Collision Of Solitary Waves Research Articles

Head-on Collision of Solitary Waves Described by the Toda Lattice Model in Granular ChainSupported by the National Natural Science Foundation of China (Grant No. 11574153).

We study the head-on collision of two solitary waves in a precompressed granular chain using the discrete element method. Our study takes the Toda chain solution as the initial condition for the simulations. The simulation covers the dynamical evolution of the collision process from the start of the incident wave to the end of the collision. The interaction has a central collision region of about five-grain width in which two solitary waves merge completely and share only one peak. Four stages, i.e., the pre-in-phase traveling stage, lag-phase collision state, lead-phase collision state, and post-in-phase traveling stage, are identified to describe the complex collision processes. Our results may be helpful for explaining the existence of long-lived solitary waves seen in the simulations by Takato and Sen [Europhys. Lett. 100 (2012) 24003].

Collisions of solitary waves in condensates beyond mean-field theory

Bright solitary waves in a Bose-Einstein condensate contain thousands of identical atoms held together despite their only weakly attractive contact interactions. They nonetheless behave like a compound object, staying whole in collisions, with their collision properties strongly affected by inter-soliton quantum coherence. We show that separate solitary waves decohere due to phase diffusion, dependent on their effective ambient temperature, after which their initial mean-field relative phases are no longer well defined or relevant for collisions. In this situation, collisions occur predominantly repulsively and can no longer be described within mean field theory. When considering the time-scales involved in recent solitary wave experiments where non-equilibrium phenomena play an important role, these features could explain the predominantly repulsive collision dynamics observed in most condensate soliton train experiments.

Numerical analysis on the effects of a submerged bottom-mounted barrier in the head-on collision of two solitary waves

The collision of solitary waves is closely related to many physical problems. In this paper, the effects of a submerged bottom-mounted barrier in the head-on collision of two solitary waves are analyzed using a self-developed two-dimensional two-phase flow model. Advanced Adaptive Mesh Refinement (AMR) grid is adopted to reduce the computational cost. The free surface is captured with the Volume of Fraction (VOF) method combined with Piecewise Linear Interface Calculation (PLIC). The Immersed Boundary (IB) method is used to account for the existence of the barrier. Several benchmark problems including solitary wave propagation, head-on collision of two solitary waves, and solitary wave past a submerged barrier are considered to validate the accuracy of the model. Then the model is utilized to simulate the head-on collision of two solitary waves with the existence of a submerged bottom-mounted barrier. Both the symmetric (two waves with equal amplitudes) and asymmetric (two waves with unequal amplitudes) head-on collisions are investigated. The results with and without the barrier are compared and analyzed in detail. Special attention is paid to the variations of the wave amplitudes and the vortex sheddings.

Head-on Collision of Ion-Acoustic Solitary Waves with Two Negative Ion Species in Electron–Positron–Ion Plasmas and Production of Rogue Waves

The interactions between ion acoustic solitary waves (IASWs) are investigated considering completely ionized electron–positron–ion (e–p–i) plasmas consisting of ions with positive and two negative species, nonthermal electrons, and positrons. Two-sided Korteweg–de Vries (KdV) and modified KdV (mKdV) equations are derived using extended Poincare–Lighthill–Kuo (ePLK) reductive perturbation method. To investigate the production of ion-acoustic rogue waves (IARWs), the rational solution of nonlinear Schrodinger equation (NLSE) is derived from the mKdV equation. Two types of plasmas containing $${\text{A}}{{{\text{r}}}^{ + }}$$ , $${{{\text{F}}}^{ - }}$$ , and $${\text{SF}}_{{\text{5}}}^{ - }$$ species, as well as $${\text{SF}}_{{\text{5}}}^{ + }$$ , $${{{\text{F}}}^{ - }}$$ , and $${\text{SF}}_{{\text{5}}}^{ - }$$ species, are taken into account to study their effects on the amplitudes and phase shifts after collision, as well as the production and properties of rogue waves (RWs). It is observed during collision that a high-amplitude wave is produced in the interaction region depending on the type and parameters of plasmas. The nonthermality of electrons and positrons, electron-to-positron temperature ratio, and the density of negative ions modify the phase shift and amplitude of the waves produced during the collision of the two solitons. The amplitude of the RW for the $${\text{A}}{{{\text{r}}}^{ + }}$$ , $${{{\text{F}}}^{ - }}$$ , and $${\text{SF}}_{{\text{5}}}^{ - }$$ plasmas is found to be larger than that for the $${\text{SF}}_{{\text{5}}}^{ + }$$ , $${{{\text{F}}}^{ - }}$$ , and $${\text{SF}}_{{\text{5}}}^{ - }$$ plasmas.

Classification of head-on collisions of ion-acoustic solitary waves in a plasma consisting of cold ions and Boltzmann electrons

Head-on collisions of ion-acoustic solitary waves in a collisionless plasma consisting of cold ions and Boltzmann electrons are studied using the particle-in-cell simulation. It is shown that the collision of solitary waves can occur under different scenarios. Solitary waves preserve or do not preserve their amplitudes and shapes after a collision, depending on their initial amplitudes. The range of initial amplitudes, at which a solitary wave preserves its identity after collisions, is established. The use of a diagram of initial amplitudes of colliding solitary waves to consider possible collision scenarios is discussed. The characteristic regions in the diagram of the initial amplitudes corresponding to different collision scenarios are determined, and a classification of head-on collisions of ion-acoustic solitary waves in a plasma is proposed.

Acceleration and Trapping of Ions upon Collision of Ion-Acoustic Solitary Waves in Plasma with Negative Ions

The phenomena occurring under head-on collision of ion-acoustic solitary waves in collisionless plasma consisting of positive and negative ions and electrons obeying the Boltzmann distribution are considered. Using particle-in-cell simulations, it is shown that large-amplitude compressive ion-acoustic solitary waves do not preserve their identity after the collision. Their amplitudes decrease and their shapes change. It is shown that the collision is accompanied by the generation of fast positive ions the velocity of which can exceed more than threefold the speed of sound. In addition, the collision is accompanied by the trapping of negative ions by the field of ion-acoustic solitary waves formed after the collision.

The conservative and fourth-order compact finite difference schemes for regularized long wave equation

In this paper, two conservative and fourth-order compact finite difference schemes are proposed and analyzed for solving the regularized long wave (RLW) equation. The first compact finite difference scheme is two-level and nonlinear implicit. The second scheme is three-level and linearized implicit. Conservations of the discrete mass and energy, and unique solvability of the numerical solutions are proved. Convergence and unconditional stability are also derived without any restrictions on the grid ratios by using discrete energy method. The optimal error estimates in norm ‖⋅‖ and ‖⋅‖L∞ are of fourth-order and second-order accuracy for the spatial and temporal step sizes, respectively. Numerical examples are presented to simulate the collision of different solitary waves and support the theoretical analysis.

Numerical analysis on the generation, propagation and interaction of solitary waves by a Harmonic Polynomial Cell Method

A numerical wave tank based on the Harmonic Polynomial Cell (HPC) method is created to study the generation, propagation and interaction of solitary waves. The HPC method has been proven to be of high accuracy and efficiency in modeling of water waves, wave–wave and wave–structure interaction within the context of potential flow. An important feature of the present HPC method is that the free surface and solid boundaries are immersed in a stationary Cartesian grid. Solitary waves with σ, i.e. amplitude to water depth ratio, up to 0.6 are generated by different methods. We demonstrate that the results based on the first-, third- and ninth-order method are less satisfactory than the fully-nonlinear method in generating solitary waves withσ>0.4. Additionally, both the head-on and overtaking collision between two solitary waves are studied. In the investigation of the phase shifts after the head-on collision, our window model successfully explain the main reason why Su and Mirie (1980)’s third-order approximation of the uniform phase shifts is inconsistent with Chen and Yeh (2014)’s experimental results and Craig et al. (2006)’s fully nonlinear numerical results. For the overtaking collision of solitary waves, the collision process and the phase shifts are numerically analyzed. Our present result also confirms Craig et al. (2006)’s category of the overtaking collision.

Numerical study on the head-on collisions of solitary waves by a harmonic polynomial cell method

A 2D fully-nonlinear numerical wave flume is created with a piston-type wave generator based on a recently developed 2D harmonic polynomial cell method (HPC), which has been demonstrated as a highly accurate and efficient approach for modelling of water waves and their interaction with marine structures within the context of potential flow. The overlapping grid technique and a Goring’s wave making method are applied to generate solitary waves. The main focus of this paper is on the head-on collisions of pairwise solitary waves with identical or different amplitudes, including the maximum run-up and spatial phase shift during the whole colliding process. As for collision between identical solitary waves with amplitude σ =0.4, our present numerical results are in excellent agreement with experimental results both in maximum run-up and phase shifts, which indicates the accuracy of HPC method in simulating the head-on collision of solitary waves. The immediate and long-term phase shifts based on wave crest trajectories were obtained with success to explain the inconsistent results of phase shifts between experimental, fully nonlinear numerical results and the third-order approximation results. It is found from the results that the maximum run-up surpasses the sum of the initial wave amplitudes. The analysis of the phase shifts shows that the phase shifts of two solitary waves due to the head-on collision vary with the window length of data and the distance between measurement point and the collision center. The general trend is that the phase shifts measured close to the collision center are larger than those far from the collision center.

Head-on Collision of Ion-Acoustic Shock and Solitary Waves in Collisionless Plasma with Pair Ions and Electrons

Unmagnetized collisionless plasma system consisting of positive- and negative ions, and electrons are considered to study the head-on collision of ion-acoustic shock waves (IASWs) and its effects on the formation of shock (monotonic and oscillatory) waves and phase shift. The soliton solution is derived from the two-sided Korteweg-de Vries Burger (KdVB) equations. The KdVB equations are obtained using extend Poincare-Lighthill-Kuo (ePLK) method. It is assumed that the negative ions are immobile and the electrons follow the Boltzmann energy distribution in the plasma. The effects of plasma parameters such as density ratios and kinematic viscosities on electrostatic shock profiles, phase shift, amplitudes, formation of shock (monotonic and oscillatory) as well as on the soliton solution are investigated. It is found that the density ratio of negative to positive ions plays a vital role on the formation of shock waves and phase shift after collision.

Conservative compact finite difference scheme for the N‐coupled nonlinear Klein–Gordon equations

In this article, a compact finite difference method is developed for the periodic initial value problem of the N‐coupled nonlinear Klein–Gordon equations. The present scheme is proved to preserve the total energy in the discrete sense. Due to the difficulty in obtaining the priori estimate from the discrete energy conservation law, the cut‐off function technique is employed to prove the convergence, which shows the new scheme possesses second order accuracy in time and fourth order accuracy in space, respectively. Additionally, several numerical results are reported to confirm our theoretical analysis. Lastly, we apply the reliable method to simulate and study the collisions of solitary waves numerically.

A new lattice hierarchy: Hamiltonian structures, symplectic map and N-fold Darboux transformation

• A new integrable lattice hierarchy is proposed. • The associated Hamiltonian structures and symplectic map are derived. • The N -fold Douboux transformations of the first two lattice systems are given. • The elastic and inelastic interaction phenomena in the lattice systems are found. A new lattice hierarchy is constructed from a discrete matrix spectral problem. By the Tu scheme technique, the associated Hamiltonian structures and infinitely many conservation laws of this hierarchy are derived. Then a symplectic map is proposed based on the Lax pair and the adjoint Lax pair. Furthermore, the N -fold Darboux transformation and explicitly exact solutions of the first two equations in the hierarchy are investigated. Finally, the density profiles of these exact solutions are presented to illustrate the overtaking collisions of solitary waves .

Solitary wave simulations of Complex Modified Korteweg-de Vries (CMKdV) Equation using Quintic Trigonometric B-Spline

The Modified form of the Complex Korteweg-de Vries (CMKdV) Equation is solved numerically using collocation method based on quintic trigonometric B-Splines. A Crank Nicolson rule is used to discretize in time. The well-known examples, propagation of bell-shaped initial pulse and collision of multi solitary waves are simulated using Matlab programme language. Computational results are examined by calculation of the accuracy of the method in terms of maximum error norm and the three conservation laws I 1 , I 2 and I 3 . Because the absolute changes of the lowest three laws are also good indicators of valid results even when the analytical solutions do not exist. A comparison with some earlier works is given

Solitary waves and localized nonlinear excitations in the strongly nonlinear β-Fermi-Pasta-Ulam-Tsingou chain

We develop new analytic descriptions for solitary waves (SWs) and SW-like objects in the non-integrable and strongly nonlinear β-Fermi-Pasta-Ulam-Tsingou (β-FPUT) system for various strengths of the harmonic term in the potential. We then show that that the collision of identical solitary waves in the decorated β-FPUT chain can lead to the formation of one or more long-lived localized nonlinear excitations (LNEs), thereby suggesting that these systems allow engineering of preferential energy distribution over extended times.

Collision of Ion-Acoustic Solitary Waves in Plasma

The process of collision of ion-acoustic solitary waves in a collisionless plasma with cold ions and Boltzmann electrons is studied using numerical simulations. It is shown that solitary waves with sufficiently large amplitudes do not preserve their identity after the collision. Their amplitudes decrease, and the shapes change. It is found that the collision is accompanied by the generation of fast ions with velocities exceeding threefold the ion sound speed.

Particle-in-cell simulation of the head-on collision of large amplitude ion-acoustic solitary waves in a collisionless plasma

The head-on collision of ion-acoustic solitary waves in a collisionless plasma with cold ions and Boltzmann electrons is studied using the particle-in-cell (PIC) simulation. It is shown that solitary waves of sufficiently large amplitudes do not retain their identity after a collision. Their amplitudes decrease and their forms change. In the collision of solitary waves, accelerated ions are formed. The ion velocities can reach three speeds of sound. Dependences of amplitudes of the potential and densities of ions and electrons after a head-on collision of identical solitary waves on their initial amplitude are presented.

Solitary wave collisions by double-dam-broken simulations with the MPS method

PurposeThe paper aims to apply a modified version of the MPS method to a double-dam-breaking test in which high dispersion zones and high natural clusterization zones are present, such as when the water column collapses into two sides and the two solitary waves collide, respectively.Design/methodology/approachThe work takes advantage of the mixed source term from the cheaper computational version of the moving particle semi-implicit (MPS) method to reduce one step from the MPS classical algorithm. The proposed test can be successfully simulated by applying modifications to the variance parameter in the Laplacian operator and gradient model.FindingsThe results show stable behavior in dispersion and clusterization zones. Also, the collision and merging produced by solitary waves was successfully simulated.Research limitations/implicationsThe main limitation in this work was the development of a comparison between the obtained results and the simulations with the original cheaper computational version of the MPS, this limitation is due to the overestimation of inter particle repulsive forces from its gradient model.Practical implicationsThe application of solitary waves is of paramount importance in a number of applications, and this stems from the fact that the interaction of solitary waves with ships and other floating structures could generate highly deformed and complex free surface flows.Social implicationsFor future work, the modified version of the MPS method can be applied in flow over sill base simulations, in close and open channels, and in simulating breaking waves to determine impact pressures by using solitary wave propagation.Originality/valueThe simulation of interaction of large groups of particles as in the case when two solitary waves collide could cause severe instability problems in pressure, causing the computer analysis to stop. MPS classical algorithm takes into account an explicit step that, in this case, may provoke the problem. For this reason, the cheaper version of MPS method is used to correctly simulate solitary wave interactions.

Head-on collision of internal waves with trapped cores

Abstract. The dynamics and energetics of a head-on collision of internal solitary waves (ISWs) with trapped cores propagating in a thin pycnocline were studied numerically within the framework of the Navier–Stokes equations for a stratified fluid. The peculiarity of this collision is that it involves trapped masses of a fluid. The interaction of ISWs differs for three classes of ISWs: (i) weakly non-linear waves without trapped cores, (ii) stable strongly non-linear waves with trapped cores, and (iii) shear unstable strongly non-linear waves. The wave phase shift of the colliding waves with equal amplitude grows as the amplitudes increase for colliding waves of classes (i) and (ii) and remains almost constant for those of class (iii). The excess of the maximum run-up amplitude, normalized by the amplitude of the waves, over the sum of the amplitudes of the equal colliding waves increases almost linearly with increasing amplitude of the interacting waves belonging to classes (i) and (ii); however, it decreases somewhat for those of class (iii). The colliding waves of class (ii) lose fluid trapped by the wave cores when amplitudes normalized by the thickness of the pycnocline are in the range of approximately between 1 and 1.75. The interacting stable waves of higher amplitude capture cores and carry trapped fluid in opposite directions with little mass loss. The collision of locally shear unstable waves of class (iii) is accompanied by the development of instability. The dependence of loss of energy on the wave amplitude is not monotonic. Initially, the energy loss due to the interaction increases as the wave amplitude increases. Then, the energy losses reach a maximum due to the loss of potential energy of the cores upon collision and then start to decrease. With further amplitude growth, collision is accompanied by the development of instability and an increase in the loss of energy. The collision process is modified for waves of different amplitudes because of the exchange of trapped fluid between colliding waves due to the conservation of momentum.

Frontal collision of internal solitary waves of first mode

The dynamics and energetics of a frontal collision of internal solitary waves (ISW) of first mode in a fluid with two homogeneous layers separated by a thin interfacial layer are studied numerically within the framework of the Navier–Stokes equations for stratified fluid. It was shown that the head-on collision of internal solitary waves of small and moderate amplitude results in a small phase shift and in the generation of dispersive wave train travelling behind the transmitted solitary wave. The phase shift grows as amplitudes of the interacting waves increase. The maximum run-up amplitude during the wave collision reaches a value larger than the sum of the amplitudes of the incident solitary waves. The excess of the maximum run-up amplitude over the sum of the amplitudes of the colliding waves grows with the increasing amplitude of interacting waves of small and moderate amplitudes whereas it decreases for colliding waves of large amplitude. Unlike the waves of small and moderate amplitudes collision of ISWs of large amplitude was accompanied by shear instability and the formation of Kelvin–Helmholtz (KH) vortices in the interface layer, however, subsequently waves again become stable. The loss of energy due to the KH instability does not exceed 5%–6%. An interaction of large amplitude ISW with even small amplitude ISW can trigger instability of larger wave and development of KH billows in larger wave. When smaller wave amplitude increases the wave interaction was accompanied by KH instability of both waves.

Head-on collision of ion acoustic solitary waves in electron-positron-ion nonthermal plasmas for weakly and highly relativistic regimes

A comparative study of the interactions between nonlinear ion acoustic solitary waves (IASWs) propagating toward each other, and the electrostatic nonlinear propagation of IASWs, both for the weakly and relativistic regimes consisting of relativistic warm ions, nonthermal electrons, and positrons, is carried out. Two-sided Korteweg–de Vries (KdV) equations are derived using the extended Poincaré-Lighthill-Kuo (PLK) method to reveal the physical issues concerned. The effects of positron concentration, ion-electron temperature ratio, electron-positron temperature ratio, relativistic streaming factor, the population of electron, and positron nonthermality on the electrostatic resonances and their phase shifts are investigated for both regimes. It is found that the plasma parameters significantly modify the phase shifts, electrostatic resonances, hump-shaped electrostatic potential profiles, and the electric fields on the nonlinear propagation characteristics of IASWs. The results obtained may be useful for clarifications of interaction between IASWs in astrophysical and laboratory plasmas, especially in pulsar magnetosphere, laser produced, inertial confinement plasmas, and pulsar relativistic winds with supernova ejecta that produce nonthermal electrons, positrons, and relativistic ions.