scholarly journals MULTIPARAMETRIC SOLUTIONS TO THE GARDNER EQUATION AND THE DEGENERATE RATIONAL CASE

2020 ◽  
Vol 0 (0) ◽  
pp. 0-0
Author(s):  
Pierre Gaillard ◽  
Keyword(s):  
Gardner Equation ◽  
Rational Case

THE EXACT TRAVELING WAVE SOLUTIONS AND THEIR BIFURCATIONS IN THE GARDNER AND GARDNER–KP EQUATIONS

2012 ◽  
Vol 22 (05) ◽  
pp. 1250126 ◽  
Author(s):  
FANG YAN ◽  
CUNCAI HUA ◽  
HAIHONG LIU ◽  
ZENGRONG LIU
Keyword(s):  
Traveling Wave ◽  
Function Method ◽  
Traveling Wave Solutions ◽  
Analytic Solutions ◽  
Auxiliary Function ◽  
Kp Equation ◽  
Gardner Equation ◽  
Exact Traveling Wave Solutions ◽  
Wave Solutions ◽  
Kp Equations

By using the method of dynamical systems, this paper studies the exact traveling wave solutions and their bifurcations in the Gardner equation. Exact parametric representations of all wave solutions as well as the explicit analytic solutions are given. Moreover, several series of exact traveling wave solutions of the Gardner–KP equation are obtained via an auxiliary function method.

Analysis of the breakage rate function for selected process parameters in quartzite milling

2012 ◽  
Vol 33 (1) ◽  
pp. 117-129 ◽  
Author(s):  
Tomasz Olejnik
Keyword(s):  
Process Parameters ◽  
Rate Function ◽  
Variable Number ◽  
Specific Rate ◽  
Gardner Equation ◽  
Contact Points ◽  
Batch System ◽  
Grinding Media ◽  
Breakage Rate ◽  
The Impact

Analysis of the breakage rate function for selected process parameters in quartzite milling The paper presents the results of studies on quartzite milling in a ball mill. The milling was conducted in a batch system, for diversified compositions of balls. The milling product was subjected to granulometrical, morphological and strength analyses. On the basis of the developed Reid's theory and using the Austin-Gardner equation, a form of the function circumscribing the specific rate of comminution of selected size fractions was determined. The values of the breakage rate function bi, j for the mill's apparatus conditions were determined. The impact was investigated for a variable number of grinding media contact points on the values of specific rate S and the values of the breakage rate function bi, j. Furthermore, the values of coefficients occurring in the equations circumscribing the specific rate of milling S and breakage parameter bi, j were determined.

Analytical and numerical studies of the variable-coefficient Gardner equation

2004 ◽  
Vol 152 (2) ◽  
pp. 449-471 ◽  
Author(s):  
O. Nakoulima ◽  
N. Zahibo ◽  
E. Pelinovsky ◽  
T. Talipova ◽  
A. Slunyaev ◽  
...  
Keyword(s):  
Variable Coefficient ◽  
Gardner Equation ◽  
Numerical Studies

Modulated periodic wavetrains in the spherical Gardner equation

Wave Motion ◽  
2021 ◽  
pp. 102844
Author(s):  
Gunay Aslanova ◽  
Ali Demirci ◽  
Semra Ahmetolan
Keyword(s):  
Gardner Equation

Solitons and (semi-)rational solutions for the (2+1)-dimensional Gardner equation

2021 ◽  
pp. 107883
Author(s):  
Han-Han Sheng ◽  
Li-Wen Xiao ◽  
Guo-Fu Yu ◽  
Yi-Ning Zhong
Keyword(s):  
Rational Solutions ◽  
Gardner Equation

Statistical properties of the internal solitary wave ensemble

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

<p>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.</p><p>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.</p><p>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.</p><p>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 “BASIS” (№ 20-1-3-3-1) is also acknowledged by ED</p>

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