ON THE LONG-TIME ASYMPTOTIC BEHAVIOR OF THE MODIFIED KORTEWEG-DE VRIES EQUATION WITH STEP-LIKE INITIAL DATA
The result's identifiers
Result code in IS VaVaI
<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F00216208%3A11320%2F20%3A10422104" target="_blank" >RIV/00216208:11320/20:10422104 - isvavai.cz</a>
Result on the web
<a href="https://verso.is.cuni.cz/pub/verso.fpl?fname=obd_publikace_handle&handle=10mWN~by2X" target="_blank" >https://verso.is.cuni.cz/pub/verso.fpl?fname=obd_publikace_handle&handle=10mWN~by2X</a>
DOI - Digital Object Identifier
<a href="http://dx.doi.org/10.1137/19M1279964" target="_blank" >10.1137/19M1279964</a>
Alternative languages
Result language
angličtina
Original language name
ON THE LONG-TIME ASYMPTOTIC BEHAVIOR OF THE MODIFIED KORTEWEG-DE VRIES EQUATION WITH STEP-LIKE INITIAL DATA
Original language description
We study the long-time asymptotic behavior of the solution q(x, t), x is an element of R, t is an element of R+, of the modified Korteweg-de Vries equation (MKdV) q(t) + 6q(2)q(x) + q(xxx) = 0 with step-like initial datum q(x,0) -> {( c- for x -> -infinity,)(c+ for x -> +infinity,) with c(-) > c(+) >= 0. For the step initial data q(x, 0) = {(c+) (c- for x <= 0,)(for x > 0) the solution develops an oscillatory region called the dispersive shock wave region that connects the two constant regions c(+) and c(-). We show that the dispersive shock wave is described by a modulated periodic traveling wave solution of the MKdV equation where the modulation parameters evolve according to a Whitham modulation equation. The oscillatory region is expanding within a cone in the (x,t) plane defined as -6c(-)(2) + 12c(+)(2) + < x/t < 4c(-)(2)+ 2c(+)(2), with t >> 1. For step-like initial data we show that the solution decomposes for long times into three main regions: (1) a region where solitons and breathers travel with positive velocities on a constant background c(+); (2) an expanding oscillatory region (that generically contains breathers); (3) a region of breathers traveling with negative velocities on the constant background c(-). When the oscillatory region does not contain breathers, the form of the asymptotic solution coincides up to a phase shift with the dispersive shock wave solution obtained for the step initial data. The phase shift depends on the solitons, the breathers, and the radiation of the initial data. This shows that the dispersive shock wave is a coherent structure that interacts in an elastic way with solitons, breathers, and radiation.
Czech name
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Czech description
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Classification
Type
J<sub>imp</sub> - Article in a specialist periodical, which is included in the Web of Science database
CEP classification
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OECD FORD branch
10101 - Pure mathematics
Result continuities
Project
—
Continuities
I - Institucionalni podpora na dlouhodoby koncepcni rozvoj vyzkumne organizace
Others
Publication year
2020
Confidentiality
S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů
Data specific for result type
Name of the periodical
SIAM Journal on Mathematical Analysis
ISSN
0036-1410
e-ISSN
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Volume of the periodical
52
Issue of the periodical within the volume
6
Country of publishing house
US - UNITED STATES
Number of pages
102
Pages from-to
5892-5993
UT code for WoS article
000600695200020
EID of the result in the Scopus database
2-s2.0-85098773267