Seminar talk, 18 May 2020: Difference between revisions
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| abstract = An arbitrary compact-support initial datum for the Korteweg-de Vries equation asymptotically splits into solitons and a radiation tail, moving in opposite direction. We give a simple method to predict the number and amplitudes of resulting solitons and some integral characteristics of the tail using only conservation laws. | | abstract = An arbitrary compact-support initial datum for the Korteweg-de Vries equation asymptotically splits into solitons and a radiation tail, moving in opposite direction. We give a simple method to predict the number and amplitudes of resulting solitons and some integral characteristics of the tail using only conservation laws. | ||
A similar technique allows to predict details of the behavior of a soliton which, while moving in non-dissipative and dispersion-constant medium encounters a finite-width barrier with varying dissipation and/or dispersion; beyond the layer dispersion is constant (but not necessarily of the same value) and dissipation is null. The process is described with a special type generalized KdV-Burgers equation | A similar technique allows to predict details of the behavior of a soliton which, while moving in non-dissipative and dispersion-constant medium encounters a finite-width barrier with varying dissipation and/or dispersion; beyond the layer dispersion is constant (but not necessarily of the same value) and dissipation is null. The process is described with a special type generalized KdV-Burgers equation <math>u_t=(u^2+f(x)u_{xx})_x</math>. | ||
The transmitted wave either retains the form of a soliton (though of different parameters) or scatters a into a number of them. And a reflection wave may be negligible or absent. This models a situation similar to a light passing from a humid air to a dry one through the vapor saturation/condensation area. Some rough estimations for a prediction of an output are given using the relative decay of the KdV conserved quantities; in particular a formula for a number of solitons in the transmitted signal is given. | The transmitted wave either retains the form of a soliton (though of different parameters) or scatters a into a number of them. And a reflection wave may be negligible or absent. This models a situation similar to a light passing from a humid air to a dry one through the vapor saturation/condensation area. Some rough estimations for a prediction of an output are given using the relative decay of the KdV conserved quantities; in particular a formula for a number of solitons in the transmitted signal is given. | ||
Revision as of 14:16, 11 May 2020
Speaker: Alexey Samokhin
Title: Using the KdV conserved quantities in problems of splitting of initial data and reflection / refraction of solitons in varying dissipation and/or dispersion media
Abstract:
An arbitrary compact-support initial datum for the Korteweg-de Vries equation asymptotically splits into solitons and a radiation tail, moving in opposite direction. We give a simple method to predict the number and amplitudes of resulting solitons and some integral characteristics of the tail using only conservation laws.
A similar technique allows to predict details of the behavior of a soliton which, while moving in non-dissipative and dispersion-constant medium encounters a finite-width barrier with varying dissipation and/or dispersion; beyond the layer dispersion is constant (but not necessarily of the same value) and dissipation is null. The process is described with a special type generalized KdV-Burgers equation .
The transmitted wave either retains the form of a soliton (though of different parameters) or scatters a into a number of them. And a reflection wave may be negligible or absent. This models a situation similar to a light passing from a humid air to a dry one through the vapor saturation/condensation area. Some rough estimations for a prediction of an output are given using the relative decay of the KdV conserved quantities; in particular a formula for a number of solitons in the transmitted signal is given.
Language: English