ON THE LONG-TIME ASYMPTOTIC BEHAVIOR OF THE MODIFIED KORTEWEG-DE VRIES EQUATION WITH STEP-LIKE INITIAL DATA
Identifikátory výsledku
Kód výsledku v 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>
Výsledek na webu
<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>
Alternativní jazyky
Jazyk výsledku
angličtina
Název v původním jazyce
ON THE LONG-TIME ASYMPTOTIC BEHAVIOR OF THE MODIFIED KORTEWEG-DE VRIES EQUATION WITH STEP-LIKE INITIAL DATA
Popis výsledku v původním jazyce
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.
Název v anglickém jazyce
ON THE LONG-TIME ASYMPTOTIC BEHAVIOR OF THE MODIFIED KORTEWEG-DE VRIES EQUATION WITH STEP-LIKE INITIAL DATA
Popis výsledku anglicky
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.
Klasifikace
Druh
J<sub>imp</sub> - Článek v periodiku v databázi Web of Science
CEP obor
—
OECD FORD obor
10101 - Pure mathematics
Návaznosti výsledku
Projekt
—
Návaznosti
I - Institucionalni podpora na dlouhodoby koncepcni rozvoj vyzkumne organizace
Ostatní
Rok uplatnění
2020
Kód důvěrnosti údajů
S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů
Údaje specifické pro druh výsledku
Název periodika
SIAM Journal on Mathematical Analysis
ISSN
0036-1410
e-ISSN
—
Svazek periodika
52
Číslo periodika v rámci svazku
6
Stát vydavatele periodika
US - Spojené státy americké
Počet stran výsledku
102
Strana od-do
5892-5993
Kód UT WoS článku
000600695200020
EID výsledku v databázi Scopus
2-s2.0-85098773267