Phase turbulence in the Complex Ginzburg--Landau equation via Kuramoto--Sivashinsky phase dynamics
We study the Complex Ginzburg--Landau initial value problem $\partial_t u=(1+i\alpha) \partial_x^2 u + u - (1+i\beta) u |u|^2$, $u(x,0)=u_0(x)$ for a complex field $u\in{\bf C}$, with $\alpha,\beta\in{\bf R}$. We consider the Benjamin--Feir linear instability region $1+\alpha\beta=-\epsilon^2$ with...
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| creator | van Baalen, Guillaume |
| description | We study the Complex Ginzburg--Landau initial value problem $\partial_t
u=(1+i\alpha) \partial_x^2 u + u - (1+i\beta) u |u|^2$, $u(x,0)=u_0(x)$ for a
complex field $u\in{\bf C}$, with $\alpha,\beta\in{\bf R}$. We consider the
Benjamin--Feir linear instability region $1+\alpha\beta=-\epsilon^2$ with
$\epsilon\ll1$ and $\alpha^2 |
| doi_str_mv | 10.48550/arxiv.math-ph/0302021 |
| format | Article |
| fullrecord | <record><control><sourceid>arxiv_GOX</sourceid><recordid>TN_cdi_arxiv_primary_math_ph_0302021</recordid><sourceformat>XML</sourceformat><sourcesystem>PC</sourcesystem><sourcerecordid>math_ph_0302021</sourcerecordid><originalsourceid>FETCH-LOGICAL-a771-90c45917e60e87b46413d217d4d52f2104cccc7755b9dfbc9224b5e1002e2f953</originalsourceid><addsrcrecordid>eNotz0FPwyAYxnEuHsz0KxgOXtmAwliPptFpbDITd29eChViSystzeqnV-eey3P7Jz-E7hhdi52UdAPx5Od1B5Mjg9vQjHLK2TXSbw5Gi6cUdWptqC32AU_O4qLvhtae8N6Hb53iByElBAMJ268Ek-8Dnj3g1xSh66eekHc_w-h8GD8XPJybZgnQ-Xq8QVcNtKO9vfwKHZ8ej8UzKQ_7l-KhJKAUIzmthcyZsltqd0qLrWCZ4UwZYSRvOKOi_p1SUurcNLrOORdaWkYpt7zJZbZC9__Zs7Qaou8gLtWfuBpcdRFnP5w-VKg</addsrcrecordid><sourcetype>Open Access Repository</sourcetype><iscdi>true</iscdi><recordtype>article</recordtype></control><display><type>article</type><title>Phase turbulence in the Complex Ginzburg--Landau equation via Kuramoto--Sivashinsky phase dynamics</title><source>arXiv.org</source><creator>van Baalen, Guillaume</creator><creatorcontrib>van Baalen, Guillaume</creatorcontrib><description>We study the Complex Ginzburg--Landau initial value problem $\partial_t
u=(1+i\alpha) \partial_x^2 u + u - (1+i\beta) u |u|^2$, $u(x,0)=u_0(x)$ for a
complex field $u\in{\bf C}$, with $\alpha,\beta\in{\bf R}$. We consider the
Benjamin--Feir linear instability region $1+\alpha\beta=-\epsilon^2$ with
$\epsilon\ll1$ and $\alpha^2<1/2$.
We show that for all $\epsilon\leq{\cal O}(\sqrt{1-2\alpha^2} L_0^{-32/37})$,
and for all initial data $u_0$ sufficiently close to 1 (up to a global phase
factor $\ed^{i \phi_0}, \phi_0\in{\bf R}$) in the appropriate space, there
exists a unique (spatially) periodic solution of space period $L_0$.
These solutions are small {\em even} perturbations of the traveling wave
solution, $u=(1+\alpha^2 s) \ed^{i \phi_0-i\beta t} \ed^{i\alpha \eta}$, and
$s,\eta$ have bounded norms in various $\L^p$ and Sobolev spaces.
We prove that $s\approx-{1/2} \eta''$ apart from ${\cal O}(\epsilon^2)$
corrections whenever the initial data satisfy this condition, and that in the
linear instability range $L_0^{-1}\leq\epsilon\leq{\cal O}(L_0^{-32/37})$, the
dynamics is essentially determined by the motion of the phase alone, and so
exhibits `phase turbulence'.
Indeed, we prove that the phase $\eta$ satisfies the Kuramoto--Sivashinsky
equation $\partial_t\eta= -\bigl({\textstyle\frac{1+\alpha^2}{2}}\bigr)
\triangle^2\eta -\epsilon^2\triangle\eta -{(1+\alpha^2)} (\eta')^2$ for times
$t_0\leq{\cal O}(\epsilon^{-52/5} L_0^{-32/5})$, while the amplitude
$1+\alpha^2 s$ is essentially constant.</description><identifier>DOI: 10.48550/arxiv.math-ph/0302021</identifier><language>eng</language><subject>Mathematics - Analysis of PDEs ; Mathematics - Mathematical Physics ; Physics - Fluid Dynamics ; Physics - Mathematical Physics</subject><creationdate>2003-02</creationdate><oa>free_for_read</oa><woscitedreferencessubscribed>false</woscitedreferencessubscribed></display><links><openurl>$$Topenurl_article</openurl><openurlfulltext>$$Topenurlfull_article</openurlfulltext><thumbnail>$$Tsyndetics_thumb_exl</thumbnail><link.rule.ids>228,230,780,885</link.rule.ids><linktorsrc>$$Uhttps://arxiv.org/abs/math-ph/0302021$$EView_record_in_Cornell_University$$FView_record_in_$$GCornell_University$$Hfree_for_read</linktorsrc><backlink>$$Uhttps://doi.org/10.48550/arXiv.math-ph/0302021$$DView paper in arXiv$$Hfree_for_read</backlink></links><search><creatorcontrib>van Baalen, Guillaume</creatorcontrib><title>Phase turbulence in the Complex Ginzburg--Landau equation via Kuramoto--Sivashinsky phase dynamics</title><description>We study the Complex Ginzburg--Landau initial value problem $\partial_t
u=(1+i\alpha) \partial_x^2 u + u - (1+i\beta) u |u|^2$, $u(x,0)=u_0(x)$ for a
complex field $u\in{\bf C}$, with $\alpha,\beta\in{\bf R}$. We consider the
Benjamin--Feir linear instability region $1+\alpha\beta=-\epsilon^2$ with
$\epsilon\ll1$ and $\alpha^2<1/2$.
We show that for all $\epsilon\leq{\cal O}(\sqrt{1-2\alpha^2} L_0^{-32/37})$,
and for all initial data $u_0$ sufficiently close to 1 (up to a global phase
factor $\ed^{i \phi_0}, \phi_0\in{\bf R}$) in the appropriate space, there
exists a unique (spatially) periodic solution of space period $L_0$.
These solutions are small {\em even} perturbations of the traveling wave
solution, $u=(1+\alpha^2 s) \ed^{i \phi_0-i\beta t} \ed^{i\alpha \eta}$, and
$s,\eta$ have bounded norms in various $\L^p$ and Sobolev spaces.
We prove that $s\approx-{1/2} \eta''$ apart from ${\cal O}(\epsilon^2)$
corrections whenever the initial data satisfy this condition, and that in the
linear instability range $L_0^{-1}\leq\epsilon\leq{\cal O}(L_0^{-32/37})$, the
dynamics is essentially determined by the motion of the phase alone, and so
exhibits `phase turbulence'.
Indeed, we prove that the phase $\eta$ satisfies the Kuramoto--Sivashinsky
equation $\partial_t\eta= -\bigl({\textstyle\frac{1+\alpha^2}{2}}\bigr)
\triangle^2\eta -\epsilon^2\triangle\eta -{(1+\alpha^2)} (\eta')^2$ for times
$t_0\leq{\cal O}(\epsilon^{-52/5} L_0^{-32/5})$, while the amplitude
$1+\alpha^2 s$ is essentially constant.</description><subject>Mathematics - Analysis of PDEs</subject><subject>Mathematics - Mathematical Physics</subject><subject>Physics - Fluid Dynamics</subject><subject>Physics - Mathematical Physics</subject><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2003</creationdate><recordtype>article</recordtype><sourceid>GOX</sourceid><recordid>eNotz0FPwyAYxnEuHsz0KxgOXtmAwliPptFpbDITd29eChViSystzeqnV-eey3P7Jz-E7hhdi52UdAPx5Od1B5Mjg9vQjHLK2TXSbw5Gi6cUdWptqC32AU_O4qLvhtae8N6Hb53iByElBAMJ268Ek-8Dnj3g1xSh66eekHc_w-h8GD8XPJybZgnQ-Xq8QVcNtKO9vfwKHZ8ej8UzKQ_7l-KhJKAUIzmthcyZsltqd0qLrWCZ4UwZYSRvOKOi_p1SUurcNLrOORdaWkYpt7zJZbZC9__Zs7Qaou8gLtWfuBpcdRFnP5w-VKg</recordid><startdate>20030210</startdate><enddate>20030210</enddate><creator>van Baalen, Guillaume</creator><scope>AKZ</scope><scope>GOX</scope></search><sort><creationdate>20030210</creationdate><title>Phase turbulence in the Complex Ginzburg--Landau equation via Kuramoto--Sivashinsky phase dynamics</title><author>van Baalen, Guillaume</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-a771-90c45917e60e87b46413d217d4d52f2104cccc7755b9dfbc9224b5e1002e2f953</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>2003</creationdate><topic>Mathematics - Analysis of PDEs</topic><topic>Mathematics - Mathematical Physics</topic><topic>Physics - Fluid Dynamics</topic><topic>Physics - Mathematical Physics</topic><toplevel>online_resources</toplevel><creatorcontrib>van Baalen, Guillaume</creatorcontrib><collection>arXiv Mathematics</collection><collection>arXiv.org</collection></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext_linktorsrc</fulltext></delivery><addata><au>van Baalen, Guillaume</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>Phase turbulence in the Complex Ginzburg--Landau equation via Kuramoto--Sivashinsky phase dynamics</atitle><date>2003-02-10</date><risdate>2003</risdate><abstract>We study the Complex Ginzburg--Landau initial value problem $\partial_t
u=(1+i\alpha) \partial_x^2 u + u - (1+i\beta) u |u|^2$, $u(x,0)=u_0(x)$ for a
complex field $u\in{\bf C}$, with $\alpha,\beta\in{\bf R}$. We consider the
Benjamin--Feir linear instability region $1+\alpha\beta=-\epsilon^2$ with
$\epsilon\ll1$ and $\alpha^2<1/2$.
We show that for all $\epsilon\leq{\cal O}(\sqrt{1-2\alpha^2} L_0^{-32/37})$,
and for all initial data $u_0$ sufficiently close to 1 (up to a global phase
factor $\ed^{i \phi_0}, \phi_0\in{\bf R}$) in the appropriate space, there
exists a unique (spatially) periodic solution of space period $L_0$.
These solutions are small {\em even} perturbations of the traveling wave
solution, $u=(1+\alpha^2 s) \ed^{i \phi_0-i\beta t} \ed^{i\alpha \eta}$, and
$s,\eta$ have bounded norms in various $\L^p$ and Sobolev spaces.
We prove that $s\approx-{1/2} \eta''$ apart from ${\cal O}(\epsilon^2)$
corrections whenever the initial data satisfy this condition, and that in the
linear instability range $L_0^{-1}\leq\epsilon\leq{\cal O}(L_0^{-32/37})$, the
dynamics is essentially determined by the motion of the phase alone, and so
exhibits `phase turbulence'.
Indeed, we prove that the phase $\eta$ satisfies the Kuramoto--Sivashinsky
equation $\partial_t\eta= -\bigl({\textstyle\frac{1+\alpha^2}{2}}\bigr)
\triangle^2\eta -\epsilon^2\triangle\eta -{(1+\alpha^2)} (\eta')^2$ for times
$t_0\leq{\cal O}(\epsilon^{-52/5} L_0^{-32/5})$, while the amplitude
$1+\alpha^2 s$ is essentially constant.</abstract><doi>10.48550/arxiv.math-ph/0302021</doi><oa>free_for_read</oa></addata></record> |
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| subjects | Mathematics - Analysis of PDEs Mathematics - Mathematical Physics Physics - Fluid Dynamics Physics - Mathematical Physics |
| title | Phase turbulence in the Complex Ginzburg--Landau equation via Kuramoto--Sivashinsky phase dynamics |
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