scholarly journals Analytical Solutions and Collision Stability Analysis of Solitary Waves in a Pre-compressed One-dimensional Granular Crystal

2021 ◽  
Author(s):  
Zhi-Guo Liu ◽  
Jinliang Zhang ◽  
Yue-Sheng Wang ◽  
Guoliang Huang
Keyword(s):  
Solitary Wave ◽  
Solitary Waves ◽  
Solitary Wave Solution ◽  
Original System ◽  
Bilinear Method ◽  
One Dimensional ◽  
Balance Principle ◽  
The Difference ◽  
Granular Crystal ◽  
The Stability

Abstract In this paper, the governing equation in a pre-compressed one-dimensional granular crystal, which was previously discussed by Nesterenko [J. Appl. Mech. Phys. 24, 733 (1983)], is solved analytically. Multiple solitary wave solutions are obtained by using the homogeneous balance principle and Hirota’s bilinear method. We analyze the difference between the original system and the KdV system and examine the collision of solitary waves in some special parameters. The dynamic behavior and stability of the double solitary waves are also studied. We find that the opposite collision between single solitary waves may be stable and thus generate a stable double solitary wave. It is concluded that the collision is a special stable double solitary wave solution. We further propose a possible way to determine the stability of multiple solitary waves qualitatively.

Solitary wave solutions, lump solutions and interactional solutions to the (2+1)-dimensional potential Kadomstev–Petviashvili equation

2019 ◽  
Vol 34 (04) ◽  
pp. 2050055
Author(s):  
Jiang-Su Geng ◽  
Hai-Qiang Zhang
Keyword(s):  
Solitary Wave ◽  
Solitary Waves ◽  
Wave Solution ◽  
Solitary Wave Solution ◽  
Lump Solution ◽  
Bilinear Method ◽  
Elastic Interactions ◽  
Petviashvili Equation ◽  
Pkp Equation ◽  
Hirota Bilinear

In this paper, the [Formula: see text]-solitary wave solution to the (2[Formula: see text]+[Formula: see text]1)-dimensional potential Kadomstev–Petviashvili (PKP) equation is obtained with the Hirota bilinear method. Via the limit technique of long wave, the [Formula: see text]-lump solution can be derived from resulting [Formula: see text]-solitary wave solution. In addition, interactional solutions consisting of lumps and solitary waves for the PKP equation are obtained, which can describe elastic interactions of lumps and solitary waves. These results are illustrated by graphics of several sample examples.

A regularized model for strongly nonlinear internal solitary waves

2009 ◽  
Vol 629 ◽  
pp. 73-85 ◽  
Author(s):  
WOOYOUNG CHOI ◽  
RICARDO BARROS ◽  
TAE-CHANG JO
Keyword(s):  
Stability Analysis ◽  
Solitary Wave ◽  
Solitary Waves ◽  
Wave Solution ◽  
Wave Amplitude ◽  
Solitary Wave Solution ◽  
Shear Instability ◽  
Internal Solitary Waves ◽  
Strongly Nonlinear ◽  
Regularized Model

The strongly nonlinear long-wave model for large amplitude internal waves in a two-layer system is regularized to eliminate shear instability due to the wave-induced velocity jump across the interface. The model is written in terms of the horizontal velocities evaluated at the top and bottom boundaries instead of the depth-averaged velocities, and it is shown through local stability analysis that internal solitary waves are locally stable to perturbations of arbitrary wavelengths if the wave amplitudes are smaller than a critical value. For a wide range of depth and density ratios pertinent to oceanic conditions, the critical wave amplitude is close to the maximum wave amplitude and the regularized model is therefore expected to be applicable to the strongly nonlinear regime. The regularized model is solved numerically using a finite-difference method and its numerical solutions support the results of our linear stability analysis. It is also shown that the solitary wave solution of the regularized model, found numerically using a time-dependent numerical model, is close to the solitary wave solution of the original model, confirming that the two models are asymptotically equivalent.

Solitary wave solutions, lump solutions and interactional solutions to the (3 + 1)-dimensional generalized B-type Kadomtsev-Petviashvili equation

2020 ◽  
Vol 34 (07) ◽  
pp. 2050053
Author(s):  
Min Gao ◽  
Hai-Qiang Zhang
Keyword(s):  
Solitary Wave ◽  
Solitary Wave Solution ◽  
Lump Solution ◽  
Solitary Wave Solutions ◽  
Bilinear Method ◽  
Long Wave ◽  
Petviashvili Equation ◽  
Dimensional Equation ◽  
Hirota Bilinear ◽  
Bkp Equation

In this paper, we investigate a [Formula: see text]-dimensional B-type Kadomtsev-Petviashvili (BKP) equation, which is a generalization of the [Formula: see text]-dimensional equation. Based on the Hirota bilinear method and the limit technique of long wave, we systematically construct a family of exact solutions of BKP equation including the [Formula: see text]-solitary wave solution, lump solution as well as interaction solution between lump waves and solitary waves.

Solitary-wave solution for a complex one-dimensional field

1976 ◽  
Vol 59 (4) ◽  
pp. 255-258 ◽  
Author(s):  
S. Sarker ◽  
S.E. Trullinger ◽  
A.R. Bishop
Keyword(s):  
Solitary Wave ◽  
Wave Solution ◽  
Solitary Wave Solution ◽  
One Dimensional

Observations of solitary wave dynamics of film flows

2001 ◽  
Vol 435 ◽  
pp. 191-215 ◽  
Author(s):  
M. VLACHOGIANNIS ◽  
V. BONTOZOGLOU
Keyword(s):  
Solitary Wave ◽  
Solitary Waves ◽  
Inelastic Interaction ◽  
Imaging Method ◽  
Glycerol Solution ◽  
Generation Process ◽  
Transient Phenomena ◽  
Water Glycerol ◽  
Front Running ◽  
The Difference

Experimental results are reported on non-stationary evolution and interactions of waves forming on water and water–glycerol solution flowing along an inclined plane. A nonlinear wave generation process leads to a large number of solitary humps with a wide variety of sizes. A uorescence imaging method is applied to capture the evolution of film height in space and time with accuracy of a few microns. Coalescence – the inelastic interaction of solitary waves resulting in a single hump – is found to proceed at a timescale correlated to the difference in height between the interacting waves. The correlation indicates that waves of similar height do not merge. Transient phenomena accompanying coalescence are reported. The front-running ripples recede during coalescence, only to reappear when the new hump recovers its teardrop shape. The tail of the resulting solitary wave develops an elevated substrate relative to the front, which decays exponentially in time; both observations about the tail confirm theoretical predictions. In experiments with water, the elevated back substrate is unstable, yielding to a tail oscillation with wavelength similar to that of the front-running ripples. This instability plays a key role in two complex interaction phenomena observed: the nucleation of a new crest between two interacting solitary humps and the splitting of a large hump (that has grown through multiple coalescence events) into solitary waves of similar size.

scholarly journals Numerical approximation to Benjamin type equations. Generation and stability of solitary waves

2020 ◽  
Author(s):  
VA Dougalis ◽  
A Duran ◽  
Dimitrios Mitsotakis
Keyword(s):  
Solitary Wave ◽  
Solitary Waves ◽  
Numerical Approximation ◽  
Solitary Wave Solutions ◽  
Time Stepping ◽  
Runge Kutta Method ◽  
Stability Of Solitary Waves ◽  
The Stability ◽  
Numerical Generation ◽  
The Waves

© 2018 Elsevier B.V. This paper is concerned with the study, by computational means, of the generation and stability of solitary-wave solutions of generalized versions of the Benjamin equation. The numerical generation of the solitary-wave profiles is accurately performed with a modified Petviashvili method which includes extrapolation to accelerate the convergence. In order to study the dynamics of the solitary waves the equations are discretized in space with a Fourier pseudospectral collocation method and a fourth-order, diagonally implicit Runge–Kutta method of composition type as time-stepping integrator. The stability of the waves is numerically studied by performing experiments with small and large perturbations of the solitary pulses as well as interactions of solitary waves.

scholarly journals Solitary Wave Dynamics Governed by the Modified FitzHugh–Nagumo Equation

2020 ◽  
Vol 15 (6) ◽  
Author(s):  
Aleksandra Gawlik ◽  
Vsevolod Vladimirov ◽  
Sergii Skurativskyi
Keyword(s):  
Solitary Wave ◽  
Solitary Waves ◽  
Traveling Wave ◽  
Propagation Velocity ◽  
Function Technique ◽  
Wave Dynamics ◽  
Wave Solutions ◽  
Nagumo Equation ◽  
The Stability ◽  
Nonlinear Wave Solutions

Abstract The paper deals with the studies of the nonlinear wave solutions supported by the modified FitzHugh–Nagumo (mFHN) system. It was proved in our previous work that the model, under certain conditions, possesses a set of soliton-like traveling wave (TW) solutions. In this paper, we show that the model has two solutions of the soliton type differing in propagation velocity. Their location in parametric space, and stability properties are considered in more details. Numerical results accompanied by the application of the Evans function technique prove the stability of fast solitary waves and instability of slow ones. A possible way of formation and annihilation of localized regimes in question is studied therein too.

Cure Monitoring of Adhesive for Composite/Metal Bonded Structure Based on Highly Nonlinear Solitary Waves

2019 ◽  
Author(s):  
Wu Bin ◽  
Li Mingzhi ◽  
Liu Xiucheng ◽  
Wang Heying ◽  
He Cunfu ◽  
...  
Keyword(s):  
Solitary Wave ◽  
Solitary Waves ◽  
Curing Process ◽  
One Dimensional ◽  
Granular Chain ◽  
Highly Nonlinear ◽  
Highly Nonlinear Solitary Waves ◽  
Simulation Results ◽  
The One ◽  
Nonlinear Solitary Waves

Abstract In this paper, a nondestructive evaluation technique based on highly nonlinear solitary waves (HNSWs) is proposed to monitor the curing process of adhesive for composite/metal bonded structure. HNSWs are mechanical waves with high energy intensity and non-distortive nature which can form and propagate in a nonlinear system, such as a one-dimensional granular chain. In the present study, a finite element model of the one-dimensional granular chain is established with the commercial software Abaqus, to study the reflection behavior of HNSWs at the interface between the particle at the end of chain and the sample. The simulation results show that the time of flight (TOF) of the primary reflected solitary wave decreases with the stiffness of the sample increases, and the amplitude ratio (AR) between the primary reflected solitary wave and the incident solitary wave increases. An HNSWs transducer based on the one-dimensional granular chain is designed and fabricated. The relationship between the characteristic parameters of the primary reflected solitary wave (TOF and AR) and the curing time of adhesive for a composite/metal bonded structure is experimentally investigated. The experiment results suggest that the TOF decreases and the AR increases as the epoxy cures. The experimental results are in good agreement with the simulation results. This study provides a new characterization method for monitoring the curing process of adhesive for composite/metal bonded structure.

Model Reduction of a Parametrically Excited Drivetrain

2012 ◽  
Author(s):  
Thomas Pumhoessel ◽  
Peter Hehenberger ◽  
Klaus Zeman
Keyword(s):  
Stability Margin ◽  
Original System ◽  
Reduced Model ◽  
Parametrically Excited ◽  
System Parameters ◽  
Additional Stability ◽  
System Equations ◽  
The Difference ◽  
Set Up ◽  
The Stability

The complexity of engineering systems is continuously increasing, resulting in mathematical models that become more and more computationally expensive. Furthermore, in model based design, for example, system parameters are subject of change, and therefore, the system equations have to be evaluated repeatedly. Hence, there is a need for providing reduced models which are as compact as possible, but still reflect the properties of the original model in a satisfactory manner. In this contribution, the reduction of differential equations with time-periodic coefficients, termed as parametrically excited systems, is investigated using the method of Proper Orthogonal Decomposition (POD). A reduced model is set up based on the solution of the original system for a certain parametric combination resonance of the difference type, resulting in an additional stability margin of the trivial solution. It is shown that the POD reduced model approximates the stability behavior of the original system much better than a modally reduced model even if system parameters are subject of change.

THE STABILITY OF THE SOLITARY WAVE SOLUTIONS TO THE GENERALIZED COMPOUND KDV EQUATION

2011 ◽  
Vol 04 (03) ◽  
pp. 475-480
Author(s):  
Xiaohua Liu ◽  
Weiguo Zhang
Keyword(s):  
Solitary Wave ◽  
Variational Method ◽  
Wave Solution ◽  
Lyapunov Functional ◽  
Solitary Wave Solution ◽  
Kdv Equation ◽  
Solitary Wave Solutions ◽  
Wave Solutions ◽  
The Stability ◽  
Lyapunov Sense

Using variational method, we investigate that the solitary wave solution u(x - ct) to the Generalized Compound Kdv Equation with two nonlinear terms is stable in the Lyapunov sense when 0 < p < 2 holds. The result is new. There shows a new method to consider the extremum of Lyapunov functional.

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