed in the Korteweg–de Vries equation (KdV) for two-soliton collisions remains applicable to the Schamel equation. However, in contrast to the KdV, we demonstrate that an algebraic Lax-categorization based on the ratio of the initial solitary wave amplitudes is not feasible for the Schamel equation. Additionally, we show that the statistical moments for two-solitary wave collisions are qualitatively similar to the KdV equation and the phase shifts after soliton interactions are close to ones in integrable KdV and modified KdV models.

Nucleation of transition waves via collisions of elastic vector solitons

In this work, we show that collisions of one type of nonlinear wave can lead to generation of a different kind of nonlinear wave. Specifically, we demonstrate the formation of topological solitons (or transition waves) via collisions of elastic vector solitons, another type of nonlinear wave, in a multistable mechanical system with coupling between translational and rotational degrees of freedom. We experimentally observe the nucleation of a phase transformation arising from colliding waves, and we numerically investigate head-on and overtaking collisions of solitary waves with vectorial properties (i.e., elastic vector solitons). Unlike KdV-type solitons, which maintain their shape despite collisions, our system shows that collisions of two vector solitons can cause nucleation of a new phase via annihilation of the vector solitons, triggering the propagation of transition waves. The propagation of these depends both on the amount of energy carried by the vector solitons and on their respective rotational directions. The observation of the initiation of transition waves with collisions of vector solitons in multistable mechanical systems is an unexplored area of fundamental nonlinear wave interactions and could also prove useful in applications involving reconfigurable structures.

Numerical Study on Collisions of Solitons of Surface Waves in Finite Water Depth

Head-on collisions between two solitary waves in the framework of the nonlinear Schrödinger (NLS) equation were investigated using the Fourier spectral method. When solitary waves undergo collision, the peak value of surface elevation (hereafter referred to as ζmax) exhibits fluctuations with increasing relative water depths k0h (where k0 is the wave number and h is the water depth). ζmax is approximately equal to the sum of the peak values of the two solitary waves with smaller wave steepness ε0 (ε0 = k0a0, a0 is the free background amplitude parameter), and it exhibits fluctuations for ε0 > 0.10. Similar results have been observed in the study of head-on collisions for four solitary waves. These results show that the water depth and wave steepness play important roles in the collision of solitary waves, and the effects of the interactions of intense wave groups are important in studies of the mechanisms and manifestations of freak oceanic waves.

A linear decoupled physical-property-preserving difference method for fractional-order generalized Zakharov system

In this paper, an efficient physical-property-preserving algorithm for the space fractional-order generalized Zakharov system is proposed and analyzed. Firstly, the space fractional-order generalized Zakharov system is reformulated as an equivalent system of equations by introducing the auxiliary equation. Then the spatial fourth-order physical-property-preserving linearly implicit difference scheme is developed for the transformed system. Subsequently, with the aid of the cut-off function and discrete energy analysis method, the underlying scheme are proven to be with the optimal order of O(Δt2+h4) in discrete L∞ and L2 norms. The main feature of the new scheme is physical-property-preserving, linearly decoupled and easy to be applied in parallel computing, especially in long time simulations and large-scale problems. At last, ample numerical results are exhibited to substantiate the efficiency and preservation properties of our scheme, and investigate the dynamic behaviors of the collision of different solitary waves with subsonic and supersonic propagation speeds, respectively.

The phase shift in the collision of solitary waves with different angles in 2D unmagnetized quantum plasma

The phase shift in the collision of solitary waves with different angles in 2D unmagnetized quantum plasma

Obliquely overtaking Collisions of electron-acoustic solitary waves of inertial cold and beam electrons in the presence of superthermal electrons

Overtaking collision of the electron-acoustic solitary waves in the four-component collisionless, magnetized plasma has been investigated. The system consists of cool inertial background electrons of temperature Tc, cool inertial electron beam of temperature Tb, hot inertialess superthermal electrons modeled by a κ-distribution of temperature Th and ions. Nonlinear Korteweg–de Vries (KdV) equation that characterizing the EASWs in such a plasma model is derived. The system admits both compressive and rarefactive solitary waves. The overtaking collision of two and three-soliton solutions is investigated using Hirota’s bilinear technique. Physical parameters influence the interchange of energy caused by overtaking collisions. These parameters comprise the magnetic field, obliquity angle, the beam, cool and hot electrons number densities and temperatures, and this causes the solitons behavior to be alternated. The amplitude, and width increase as the superthermal electrons temperature increases or magnetic field decreases. The system parameters have an effect on the phase shifts as well. The current research is an attempt to shed light on the characteristics of N-soliton in the four-component collisionless, magnetized plasma that are applicable in several astrophysical plasma environments such as the Earth’s magnetosphere.

Solitary waves explore the quantum-to-classical transition

How the classical appearance of our environment arises from the underlying quantum many-body theory is an open fundamental question. We propose that phenomena involved in the quantum-to-classical transition can be probed in collisions of bright solitary waves in Bose-Einstein condensates, where thousands of atoms form a large compound object at ultra cold temperatures. For the experimentally most relevant quasi-1D regime, where integrability is broken through effective three-body interactions, we find that ensembles of solitary waves exhibit complex interplay between phase coherence and entanglement generation in beyond mean-field simulations using the truncated Wigner method: An initial state of two solitons with a well-defined relative phase looses that phase coherence in the ensemble, with its single-particle two-mode density matrix exhibiting similar dynamics as a decohering two-mode superposition. This apparent decoherence is a prerequisite for the formation of entangled superpositions of different atom numbers in a subsequent soliton collision. The necessity for the solitons to first decohere is explained based on the underlying phase-space of the quintic mean-field equation. We show elsewhere that superpositions of different atom numbers later further evolve into spatially entangled solitons. Loss of ensemble phase coherence followed by system internal entanglement generation appear in an unusual order in this closed system, compared to a typical open quantum system.

Solitary wave collisions for the Whitham equation

Solitary wave collisions for the Whitham equation

Numerical Simulations of Cross-boundary Layer Transport by Head-on Internal Solitary Wave Collisions

Numerical Simulations of Cross-boundary Layer Transport by Head-on Internal Solitary Wave Collisions

Numerical simulation of a generalized nonlinear derivative Schrödinger equation

<abstract><p>In this paper, three nonlinear finite difference schemes are proposed for solving a generalized nonlinear derivative Schrödinger equation which exposits the propagation of ultrashort pulse through optical fiber and has been illustrated to admit exact soliton-solutions. Two of the three schemes are two-level ones and the third scheme is a three-level one. It is proved that the two-level schemes only preserve the total mass or the total energy in the discrete sense and the three-level scheme preserves both the total mass and total energy. Furthermore, many numerical results are presented to test the conservative properties and convergence rates of the proposed schemes. Several dynamical behaviors including solitary-wave collisions and the first-order rogue wave solution are also simulated, which further illustrates the effectiveness of the proposed method for the generalized nonlinear derivative Schrödinger equation.</p></abstract>

On the Rosenau equation: Lie symmetries, periodic solutions and solitary wave dynamics

In this paper, we first consider the Rosenau equation with the quadratic nonlinearity and identify its Lie symmetry algebra. We obtain reductions of the equation to ODEs, and find periodic analytical solutions in terms of elliptic functions. Then, considering a general power-type nonlinearity, we prove the non-existence of solitary waves for some parameters using Pohozaev type identities. The Fourier pseudo-spectral method is proposed for the Rosenau equation with this single power type nonlinearity. In order to investigate the solitary wave dynamics, we generate the initial solitary wave profile by using the Petviashvili’s method. Then the evolution of the single solitary wave and overtaking collision of solitary waves are investigated by various numerical experiments.

Experimental observation of O-solitary waves in shallow water

Solitary waves have been well studied in various fields, including water waves, optical communications, plasmas, and biophysics. We present experimental results that a two-dimensional solitary wave, with its crest shaped like the letter “O,” occurs around an island terrain in shallow water. We name it the O-soliton (a cylindrical-type soliton) in this paper. This is the first experimental observation of the O-soliton including both outward and inward propagation in shallow water. O-solitons are observed with stable and periodic occurrences under typical coastal conditions. A water column at its center is higher than eight times the initial mean wave height measured by six wave gauges arranged in a regular hexagon. A one-dimension solitary wave incurs a phase shift when it collides with the O-soliton. However, no phase shift is observed in the O-soliton itself. Furthermore, when it collapses to its center point, it remains in this state for a short period. This is analogous to the pausing of the wave crest during a head-on collision of solitary waves or the reflection of a solitary wave at a vertical wall. These results also enrich the general understanding of solitary waves, whose repeated occurrence suggests that they may naturally occur in the ocean, and motivate analogous studies in other fields of physics.

On solitary-wave solutions of Boussinesq/Boussinesq systems for internal waves

In this paper we consider a three-parameter system of Boussinesq/Boussinesq type, modeling the propagation of internal waves. Some theoretical and numerical properties of the systems were previously analyzed by the authors. As a second part of the study, the present paper is concerned with the analysis of existence and the numerical simulation of some issues of the dynamics of solitary-wave solutions. Standard theories are used to derive several results of existence of classical and generalized solitary waves, depending on the parameters of the models. A numerical procedure based on a Fourier collocation approximation for the ode system of the solitary wave profiles with periodic boundary conditions, and on the iterative solution of the resulting fixed-point equations with the Petviashvili scheme combined with vector extrapolation techniques, is used to generate numerically approximations of solitary waves. These are an essential part of a computational study of the dynamics of the solitary waves, both classical and generalized. Using a full discretization based on spectral approximation in space of the corresponding periodic initial-value problem for the systems, and a fourth-order Runge–Kutta method of composition type as time integrator, we explore the evolution of small and large perturbations of the computed solitary-wave profiles, and we study computationally the collisions of solitary waves as well as the resolution of initial data into trains of solitary waves.

Viscous Flow Evolution and Boundary Layer Characteristics during the Head-On Collision of Solitary Waves

Lee, T.-C.; Chang, C.-H., and Tang, C.-J., 2021. Viscous flow evolution and boundary layer characteristics during the head-on collision of solitary waves. Journal of Coastal Research, 37(3), 544–556. Coconut Creek (Florida), ISSN 0749-0208.Viscous shear flow near the bottom during the head-on collision of two solitary waves is investigated using a stream function–vorticity model. Fully nonlinear free-surface conditions are satisfied numerically at every time step. The bottom boundary-layer flow region is locally refined to explore its physical characteristics through detailed vorticity transport in the shear layer beneath the two solitary waves during their collision. Both symmetric and asymmetric head-on collisions are examined and discussed. The wave deformations are compared with the results of other studies for validation, and the fluid mechanisms in the flow field are illustrated with streamline, equivorticity line, path line, and timeline plots to reveal the interaction processes. It is revealed that the wave interaction behaves nonlinearly not only at the free surface but also in the vorticity exchanges near the bottom.

Numerical investigations of shallow water waves via generalized equal width (GEW) equation

In this article, a mathematical model representing solution of the nonlinear generalized equal width (GEW) equation has been considered. Here we aim to investigate solutions of GEW equation using a numerical scheme by using sextic B-spline Subdomain finite element method. At first Galerkin finite element method is proposed and a priori bound has been established. Then a semi-discrete and a Crank-Nicolson Galerkin finite element approximation have been studied respectively. In addition to that a powerful Fourier series analysis has been performed and indicated that our method is unconditionally stable. Finally, proficiency and practicality of the method have been demonstrated by illustrating it on two important problems of the GEW equation including propagation of single solitons and collision of double solitary waves. The performance of the numerical algorithm has been demonstrated for the motion of single soliton by computing L∞ and L2 norms and for the other problem computing three invariant quantities I1,I2 and I3. The presented numerical algorithm has been compared with other established schemes and it is observed that the presented scheme is shown to be effectual and valid.