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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Cambridge University Press
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| Series: | European Journal of Applied Mathematics |
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| author | Engui Fan Gaozhan Li Yiling Yang |
| author_facet | Engui Fan Gaozhan Li Yiling Yang |
| author_sort | Engui Fan |
| collection | DOAJ |
| description | 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. |
| format | Article |
| id | doaj-art-9adfb8a7eb0840ed8a3b5830248f4bbc |
| institution | OA Journals |
| issn | 0956-7925 1469-4425 |
| language | English |
| publisher | Cambridge University Press |
| record_format | Article |
| series | European Journal of Applied Mathematics |
| spelling | doaj-art-9adfb8a7eb0840ed8a3b5830248f4bbc2025-08-20T02:24:59ZengCambridge University PressEuropean Journal of Applied Mathematics0956-79251469-442514410.1017/S0956792525000178On the long-time asymptotics of the modified Camassa–Holm equation with step-like initial dataEngui Fan0Gaozhan Li1Yiling Yang2https://orcid.org/0000-0001-6628-4745School of Mathematical Sciences and Key Laboratory of Mathematics for Nonlinear Science, Fudan University, Shanghai, P.R. ChinaSchool of Mathematical Sciences and Key Laboratory of Mathematics for Nonlinear Science, Fudan University, Shanghai, P.R. ChinaCollege of Mathematics and Statistics, Chongqing University, Chongqing, P.R. ChinaWe 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.https://www.cambridge.org/core/product/identifier/S0956792525000178/type/journal_articlemodified Camassa–Holm equationstep-like initial dataRiemann–Hilbert problemlong-time asymptoticsairy function35Q1541A6037K1535C20 |
| spellingShingle | Engui Fan Gaozhan Li Yiling Yang On the long-time asymptotics of the modified Camassa–Holm equation with step-like initial data European Journal of Applied Mathematics modified Camassa–Holm equation step-like initial data Riemann–Hilbert problem long-time asymptotics airy function 35Q15 41A60 37K15 35C20 |
| title | On the long-time asymptotics of the modified Camassa–Holm equation with step-like initial data |
| title_full | On the long-time asymptotics of the modified Camassa–Holm equation with step-like initial data |
| title_fullStr | On the long-time asymptotics of the modified Camassa–Holm equation with step-like initial data |
| title_full_unstemmed | On the long-time asymptotics of the modified Camassa–Holm equation with step-like initial data |
| title_short | On the long-time asymptotics of the modified Camassa–Holm equation with step-like initial data |
| title_sort | on the long time asymptotics of the modified camassa holm equation with step like initial data |
| topic | modified Camassa–Holm equation step-like initial data Riemann–Hilbert problem long-time asymptotics airy function 35Q15 41A60 37K15 35C20 |
| url | https://www.cambridge.org/core/product/identifier/S0956792525000178/type/journal_article |
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