On the long-time asymptotics of the modified Camassa–Holm equation with step-like initial data
We study the long-time asymptotics for the solution of the modified Camassa–Holm (mCH) equation with step-like initial data. \begin{align*} &m_{t}+\left (m\left (u^{2}-u_{x}^{2}\right )\right )_{x}=0, \quad m=u-u_{xx}, \\[3pt] & {u(x,0)=u_0(x)\to \left \{ \begin{array}{l@{\quad}l} 1/c_+,...
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| Main Authors: | , , |
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| Format: | Article |
| Language: | English |
| Published: |
Cambridge University Press
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| Series: | European Journal of Applied Mathematics |
| Subjects: | |
| Online Access: | https://www.cambridge.org/core/product/identifier/S0956792525000178/type/journal_article |
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| Summary: | We study the long-time asymptotics for the solution of the modified Camassa–Holm (mCH) equation with step-like initial data.
\begin{align*} &m_{t}+\left (m\left (u^{2}-u_{x}^{2}\right )\right )_{x}=0, \quad m=u-u_{xx}, \\[3pt] & {u(x,0)=u_0(x)\to \left \{ \begin{array}{l@{\quad}l} 1/c_+, &\ x\to +\infty, \\[3pt] 1/c_-, &\ x\to -\infty, \end{array}\right .} \end{align*}
where
$c_+$
and
$c_-$
are two positive constants. It is shown that the solution of the step-like initial problem can be characterised via the solution of a matrix Riemann–Hilbert (RH) problem in the new scale
$(y,t)$
. A double coordinate
$(\xi, c)$
with
$c=c_+/c_-$
is adopted to divide the half-plane
$\{ (\xi, c)\,:\, \xi \in \mathbb{R}, \ c\gt 0, \ \xi =y/t\}$
into four asymptotic regions. Further applying the Deift–Zhou steepest descent method, we derive the long-time asymptotic expansions of the solution
$u(y,t)$
in different space-time regions with appropriate g-functions. The corresponding leading asymptotic approximations are given with the slow/fast decay step-like background wave in genus-0 regions and elliptic waves in genus-2 regions. The second term of the asymptotics is characterised by the Airy function or parabolic cylinder model. Their residual error order is
$\mathcal{O}(t^{-2})$
or
$\mathcal{O}(t^{-1})$
, respectively. |
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| ISSN: | 0956-7925 1469-4425 |