scholarly journals Brief communication: Multiscaled solitary waves

2017 ◽  
Vol 24 (4) ◽  
pp. 695-700 ◽  
Author(s):  
Oleg G. Derzho
Keyword(s):  
Structural Stability ◽  
Solitary Waves ◽  
Vortex Core ◽  
Multiple Scales ◽  
Kdv Equation ◽  
Higher Order ◽  
Internal Solitary Waves ◽  
Density Profiles ◽  
The Core ◽  
Stability Of Waves

Abstract. It is analytically shown how competing nonlinearities yield multiscaled structures for internal solitary waves in stratified shallow fluids. These solitary waves only exist for large amplitudes beyond the limit of applicability of the Korteweg–de Vries (KdV) equation or its usual extensions. The multiscaling phenomenon exists or does not exist for almost identical density profiles. The trapped core inside the wave prevents the appearance of such multiple scales within the core area. The structural stability of waves of large amplitudes is briefly discussed. Waves of large amplitudes displaying quadratic, cubic and higher-order nonlinear terms have stable and unstable branches. Multiscaled waves without a vortex core are shown to be structurally unstable. It is anticipated that multiscaling phenomena will exist for solitary waves in various physical contexts.

scholarly journals Multiscaled Solitary Waves

2017 ◽  
Author(s):  
Oleg G. Derzho
Keyword(s):  
Solitary Waves ◽  
Multiple Scales ◽  
Kdv Equation ◽  
Core Area ◽  
Internal Solitary Waves ◽  
Density Profiles ◽  
The Core ◽  
The Kdv Equation

Abstract. It is analytically shown how competing nonlinearities yield new multiscaled (multi humped) structures for internal solitary waves in shallow fluids. These solitary waves only exist for large amplitudes beyond the limit of applicability of the KdV equation or its usual extensions. Multiscaling phenomenon exists or do not exist for almost identical density profiles. Trapped core inside the wave prevents appearance of such multiple scales within the core area. It is anticipated that multiscaling phenomena exist for solitary waves in various physical origins.

scholarly journals Higher-order Korteweg-de Vries models for internal solitary waves in a stratified shear flow with a free surface

2002 ◽  
Vol 9 (3/4) ◽  
pp. 221-235 ◽  
Author(s):  
R. Grimshaw ◽  
E. Pelinovsky ◽  
O. Poloukhina
Keyword(s):  
Free Surface ◽  
Shear Flow ◽  
Solitary Waves ◽  
Vries Equation ◽  
Wave Theory ◽  
Higher Order ◽  
Internal Solitary Waves ◽  
Long Wave ◽  
De Vries ◽  
Stratified Shear Flow

Abstract. A higher-order extension of the familiar Korteweg-de Vries equation is derived for internal solitary waves in a density- and current-stratified shear flow with a free surface. All coefficients of this extended Korteweg-de Vries equation are expressed in terms of integrals of the modal function for the linear long-wave theory. An illustrative example of a two-layer shear flow is considered, for which we discuss the parameter dependence of the coefficients in the extended Korteweg-de Vries equation.

Mode-2 internal solitary waves offshore Central America discovered by seismic oceanography method

2020 ◽  
Author(s):  
Wenhao Fan ◽  
Haibin Song ◽  
Yi Gong ◽  
Shaoqing Sun ◽  
Kun Zhang
Keyword(s):  
Phase Velocity ◽  
Central America ◽  
Solitary Waves ◽  
Kdv Equation ◽  
Seismic Survey ◽  
Internal Solitary Waves ◽  
Acquisition Time ◽  
Apparent Velocity ◽  
Seismic Oceanography ◽  
Mode 2

<p>In the past, most of the internal solitary waves (ISWs) found by seismic oceanography (SO) method were mode-1 ISWs. We discover many mode-2 ISWs in the Pacific coast of Central America by using SO method for the first time. These mode-2 ISWs are convex mode-2 ISWs with the maximum amplitudes of about 10 m, and most of them are ISWs with smaller amplitudes. The pycnocline for the mode-2 ISWs on the shelf (ISW3) is displaced 6.4% of the total seawater depth from the mid-depth of the total seawater. The deviation is large, and it shows a strong asymmetry feature of the peaks and troughs on the seismic profile. This is consistent with the results of previous numerical simulation. Observing the changes in the fine structure of mode-2 ISWs packet through pre-stack migration, it was found that the overall waveform of the three mode-2 ISWs (ISW1, ISW2, and ISW3) on the shelf during the acquisition time period of about 40 seconds is stable. The apparent phase velocity of these mode-2 ISWs calculated by the pre-stack migration section using the Common Offset Gathers is about 0.5 m/s, and their apparent propagation directions are from SW to NE along the seismic line (44 ° N, 0° pointing north). The vertical amplitude distribution and estimated apparent velocities of these mode-2 ISWs are basically consistent with the theoretical values ​​calculated from the KdV equation. By analyzing the apparent velocities of the three mode-2 ISWs (ISW1, ISW3, and ISW5) with relatively small apparent velocity errors, it is found that the apparent velocity of mode-2 ISWs generally increases with the increasing depth of seawater. In addition, the apparent phase velocity of the mode-2 ISWs with a larger maximum amplitude is generally larger. Based on the analysis of hydrological data in the study area, it was found that a strong anticyclone developed on the northwest side of the seismic survey line and a weaker anticyclone developed on the southeast side. These anticyclones will increase the depth of the thermocline in the surrounding seawater. According to previous studies, the deepening of the thermocline (pycnocline) maybe conducive to the generation of mode-2 ISWs.</p>

Interaction of Fully-Nonlinear Internal Solitary Waves of Opposite Polarity

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

<p>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.  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.<span> </span></p>

Higher-Order Nonlinear Effects on Wave Structures in a Multispecies Plasma with Nonisothermal Electrons

2010 ◽  
Vol 65 (4) ◽  
pp. 315-328 ◽  
Author(s):  
Tarsem Singh Gill ◽  
Parveen Bala ◽  
Harvinder Kaur
Keyword(s):  
Solitary Waves ◽  
Kdv Equation ◽  
Negative Ions ◽  
Nonlinear Effects ◽  
Higher Order ◽  
Relative Temperature ◽  
Generalized Kdv Equation ◽  
Ion Acoustic Solitary Waves ◽  
Ion Acoustic ◽  
Fast Ion

In the present investigation, we have studied ion-acoustic solitary waves in a plasma consisting of warm positive and negative ions and nonisothermal electron distribution. We have used reductive perturbation theory (RPT) and derived a dispersion relation which supports only two ion-acoustic modes, viz. slow and fast. The expression for phase velocities of these modes is observed to be a function of parameters like nonisothermality, charge and mass ratio, and relative temperature of ions. A modified Korteweg-de Vries (KdV) equation with a (1+1/2) nonlinearity, also known as Schamel-mKdV model, is derived. RPT is further extended to include the contribution of higher-order terms. The results of numerical computation for such contributions are shown in the form of graphs in different parameter regimes for both, slow and fast ion-acoustic solitary waves having several interesting features. For the departure from the isothermally distributed electrons, a generalized KdV equation is derived and solved. It is observed that both rarefactive and compressive solitons exist for the isothermal case. However, nonisothermality supports only the compressive type of solitons in the given parameter regime.

scholarly journals A model for large-amplitude internal solitary waves with trapped cores

2010 ◽  
Vol 17 (4) ◽  
pp. 303-318 ◽  
Author(s):  
K. R. Helfrich ◽  
B. L. White
Keyword(s):  
Solitary Waves ◽  
Large Amplitude ◽  
Numerical Solutions ◽  
Initial Conditions ◽  
Matching Condition ◽  
Shear Instability ◽  
Internal Solitary Waves ◽  
Ambient Flow ◽  
The Core ◽  
Core Boundary

Abstract. Large-amplitude internal solitary waves in continuously stratified systems can be found by solution of the Dubreil-Jacotin-Long (DJL) equation. For finite ambient density gradients at the surface (bottom) for waves of depression (elevation) these solutions may develop recirculating cores for wave speeds above a critical value. As typically modeled, these recirculating cores contain densities outside the ambient range, may be statically unstable, and thus are physically questionable. To address these issues the problem for trapped-core solitary waves is reformulated. A finite core of homogeneous density and velocity, but unknown shape, is assumed. The core density is arbitrary, but generally set equal to the ambient density on the streamline bounding the core. The flow outside the core satisfies the DJL equation. The flow in the core is given by a vorticity-streamfunction relation that may be arbitrarily specified. For simplicity, the simplest choice of a stagnant, zero vorticity core in the frame of the wave is assumed. A pressure matching condition is imposed along the core boundary. Simultaneous numerical solution of the DJL equation and the core condition gives the exterior flow and the core shape. Numerical solutions of time-dependent non-hydrostatic equations initiated with the new stagnant-core DJL solutions show that for the ambient stratification considered, the waves are stable up to a critical amplitude above which shear instability destroys the initial wave. Steadily propagating trapped-core waves formed by lock-release initial conditions also agree well with the theoretical wave properties despite the presence of a "leaky" core region that contains vorticity of opposite sign from the ambient flow.

EXACT SOLUTIONS OF COUPLED KdV EQUATION DERIVED FROM THE COUPLED NLS EQUATION USING MULTIPLE SCALES METHOD

2011 ◽  
Vol 25 (29) ◽  
pp. 4021-4028
Author(s):  
FILIZ TAŞCAN ◽  
AHMET BEKIR
Keyword(s):  
Exact Solutions ◽  
Multiple Scales ◽  
Kdv Equation ◽  
Higher Order ◽  
Multiple Scales Method ◽  
Nonlinear Schrödinger Equations ◽  
Schrödinger Equations ◽  
Nonlinear Schrödinger ◽  
Nonlinear Schrodinger Equations ◽  
Coupled Kdv Equation

In this paper, we have considered a generalized coupled higher-order nonlinear Schrödinger equations. We applied multiple scales method to coupled higher-order nonlinear Schrödinger equations to derive the coupled KdV equation. Then, the extended tanh method is applied to derived coupled KdV equation for exact solutions. The obtained results include solitary wave solutions.

Strongly dispersive internal solitary waves transformation over slope-shelf topography

2020 ◽  
pp. 2150031
Author(s):  
Changhong Zhi ◽  
Ke Chen ◽  
Yun-Xiang You
Keyword(s):  
Solitary Waves ◽  
Kdv Equation ◽  
Bottom Topography ◽  
Finite Depth ◽  
Variable Coefficients ◽  
Internal Solitary Waves ◽  
Initial Amplitude ◽  
Layer System ◽  
Amplitude Variation ◽  
Coupling Terms

The evolution of strongly dispersive internal solitary waves (ISWs) over slope-shelf topography is studied in a two-layer system of finite depth. We consider the high-order vmeKdV model extending the Korteweg-de Vries (KdV) equation with coupling terms of [Formula: see text] order to treat the strong dispersion in the problem which has variable coefficients to adapt the varying bottom topography. The strongly dispersive initial ISW is characterized by the meKdV equation according to the comparison with experiments and can be propagated by the vmeKdV equation according to the comparison between vmeKdV and vKdV theories. The vmeKdV equation is numerically implemented adopting the finite difference scheme. Three dimensionless ISW amplitudes [Formula: see text], 1.136, 1.41 and two slope inclinations [Formula: see text], 1/10 are considered. The deformation of the ISW is observed when a wave propagates past over the slope. The balancing of shoaling effect and energy dispersion determine the amplitude variation. In the cases of mild or steeper slopes, the terminal wave has a stable profile and amplitude, commonly consistent to the meKdV profile with smaller amplitude. In a particular case of mild slope with very small initial amplitude, the terminal wave amplitude grows larger than the original value.

Higher-order contributions to ion-acoustic solitary waves in a warm multicomponent plasma with an electron beam

2000 ◽  
Vol 63 (2) ◽  
pp. 139-155 ◽  
Author(s):  
W. M. MOSLEM
Keyword(s):  
Electron Beam ◽  
Solitary Waves ◽  
Kdv Equation ◽  
Negative Ions ◽  
Type Equation ◽  
Higher Order ◽  
Second Order ◽  
Negative Ion ◽  
Ion Acoustic Solitary Waves ◽  
Ion Acoustic

Higher-order contributions in reductive perturbation theory are studied for small- but finite-amplitude ion-acoustic solitary waves in a warm plasma with negative-ion, positron and electron constituents traversed by a warm electron beam (with different temperatures and pressures). The basic set of fluid equations are reduced to a Korteweg–de Vries (KdV) equation for the first-order perturbed potential and a linear inhomogeneous KdV-type equation for the second-order perturbed potential. At the critical negative-ion density, the coefficient of the nonlinear term in the KdV equation vanishes. A new set of stretched coordinates is then used to derive a modified KdV equation and a linear inhomogeneous modified KdV-type equation at the critical density of negative ions for the second-order perturbed potential. Stationary solutions of the coupled equations, for both cases, are obtained using a renormalization method.

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