Collision Effects of Solitary Waves for the Gardner Equation

Statistical properties of the internal solitary wave ensemble

10.5194/egusphere-egu21-1830 ◽

2021 ◽

Author(s):

Tatiana Talipova ◽

Ekaterina Didenkulova ◽

Anna Kokorina ◽

Efim Pelinovsky

Keyword(s):

Wave Propagation ◽

Solitary Wave ◽

Solitary Waves ◽

Internal Solitary Wave ◽

Statistical Moments ◽

Nonlinear Propagation ◽

Positive Sign ◽

Cubic Nonlinearity ◽

Gardner Equation ◽

Water Stratification

Internal solitary wave ensembles are often observed on the ocean shelves. The long internal baroclinic tide is generated by a barotropic tide on the shelf edges, and then transforms into the soliton-like wave packets during the nonlinear propagation to the beach. The tide is a periodic process and the solitary wave ensemble appears on the shelf usually each semi-diurnal period of 12.4 hours. This process is very sensitive to the variation of the tide characteristics and the hydrology.We study the propagation of the soliton ensembles numerically in the framework of the spatial form of the Gardner equation (i.e., the Korteweg-de Vries equation with both, quadratic and cubic nonlinearities) assuming horizontally uniform background and applying periodic conditions in time. The water stratification and the local depth are taken similar to the conditions of the north-western Australian shelf, where the stratification admits the existence of solitons but not breathers. The numerical simulation is performed using the Gardner equation with the negative sign of the cubic nonlinearity. For the study of the statistic properties of the solitary waves we use the ensemble of 50 realizations with the same set of 13 solitary waves which are located randomly. The histograms of the wave amplitudes change as the waves travel. The histogram variations become significant after 50 km of the wave propagation. The third (skewness) and the fourth (kurtosis) statistical moments are computed versus the travel distance. It is shown that the both moments decrease by 20% when the solitary wave groups travel for about 150 km.A similar simulation is conducted for a variable background within the framework of the variable-coefficient Gardner equation. At some location the water stratification corresponds to the positive sign of the local coefficient of the cubic nonlinearity, and then internal breathers may exist. The wave propagation in horizontally inhomogeneous hydrology leads to the occurrence of complicated patterns of solitons and breathers; in the course of the transformation they can disintegrate or form internal rogue waves. Under these conditions the statistical moments of the wave field are essentially different from case when the breather-like waves cannot occur.The research was supported by the RFBR grants No 19-05-00161 (TT and EP) and 19-35-60022 (ED). The Foundation for the Advancement of Theoretical Physics and Mathematics &#8220;BASIS&#8221; (&#8470; 20-1-3-3-1) is also acknowledged by ED

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The transformation of an interfacial solitary wave of elevation at a bottom step

Nonlinear Processes in Geophysics ◽

10.5194/npg-16-33-2009 ◽

2009 ◽

Vol 16 (1) ◽

pp. 33-42 ◽

Cited By ~ 23

Author(s):

V. Maderich ◽

T. Talipova ◽

R. Grimshaw ◽

E. Pelinovsky ◽

B. H. Choi ◽

...

Keyword(s):

Solitary Wave ◽

Solitary Waves ◽

Numerical Study ◽

Nonlinear Effects ◽

Ocean Model ◽

Gardner Equation ◽

Fully Nonlinear ◽

Layer Flow ◽

The One ◽

Bottom Step

Abstract. In this paper we study the transformation of an internal solitary wave at a bottom step in the framework of two-layer flow, for the case when the interface lies close to the bottom, and so the solitary waves are elevation waves. The outcome is the formation of solitary waves and dispersive wave trains in both the reflected and transmitted fields. We use a two-pronged approach, based on numerical simulations of the fully nonlinear equations using a version of the Princeton Ocean Model on the one hand, and a theoretical and numerical study of the Gardner equation on the other hand. In the numerical experiments, the ratio of the initial wave amplitude to the layer thickness is varied up one-half, and nonlinear effects are then essential. In general, the characteristics of the generated solitary waves obtained in the fully nonlinear simulations are in reasonable agreement with the predictions of our theoretical model, which is based on matching linear shallow-water theory in the vicinity of a step with solutions of the Gardner equation for waves far from the step.

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Cylindrical and spherical solitary waves in a dusty non-thermal plasma

Journal of Plasma Physics ◽

10.1017/s0022377812000475 ◽

2012 ◽

Vol 78 (6) ◽

pp. 629-634 ◽

Cited By ~ 3

Author(s):

S. ISLAM ◽

A.A. MAMUN ◽

A. MANNAN

Keyword(s):

Solitary Waves ◽

Thermal Plasma ◽

Reductive Perturbation Method ◽

Charged Dust ◽

Laboratory Plasma ◽

Gardner Equation ◽

Modified Gardner Equation ◽

Non Thermal Plasma ◽

Basic Characteristics ◽

Basic Features

AbstractA theoretical investigation of the basic characteristics of cylindrical and spherical dust-ion-acoustic (DIA) solitary waves (SWs) is made in a dusty non-thermal plasma, whose constituents are non-thermal electrons, inertial ions, and arbitrarily charged stationary dust. The reductive perturbation method is used to derive the modified Gardner equation. The latter is numerically analyzed for both positively and negatively charged dust. The basic features of cylindrical and spherical DIA SWs, which are found to exist in such a dusty non-thermal plasma, are identified. The implications of our results to both space and laboratory plasma situations are also discussed briefly.

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Interaction of Fully-Nonlinear Internal Solitary Waves of Opposite Polarity

10.5194/egusphere-egu21-3045 ◽

2021 ◽

Author(s):

Kevin Lamb

Keyword(s):

Wave Field ◽

Solitary Waves ◽

Kdv Equation ◽

Nonlinear Coefficient ◽

Opposite Polarity ◽

Internal Solitary Waves ◽

Linear Wave ◽

Gardner Equation ◽

Fully Nonlinear ◽

The Kdv Equation

Previous studies have suggested that fully nonlinear internal solitary waves (ISWs) are very soliton-like as the interaction of two ISWs results in only very small changes in amplitude of the interacting ISWs and in the production of a very small amplitude wave train. Previous studies have, however, considered ISWs with the polarity predicted by the sign of the quadratic nonlinear coefficient of the KdV equation. The Gardner equation, which is an extension of the KdV equation that includes a cubic nonlinear term, has ISWs of two polarities (i.e., waves of depression and elevation) when the cubic coefficient of the Gardner equation is positive. These waves are soliton solutions of the Gardner equations. &#160;In this talk I will discuss the interaction of ISWs of opposite polarity in continuous asymmetric three layer stratifications. Regions in parameter space where ISWs of opposite polarity exist will be discussed and I will demonstrate via fully nonlinear numerical simulations that the interaction of ISWs of opposite polarity waves are far from soliton-like: their interaction can result in very large changes in wave amplitude and may produce a very complicated wave field with multiple large ISWs, a large linear wave field and breather-like waves.&#160;

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The Orbital Stability of Solitary Wave Solutions for the Generalized Gardner Equation and the Influence Caused by the Interactions between Nonlinear Terms

Complexity ◽

10.1155/2019/4209275 ◽

2019 ◽

Vol 2019 ◽

pp. 1-17

Author(s):

Wei-Guo Zhang ◽

Xing-Qian Ling ◽

Xiang Li ◽

Shao-Wei Li

Keyword(s):

Solitary Wave ◽

Numerical Simulations ◽

Hypergeometric Function ◽

Solitary Waves ◽

Orbital Stability ◽

Solitary Wave Solutions ◽

Gaussian Hypergeometric Function ◽

Gardner Equation ◽

Wave Solutions ◽

Generalized Gardner Equation

In this paper, the orbital stability of solitary wave solutions for the generalized Gardner equation is investigated. Firstly, according to the theory of orbital stability of Grillakis-Shatah-Strauss, a general conclusion is given to determine the orbital stability of solitary wave solutions. Furthermore, on the basis of the two bell-shaped solitary wave solutions of the equation, the explicit expressions of the orbital stability discriminants are deduced to give the orbitally stable and instable intervals for the two solitary waves as the wave velocity changing. Moreover, the influence caused by the interaction between two nonlinear terms is also discussed. From the conclusion, it can be seen that the influences caused by this interaction are apparently when 0<p<4, which shows the complexity of this system with two nonlinear terms. Finally, by deriving the orbital stability discriminant d′′(c) in the form of Gaussian hypergeometric function, the numerical simulations of several main conclusions are given in this paper.

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Numerical experiments for long nonlinear internal waves via Gardner equation with dual-power law nonlinearity

International Journal of Modern Physics C ◽

10.1142/s0129183119500669 ◽

2019 ◽

Vol 30 (09) ◽

pp. 1950066

Author(s):

Turgut Ak

Keyword(s):

Internal Waves ◽

Solitary Waves ◽

Numerical Experiments ◽

Test Problems ◽

Gardner Equation ◽

Von Neumann ◽

B Splines ◽

Nonlinear Internal Waves ◽

Physical Problems ◽

The Stability

This paper studies Gardner equation, which represents long nonlinear internal waves. The collocation method based on B-splines is applied to the equation. The stability of the proposed numerical scheme is analyzed by using von Neumann theory. To observe some physical properties of long nonlinear internal waves, three test problems which contain the propagation of solitary waves, the interaction of solitary waves and evolution of solitons are considered. Also, the effect of nonlinearity on physical problems is investigated. In order to see this effect clearly, the same parameters are used during the computation for different degrees of nonlinearity in each problem.

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