Head-on 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.

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.

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.

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].

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.

Higher order dispersive effects in regularized Boussinesq equation

In this paper, we consider the higher order Boussinesq (HBq) equation which models the bi-directional propagation of longitudinal waves in various continuous media. The equation contains the higher order effects of frequency dispersion. The present study is devoted to the numerical investigation of the HBq equation. For this aim a numerical scheme combining the Fourier pseudo-spectral method in space and a Runge–Kutta method in time is constructed. The convergence of semi-discrete scheme is proved in an appropriate Sobolev space. To investigate the higher order dispersive effects and nonlinear effects on the solutions of HBq equation, propagation of single solitary wave, head-on collision of solitary waves and blow-up solutions are considered.

On head-on collision between two solitary waves in shallow water: the use of the extended PLK method

In the present work, we examined the head-on collision of solitary waves in shallow water theory, through the use of extended Poincare–Lighthill–Kuo (PLK) method based on the combination of reductive perturbation method with strained coordinates. Motivated with the result obtained by Ozden and Demiray (Int J Nonlinear Mech 69:66–70, 2015), we introduced a set of stretched coordinates that include some unknown functions which are to be determined so as to remove secularities that might occur in the solution. By expanding these unknown functions and the field variables into power series in the smallness parameter \(\epsilon \), introducing them into the field equations and imposing the conditions to remove the secularities, we obtained some evolution equations. By seeking a progressive wave solution to these evolution equations, we determined the speed correction terms and the phase-shift functions. The result obtained here is exactly the same with found by Ozden and Demiray (Int J Nonlinear Mech 69:66–70, 2015), wherein the analysis employed by Su and Mirie (J Fluid Mech 98:509–525, 1980) is utilized.

Kawahara solitons in Boussinesq equations using a robust Christov–Galerkin spectral method

We develop a robust Christov–Galerkin spectral technique for computing interacting localized wave solutions of and fourth and sixth-order generalized wave equations. To this end, a special complete orthonormal system of functions in L2(-∞,∞) is used whose rate of convergence is shown to be exponential for the cases under consideration. For the time-stepping, an implicit algorithm is chosen which makes use of the banded structure of the matrices representing the different spatial derivatives.As featuring examples, the head-on collision of solitary waves is investigated for a sixth-order generalized Boussinesq equation and a fourth-order Boussinesq type equation with a linear term. Its solutions comprise monotone shapes (sech-es) and damped oscillatory shapes (Kawahara solitons). The numerical results are validated against published data in the literature using the method of variational imbedding.

Head-on collision of solitary waves in coupled Korteweg–de Vries systems modeling nonlinear transmission lines

The extended Poincaré–Lighthill–Kuo (PLK) method is applied to characterize head-on collisions of solitary waves in a coupled Korteweg–de Vries (KdV) system that has multiple modes supporting solitons. As a simple physically realizable system, we investigate two coupled electrical nonlinear transmission lines (NLTLs), and the proposed method successfully leads to the collision-induced phase shifts and the wave equation that governs the dynamics of the pulses generated by colliding solitary waves.

Painlevé integrability of a generalized fifth-order KdV equation with variable coefficients: Exact solutions and their interactions

By means of singularity structure analysis, the integrability of a generalized fifth-order KdV equation is investigated. It is proven that this equation passes the Painlevé test for integrability only for three distinct cases. Moreover, the multi-soliton solutions are presented for this equation under three sets of integrable conditions. Finally, by selecting appropriate parameters, we analyze the evolution of two solitons, which is especially interesting as it may describe the overtaking and the head-on collisions of solitary waves of different shapes and different types.

Head-on collision of solitary waves in fluid-filled elastic tubes

In this work, treating the arteries as a thin walled, prestressed thin elastic tube and the blood as an inviscid fluid, we have studied the propagation of nonlinear waves, in the longwave approximation, through the use of extended PLK perturbation method. The results show that, up to O ( ϵ 2 ) , the head-on collision of two solitary waves is elastic and the solitary waves preserve their original properties after the collision. The leading-order analytical phase shifts and the trajectories of two solitons after the collision are derived explicitly.

Soliton solutions of two bidirectional sixth-order partial differential equations belonging to the KP hierarchy

In this letter, we analyse two bidirectional sixth-order partial differential equations, which are reductions in (1 + 1) dimensions of equations belonging to the KP hierarchy. They have fourth-order and fifth-order Lax pairs, respectively. We derive their Backlund transformations and, from the nonlinear superposition formula, we can build their soliton solutions like a Grammian. The interesting dynamics of these solitons is that they may describe not only the overtaking collision but also the head-on collision of solitary waves of different type and shape.

CKP and BKP Equations Related to the Generalized Quartic Hénon–Heiles Hamiltonian

In their classification of soliton equations from a group theoretical standpoint according to the representation of infinite Lie algebras, Jimbo and Miwa listed bilinear equations of low degree for the KP and the modified KP hierarchies. In this list, we consider the (1+1)-dimensional reductions of three particular equations of special interest for establishing some new links with the generalized Henon–Heiles Hamiltonian, possibly useful for integrating the latter with functions having the Painleve property. Two of those partial differential equations have N-soliton solutions that, as for the Kaup–Kupershmidt equation, can be written as the logarithmic derivative of a Grammian. Moreover, they can describe head-on collisions of solitary waves of different type and shape.

Numerical simulations of large-amplitude internal solitary waves

A numerical model based on the incompressible two-dimensional Navier–Stokes equations in the Boussinesq approximation is used to study mode-2 internal solitary waves propagating on a pycnocline between two deep layers of different densities. Numerical experiments on the collapse of an initially mixed region reveal a train of solitary waves with the largest leading wave enclosing an intrusional ‘bulge’. The waves gradually decay as they propagate along the horizontal direction, with a corresponding reduction in the size of the bulge. When the normalized wave amplitude, a, falls below the critical value ac=1.18, the wave is no longer able to transport mixed fluid as it propagates away from the mixed region, and a sharp-nosed intrusion is left behind. The wave structure is studied using a Lagrangian particle tracking scheme which shows that for small amplitudes the bulges have a well-defined elliptic shape. At larger amplitudes, the bulge entrains and mixes fluid from the outside while instabilities develop in the rear part of the bulge. Results are obtained for different wave amplitudes ranging from small-amplitude ‘regular’ waves with a=0.7 to highly nonlinear unstable waves with a=3.8. The dependence of the wave speed and wavelength on amplitude is measured and compared with available experimental data and theoretical predictions. Consistent with experiments, the wave speed increases almost linearly with amplitude at small values of a. As a becomes large, the wave speed increases with amplitude at a smaller rate, which gradually approaches the asymptotic limit for a two-fluid model. Results show that in the parameter range considered the wave amplitude decreases linearly with time at a rate inversely proportional to the Reynolds number. Numerical experiments are also conducted on the head-on collision of solitary waves. The simulations indicate that the waves experience a negative phase shift during the collision, in accordance with experimental observations. Computations are used to determine the dependence of the phase shift on the wave amplitude.

A Boussinesq system for two-way propagation of nonlinear dispersive waves

In this report, we study the system N t + W x + (NW) x − 1 6 N xxt = 0 , W t + N x + WW x − 1 6 W xxt = 0 , (∗) which describes approximately the two-dimensional propagation of surface waves in a uniform horizontal channel of length L 0 filled with an irrotational, incompressible, inviscid fluid which in its undisturbed state has depth h. The non-dimensional variables N( x, t) and W( x, t) represent the deviation of the water surface from its undisturbed position and the horizontal velocity at water level 2 3h , respectively. The natural initial-boundary-value problem corresponding to the situation wherein the channel is fitted with a wavemaker at both ends is formulated and analyzed theoretically. We then present a numerical algorithm for the approximation of solutions of the system (∗) and prove the algorithm is fourth-order accurate both in time and in space, is unconditionally stable, and has optimal computational complexity, which is to say the operation cost per time step is of order M where M is the number of grid points in the spatial discretization. Further, we implement the algorithm as a computer code and use it to study head-on collisions of solitary waves. Our numerical simulations are compared with existing theoretical, numerical and experimental results. We have tentatively concluded that the system (∗) is a good candidate for modeling two-dimensional surface water waves.

INELASTIC INTERACTION OF BOUSSINESQ SOLITONS

Two improved versions of Boussinesq equation (Boussinesq paradigm) have been considered which are well-posed (correct in the sense of Hadamard) as an initial value problem: the Proper Boussinesq Equation (PBE) and the Regularized Long Wave Equation (RLWE). Fully implicit difference schemes have been developed strictly representing, on difference level, the conservation or balance laws for the mass, pseudoenergy or pseudomomentum of the wave system. Thresholds of possible nonlinear blow-up are identified for both PBE and RLWE. The head-on collisions of solitary waves of the sech type (Boussinesq solitons) have been investigated. They are subsonic and negative (surface depressions) for PBE and supersonic and positive (surface elevations) for RLWE. The numerically recovered sign and sizes of the phase shifts are in very good quantitative agreement with analytical results for the two-soliton solution of PBE. The subsonic surface elevations are found to be not permanent but to gradually transform into oscillatory pulses whose support increases and amplitude decreases with time although the total pseudoenergy is conserved within 10−10. The latter allows us to claim that the pulses are solitons despite their “aging” (which is felt on times several times the time-scale of collision). For supersonic phase speeds, the collision of Boussinesq solitons has inelastic character exhibiting not only a significant phase shift but also a residual signal of sizable amplitude but negligible pseudoenergy. The evolution of the residual signal is investigated numerically for very long times.

Oblique and head-on collisions of solitary waves in Marangoni–Bénard convection

In Marangoni–Bénard convection with mass transfer from gas phase to liquid phase, it is found that solitary waves, excited and sustained by Marangoni stresses, are very stable during interactions with others, keeping their original shapes and celerities after interactions, thus experiencing at most a phase shift. Both positive and negative phase shifts have been observed for different types of oblique interaction. A ‘‘critical’’ angle, at which zero phase shift occurs as a result of the interaction, has been defined to delineate regions of negative and positive phase shifts.