Perturbation Theory for the Compound
Soliton of the Gardner’s Equation;
Their Interaction and Evolution
in a Media with Variable Parameters
Irina A. Soustova, Konstantin A. Gorshkov, Alexey V. Ermoshkin,
Lev A. Ostrovsky and Yuliya I. Troitskaya
Introduction
The results of studying the interaction and propagation of compound solitons in the
media with variable parameters are reported. Such solitons are encountered in the
systems, in which the waves in the form of kinks (field jumps) may exist in addition
to the solitary waves. These solitons are referred to as compound as they may be
represented as an ensemble of interacting kinks which makes possible reduced
description of their evolution to the evolution of kinks. This approach, in contrast to
the traditional one [3], when the solitons are considered as an integral formation,
permits studying non-quasi-stationary processes related to the interaction of compound solitons of strongly differing sizes and velocities, as well as to their propagation in the media with variable parameters. Some of these processes will be
considered below within the framework of the Gardner equation with variable
coefficients
Φ t + ΦðαðtÞ − μðtÞΦÞΦ x + βΦ xxx = 0
ð1Þ
This equation is frequently used for describing internal waves in the shelf region
of the oceans and seas. Hence, the presented examples are of practical interest [4,
14, 16, 15, 17–19].
This contribution is a brief review of the results of an approximate description of
the evolution and interaction of composite solitons obtained by the authors in 2001–
2016. As one of the applications of the theory, the features of the evolution of
intense internal waves in the shelf zone of the ocean are analyzed.
I. A. Soustova ( ✉ ) ⋅ K. A. Gorshkov ⋅ A. V. Ermoshkin ⋅ L. A. Ostrovsky
Y. I. Troitskaya
Institute of Applied Physics, Russian Academy of Sciences, Nizhny Novgorod, Russia
e-mail: soustova@hydro.appl.sci-nnov.ru
© Springer International Publishing AG, part of Springer Nature 2018
M. G. Velarde et al. (eds.), The Ocean in Motion, Springer Oceanography,
https://doi.org/10.1007/978-3-319-71934-4_19
279
Soliton of the Gardner’s Equation;
Their Interaction and Evolution
in a Media with Variable Parameters
Irina A. Soustova, Konstantin A. Gorshkov, Alexey V. Ermoshkin,
Lev A. Ostrovsky and Yuliya I. Troitskaya
Introduction
The results of studying the interaction and propagation of compound solitons in the
media with variable parameters are reported. Such solitons are encountered in the
systems, in which the waves in the form of kinks (field jumps) may exist in addition
to the solitary waves. These solitons are referred to as compound as they may be
represented as an ensemble of interacting kinks which makes possible reduced
description of their evolution to the evolution of kinks. This approach, in contrast to
the traditional one [3], when the solitons are considered as an integral formation,
permits studying non-quasi-stationary processes related to the interaction of compound solitons of strongly differing sizes and velocities, as well as to their propagation in the media with variable parameters. Some of these processes will be
considered below within the framework of the Gardner equation with variable
coefficients
Φ t + ΦðαðtÞ − μðtÞΦÞΦ x + βΦ xxx = 0
ð1Þ
This equation is frequently used for describing internal waves in the shelf region
of the oceans and seas. Hence, the presented examples are of practical interest [4,
14, 16, 15, 17–19].
This contribution is a brief review of the results of an approximate description of
the evolution and interaction of composite solitons obtained by the authors in 2001–
2016. As one of the applications of the theory, the features of the evolution of
intense internal waves in the shelf zone of the ocean are analyzed.
I. A. Soustova ( ✉ ) ⋅ K. A. Gorshkov ⋅ A. V. Ermoshkin ⋅ L. A. Ostrovsky
Y. I. Troitskaya
Institute of Applied Physics, Russian Academy of Sciences, Nizhny Novgorod, Russia
e-mail: soustova@hydro.appl.sci-nnov.ru
© Springer International Publishing AG, part of Springer Nature 2018
M. G. Velarde et al. (eds.), The Ocean in Motion, Springer Oceanography,
https://doi.org/10.1007/978-3-319-71934-4_19
279