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On the Long-Time Asymptotic Behavior of the Modified Korteweg--de Vries Equation with Step-like Initial Data
SIAM Journal on Mathematical Analysis ( IF 2 ) Pub Date : 2020-11-24 , DOI: 10.1137/19m1279964
Tamara Grava , Alexander Minakov

SIAM Journal on Mathematical Analysis, Volume 52, Issue 6, Page 5892-5993, January 2020.
We study the long-time asymptotic behavior of the solution $q(x,t) $, $x\in\mathbb{R}$, $t\in\mathbb{R}^+$, of the modified Korteweg--de Vries equation (MKdV) $q_t+6q^2q_x+q_{xxx}=0$ with step-like initial datum $\scriptsize q(x,0)\to \Big\{\begin{array}{@{}l@{}l@{}} c_-\quad& {for $x\to-\infty$},\\ c_+\quad& {for $x\to +\infty$}, \end{array}$ with $c_->c_+\geq 0$. For the step initial data $\scriptsize q(x,0)= \Big\{\begin{array}{@{}l@{}l@{}} c_-\quad& {for $xłeq0$},\\ c_+\quad& {for $x>0$}, \end{array} $ 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<\frac{x}{t}<4c_{-}^2+2c_{+}^2,$ with $t\gg 1$. 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.


中文翻译:

具有逐步初始数据的修正Korteweg-de Vries方程的长期渐近行为

SIAM数学分析杂志,第52卷,第6期,第5892-5993页,2020年1月。
我们研究了修正的Korteweg的解$ q(x,t)$,$ x \ in \ mathbb {R} $,$ t \ in \ mathbb {R} ^ + $的长期渐近行为- de Vries方程(MKdV)$ q_t + 6q ^ 2q_x + q_ {xxx} = 0 $,步阶为初始基准$ \ scriptsize q(x,0)\ to \ Big \ {\ begin {array} {@ {} l @ {} l @ {}} c _- \ quad&{用于$ x \ to- \ infty $},\\ c _ + \ quad&{用于$ x \ to + \ infty $},\ end {array} $ $ c _-> c _ + \ geq 0 $。对于该步骤,初始数据$ \ scriptsize q(x,0)= \ Big \ {\ begin {array} {@ {} l @ {} l @ {}} c _- \ quad&{for $xłeq0$},\\ c _ + \ quad&{for $ x> 0 $},\ end {array} $该解决方案产生了一个称为分散冲击波区域的振荡区域,该区域将两个恒定区域$ c _ + $和$ c _- $连接起来。我们表明,色散冲击波由MKdV方程的调制周期行波解描述,其中调制参数根据Whitham调制方程演化。振荡区域在定义为$ -6c _ {-} ^ 2 + 12c _ {+} ^ 2 <\ frac {x} {t} <4c _ {-} ^的$(x,t)$平面中的圆锥体内扩展。 2 + 2c _ {+} ^ 2,$与$ t \ gg 1 $。对于类似阶跃的初始数据,我们表明该解决方案可长期分解为三个主要区域:(1)在恒定背景$ c _ + $下,孤子和呼吸器以正速度传播的区域;(2)不断扩大的振荡区域(通常包含呼吸器);(3)在恒定背景$ c _- $上以负速度运动的呼吸区域。当振荡区域不包含呼吸器时,渐近解的形式与为阶跃初始数据获得的色散冲击波解的相位偏移一致。相移取决于孤子,通气孔和初始数据的辐射。这表明色散冲击波是一个相干的结构,它与孤子,呼吸器和辐射以弹性方式相互作用。
更新日期:2020-11-25
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