<?xml version="1.0" encoding="utf-8"?>
<TEI xmlns="http://www.tei-c.org/ns/1.0" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" xmlns:hal="http://hal.archives-ouvertes.fr/" xmlns:gml="http://www.opengis.net/gml/3.3/" xmlns:gmlce="http://www.opengis.net/gml/3.3/ce" version="1.1" xsi:schemaLocation="http://www.tei-c.org/ns/1.0 http://api.archives-ouvertes.fr/documents/aofr-sword.xsd">
  <teiHeader>
    <fileDesc>
      <titleStmt>
        <title>HAL TEI export of hal-01208479</title>
      </titleStmt>
      <publicationStmt>
        <distributor>CCSD</distributor>
        <availability status="restricted">
          <licence target="https://creativecommons.org/publicdomain/zero/1.0/">CC0 1.0 - Universal</licence>
        </availability>
        <date when="2026-05-24T21:17:42+02:00"/>
      </publicationStmt>
      <sourceDesc>
        <p part="N">HAL API Platform</p>
      </sourceDesc>
    </fileDesc>
  </teiHeader>
  <text>
    <body>
      <listBibl>
        <biblFull>
          <titleStmt>
            <title xml:lang="en">Amplitude equations for weakly nonlinear surface waves in variational problems</title>
            <author role="aut">
              <persName>
                <forename type="first">Sylvie</forename>
                <surname>Benzoni-Gavage</surname>
              </persName>
              <email type="md5">7e23f6fbb4a1735621ac4ba210cdf602</email>
              <email type="domain">math.univ-lyon1.fr</email>
              <idno type="idhal" notation="string">sylvie-benzoni-gavage</idno>
              <idno type="idhal" notation="numeric">14558</idno>
              <idno type="halauthorid" notation="string">6832-14558</idno>
              <idno type="ORCID">https://orcid.org/0000-0003-4302-8567</idno>
              <idno type="IDREF">https://www.idref.fr/078813522</idno>
              <affiliation ref="#struct-521754"/>
              <affiliation ref="#struct-394163"/>
            </author>
            <author role="aut">
              <persName>
                <forename type="first">Jean-François</forename>
                <surname>Coulombel</surname>
              </persName>
              <email type="md5">7c7c919c47204ebbce4c9c1825ae6c86</email>
              <email type="domain">math.univ-toulouse.fr</email>
              <idno type="idhal" notation="string">jean-francois-coulombel</idno>
              <idno type="idhal" notation="numeric">20049</idno>
              <idno type="halauthorid" notation="string">886-20049</idno>
              <idno type="IDREF">https://www.idref.fr/078813506</idno>
              <idno type="ORCID">https://orcid.org/0009-0003-4116-9443</idno>
              <affiliation ref="#struct-394163"/>
            </author>
            <editor role="depositor">
              <persName>
                <forename>Sylvie</forename>
                <surname>Benzoni-Gavage</surname>
              </persName>
              <email type="md5">7e23f6fbb4a1735621ac4ba210cdf602</email>
              <email type="domain">math.univ-lyon1.fr</email>
            </editor>
            <funder ref="#projanr-36344"/>
            <funder ref="#projanr-49748"/>
            <funder ref="#projanr-38048"/>
          </titleStmt>
          <editionStmt>
            <edition n="v1" type="current">
              <date type="whenSubmitted">2015-10-05 10:02:50</date>
              <date type="whenModified">2026-04-23 14:50:05</date>
              <date type="whenReleased">2015-10-05 14:12:52</date>
              <date type="whenProduced">2015-09-14</date>
              <date type="whenEndEmbargoed">2015-10-05</date>
              <ref type="file" target="https://hal.science/hal-01208479v1/document">
                <date notBefore="2015-10-05"/>
              </ref>
              <ref type="file" subtype="author" n="1" target="https://hal.science/hal-01208479v1/file/BC2-def.pdf" id="file-1208479-1289580">
                <date notBefore="2015-10-05"/>
              </ref>
              <ref type="externalLink" target="http://arxiv.org/pdf/1510.01119"/>
            </edition>
            <respStmt>
              <resp>contributor</resp>
              <name key="128924">
                <persName>
                  <forename>Sylvie</forename>
                  <surname>Benzoni-Gavage</surname>
                </persName>
                <email type="md5">7e23f6fbb4a1735621ac4ba210cdf602</email>
                <email type="domain">math.univ-lyon1.fr</email>
              </name>
            </respStmt>
          </editionStmt>
          <publicationStmt>
            <distributor>CCSD</distributor>
            <idno type="halId">hal-01208479</idno>
            <idno type="halUri">https://hal.science/hal-01208479</idno>
            <idno type="halBibtex">benzonigavage:hal-01208479</idno>
            <idno type="halRefHtml">&lt;i&gt;Shocks, Singularities and Oscillations in Nonlinear Optics and Fluid Mechanics.&lt;/i&gt;, Sep 2015, Rome, Italy</idno>
            <idno type="halRef">Shocks, Singularities and Oscillations in Nonlinear Optics and Fluid Mechanics., Sep 2015, Rome, Italy</idno>
            <availability status="restricted">
              <licence target="https://creativecommons.org/licenses/by-nc-sa/4.0/">CC BY-NC-SA 4.0 - Attribution - Non-commercial use - ShareAlike<ref corresp="#file-1208479-1289580"/></licence>
            </availability>
          </publicationStmt>
          <seriesStmt>
            <idno type="stamp" n="UNIV-ST-ETIENNE">Université Jean Monnet - Saint-Etienne</idno>
            <idno type="stamp" n="UNIV-NANTES">Université de Nantes</idno>
            <idno type="stamp" n="LMJL">Laboratoire de Mathématiques Jean Leray</idno>
            <idno type="stamp" n="CNRS">CNRS - Centre national de la recherche scientifique</idno>
            <idno type="stamp" n="ICJ" corresp="UNIV-LYON1">Institut Camille Jordan</idno>
            <idno type="stamp" n="UNIV-LYON1">Université Claude Bernard - Lyon I</idno>
            <idno type="stamp" n="INSA-LYON">Institut National des Sciences Appliquées de Lyon</idno>
            <idno type="stamp" n="EC-LYON">Ecole Centrale de Lyon</idno>
            <idno type="stamp" n="FMPL">Fédération de recherche Mathématiques des Pays de Loire</idno>
            <idno type="stamp" n="INSMI">CNRS-INSMI - INstitut des Sciences Mathématiques et de leurs Interactions</idno>
            <idno type="stamp" n="CHL">Centre Henri Lebesgue</idno>
            <idno type="stamp" n="TDS-MACS">Réseau de recherche en Théorie des Systèmes Distribués, Modélisation, Analyse et Contrôle des Systèmes</idno>
            <idno type="stamp" n="ICJ-MMCS" corresp="ICJ">Modélisation mathématique, calcul scientifique</idno>
            <idno type="stamp" n="LABEX-MILYON">Laboratoire d'excellence en Mathématiques et informatique fondamentale de Lyon</idno>
            <idno type="stamp" n="INSA-GROUPE">Groupe INSA</idno>
            <idno type="stamp" n="UDL">UDL</idno>
            <idno type="stamp" n="UNIV-LYON">Université de Lyon</idno>
            <idno type="stamp" n="ANR">ANR</idno>
            <idno type="stamp" n="NANTES-UNIVERSITE">Nantes Université</idno>
            <idno type="stamp" n="UNIV-NANTES-AV2022">Université de Nantes</idno>
          </seriesStmt>
          <notesStmt>
            <note type="audience" n="2">International</note>
            <note type="invited" n="1">Yes</note>
            <note type="popular" n="0">No</note>
            <note type="peer" n="1">Yes</note>
            <note type="proceedings" n="1">Yes</note>
          </notesStmt>
          <sourceDesc>
            <biblStruct>
              <analytic>
                <title xml:lang="en">Amplitude equations for weakly nonlinear surface waves in variational problems</title>
                <author role="aut">
                  <persName>
                    <forename type="first">Sylvie</forename>
                    <surname>Benzoni-Gavage</surname>
                  </persName>
                  <email type="md5">7e23f6fbb4a1735621ac4ba210cdf602</email>
                  <email type="domain">math.univ-lyon1.fr</email>
                  <idno type="idhal" notation="string">sylvie-benzoni-gavage</idno>
                  <idno type="idhal" notation="numeric">14558</idno>
                  <idno type="halauthorid" notation="string">6832-14558</idno>
                  <idno type="ORCID">https://orcid.org/0000-0003-4302-8567</idno>
                  <idno type="IDREF">https://www.idref.fr/078813522</idno>
                  <affiliation ref="#struct-521754"/>
                  <affiliation ref="#struct-394163"/>
                </author>
                <author role="aut">
                  <persName>
                    <forename type="first">Jean-François</forename>
                    <surname>Coulombel</surname>
                  </persName>
                  <email type="md5">7c7c919c47204ebbce4c9c1825ae6c86</email>
                  <email type="domain">math.univ-toulouse.fr</email>
                  <idno type="idhal" notation="string">jean-francois-coulombel</idno>
                  <idno type="idhal" notation="numeric">20049</idno>
                  <idno type="halauthorid" notation="string">886-20049</idno>
                  <idno type="IDREF">https://www.idref.fr/078813506</idno>
                  <idno type="ORCID">https://orcid.org/0009-0003-4116-9443</idno>
                  <affiliation ref="#struct-394163"/>
                </author>
              </analytic>
              <monogr>
                <title level="m">Springer INdAM Series</title>
                <meeting>
                  <title>Shocks, Singularities and Oscillations in Nonlinear Optics and Fluid Mechanics.</title>
                  <date type="start">2015-09-14</date>
                  <date type="end">2015-09-18</date>
                  <settlement>Rome</settlement>
                  <country key="IT">Italy</country>
                </meeting>
                <editor>Colombini</editor>
                <editor>F. and Del Santo</editor>
                <editor>D. and Lannes</editor>
                <editor>D.</editor>
                <imprint/>
              </monogr>
              <idno type="arxiv">1510.01119</idno>
            </biblStruct>
          </sourceDesc>
          <profileDesc>
            <langUsage>
              <language ident="en">English</language>
            </langUsage>
            <textClass>
              <keywords scheme="author">
                <term xml:lang="en">Oseen–Frank energy</term>
                <term xml:lang="en">Hamiltonian structure</term>
                <term xml:lang="en">amplitude equation</term>
                <term xml:lang="en">weakly nonlinear expansion</term>
                <term xml:lang="en">surface wave</term>
                <term xml:lang="en">non-local Burg-ers equation</term>
                <term xml:lang="en">non-local Hamilton–Jacobi equation</term>
                <term xml:lang="en">phase boundaries</term>
              </keywords>
              <classCode scheme="halDomain" n="math.math-ap">Mathematics [math]/Analysis of PDEs [math.AP]</classCode>
              <classCode scheme="halTypology" n="COMM">Conference papers</classCode>
              <classCode scheme="halOldTypology" n="COMM">Conference papers</classCode>
              <classCode scheme="halTreeTypology" n="COMM">Conference papers</classCode>
            </textClass>
            <abstract xml:lang="en">
              <p>Among hyperbolic Initial Boundary Value Problems (IBVP), those coming from a variational principle 'generically' admit linear surface waves, as was shown by Serre [J. Funct. Anal. 2006]. At the weakly nonlinear level, the behavior of surface waves is expected to be governed by an amplitude equation that can be derived by means of a formal asymptotic expansion. Amplitude equations for weakly nonlinear surface waves were introduced by Lardner [Int. J. Engng Sci. 1983], Parker and co-workers [J. Elasticity 1985] in the framework of elasticity, and by Hunter [Contemp. Math. 1989] for abstract hyperbolic problems. They consist of nonlocal evolution equations involving a complicated, bilinear Fourier multiplier in the direction of propagation along the boundary. It was shown by the authors in an earlier work [Arch. Ration. Mech. Anal. 2012] that this multiplier, or kernel, inherits some algebraic properties from the original IBVP. These properties are crucial for the (local) well-posedness of the amplitude equation, as shown together with Tzvetkov [Adv. Math., 2011]. Properties of amplitude equations are revisited here in a somehow simpler way, for surface waves in a variational setting. Applications include various physical models, from elasticity of course to the director-field system for liquid crystals introduced by Saxton [Contemp. Math. 1989] and studied by Austria and Hunter [Commun. Inf. Syst. 2013]. Similar properties are eventually shown for the amplitude equation associated with surface waves at reversible phase boundaries in compressible fluids, thus completing a work initiated by Benzoni-Gavage and Rosini [Comput. Math. Appl. 2009].</p>
            </abstract>
          </profileDesc>
        </biblFull>
      </listBibl>
    </body>
    <back>
      <listOrg type="structures">
        <org type="researchteam" xml:id="struct-521754" status="VALID">
          <orgName>Modélisation mathématique, calcul scientifique</orgName>
          <orgName type="acronym">MMCS</orgName>
          <desc>
            <address>
              <addrLine>Institut Camille JordanVilleurbane</addrLine>
              <country key="FR"/>
            </address>
            <ref type="url">http://math.univ-lyon1.fr/spip.php?article13</ref>
          </desc>
          <listRelation>
            <relation active="#struct-193738" type="direct"/>
            <relation active="#struct-126765" type="indirect"/>
            <relation active="#struct-301088" type="indirect"/>
            <relation active="#struct-194495" type="indirect"/>
            <relation active="#struct-219748" type="indirect"/>
            <relation active="#struct-301232" type="indirect"/>
            <relation active="#struct-300284" type="indirect"/>
            <relation active="#struct-1327915" type="indirect"/>
            <relation name="UMR5208" active="#struct-441569" type="indirect"/>
          </listRelation>
        </org>
        <org type="researchteam" xml:id="struct-394163" status="OLD">
          <orgName>Equations aux dérivées partielles</orgName>
          <date type="end">2021-12-31</date>
          <desc>
            <address>
              <country key="FR"/>
            </address>
          </desc>
          <listRelation>
            <relation active="#struct-37" type="direct"/>
            <relation active="#struct-97058" type="indirect"/>
            <relation active="#struct-93263" type="indirect"/>
            <relation name="UMR6629" active="#struct-441569" type="indirect"/>
          </listRelation>
        </org>
        <org type="laboratory" xml:id="struct-193738" status="VALID">
          <idno type="IdRef">111872227</idno>
          <idno type="ISNI">0000000403832988</idno>
          <idno type="RNSR">200511878U</idno>
          <idno type="ROR">https://ror.org/01ekw7f87</idno>
          <orgName>Institut Camille Jordan</orgName>
          <orgName type="acronym">ICJ</orgName>
          <date type="start">2005-01-01</date>
          <desc>
            <address>
              <addrLine>Bât. Jean Braconnier 43 bd du 11 novembre 1918 69622 VILLEURBANNE CEDEX</addrLine>
              <country key="FR"/>
            </address>
            <ref type="url">http://math.univ-lyon1.fr/</ref>
          </desc>
          <listRelation>
            <relation active="#struct-126765" type="direct"/>
            <relation active="#struct-301088" type="indirect"/>
            <relation active="#struct-194495" type="direct"/>
            <relation active="#struct-219748" type="direct"/>
            <relation active="#struct-301232" type="indirect"/>
            <relation active="#struct-300284" type="direct"/>
            <relation active="#struct-1327915" type="indirect"/>
            <relation name="UMR5208" active="#struct-441569" type="direct"/>
          </listRelation>
        </org>
        <org type="institution" xml:id="struct-126765" status="VALID">
          <idno type="ROR">https://ror.org/05s6rge65</idno>
          <orgName>École Centrale de Lyon</orgName>
          <orgName type="acronym">ECL</orgName>
          <desc>
            <address>
              <addrLine>36 avenue Guy de Collongue - 69134 Ecully cedex</addrLine>
              <country key="FR"/>
            </address>
            <ref type="url">http://www.ec-lyon.fr</ref>
          </desc>
          <listRelation>
            <relation active="#struct-301088" type="direct"/>
          </listRelation>
        </org>
        <org type="regroupinstitution" xml:id="struct-301088" status="VALID">
          <idno type="ROR">https://ror.org/01rk35k63</idno>
          <orgName>Université de Lyon</orgName>
          <desc>
            <address>
              <addrLine>92 rue Pasteur - CS 30122, 69361 Lyon Cedex 07</addrLine>
              <country key="FR"/>
            </address>
            <ref type="url">https://www.universite-lyon.fr/</ref>
          </desc>
        </org>
        <org type="institution" xml:id="struct-194495" status="VALID">
          <idno type="IdRef">026402823</idno>
          <idno type="ISNI">0000000121686185</idno>
          <idno type="ROR">https://ror.org/029brtt94</idno>
          <orgName>Université Claude Bernard Lyon 1</orgName>
          <orgName type="acronym">UCBL</orgName>
          <desc>
            <address>
              <addrLine>43, boulevard du 11 novembre 1918, 69622 Villeurbanne cedex</addrLine>
              <country key="FR"/>
            </address>
            <ref type="url">http://www.univ-lyon1.fr/</ref>
          </desc>
          <listRelation>
            <relation active="#struct-301088" type="direct"/>
          </listRelation>
        </org>
        <org type="institution" xml:id="struct-219748" status="VALID">
          <idno type="IdRef">052444724</idno>
          <idno type="ISNI">0000 0001 2292 0228</idno>
          <orgName>Institut National des Sciences Appliquées de Lyon</orgName>
          <orgName type="acronym">INSA Lyon</orgName>
          <date type="start">1957-03-18</date>
          <desc>
            <address>
              <addrLine>20 Avenue Albert Einstein, 69621 Villeurbanne cedex</addrLine>
              <country key="FR"/>
            </address>
            <ref type="url">http://www.insa-lyon.fr/</ref>
          </desc>
          <listRelation>
            <relation active="#struct-301088" type="direct"/>
            <relation active="#struct-301232" type="direct"/>
          </listRelation>
        </org>
        <org type="regroupinstitution" xml:id="struct-301232" status="VALID">
          <idno type="IdRef">162105150</idno>
          <orgName>Institut National des Sciences Appliquées</orgName>
          <orgName type="acronym">INSA</orgName>
          <desc>
            <address>
              <country key="FR"/>
            </address>
          </desc>
        </org>
        <org type="institution" xml:id="struct-300284" status="VALID">
          <idno type="IdRef">028209966</idno>
          <idno type="ISNI">0000 0001 2158 1682</idno>
          <idno type="ROR">https://ror.org/04yznqr36</idno>
          <idno type="Wikidata">Q623154</idno>
          <orgName>Université Jean Monnet - Saint-Étienne</orgName>
          <orgName type="acronym">UJM</orgName>
          <date type="start">1969-03-27</date>
          <date type="end">2025-01-01</date>
          <desc>
            <address>
              <addrLine>10, Rue Tréfilerie – CS 8230142023 Saint-Étienne Cedex 2</addrLine>
              <country key="FR"/>
            </address>
            <ref type="url">https://www.univ-st-etienne.fr/</ref>
          </desc>
          <listRelation>
            <relation active="#struct-1327915" type="direct"/>
          </listRelation>
        </org>
        <org type="regroupinstitution" xml:id="struct-1327915" status="VALID">
          <idno type="IdRef">285395831</idno>
          <orgName>Université Jean Monnet (EPSCPE)</orgName>
          <orgName type="acronym">UJM EPE</orgName>
          <date type="start">2025-01-01</date>
          <desc>
            <address>
              <addrLine>10 rue Tréfilerie42100 Saint-Etienne</addrLine>
              <country key="FR"/>
            </address>
            <ref type="url">https://www.univ-st-etienne.fr/fr/index.html</ref>
          </desc>
        </org>
        <org type="regroupinstitution" xml:id="struct-441569" status="VALID">
          <idno type="IdRef">02636817X</idno>
          <idno type="ISNI">0000000122597504</idno>
          <idno type="ROR">https://ror.org/02feahw73</idno>
          <orgName>Centre National de la Recherche Scientifique</orgName>
          <orgName type="acronym">CNRS</orgName>
          <date type="start">1939-10-19</date>
          <desc>
            <address>
              <country key="FR"/>
            </address>
            <ref type="url">https://www.cnrs.fr/</ref>
          </desc>
        </org>
        <org type="laboratory" xml:id="struct-37" status="OLD">
          <idno type="IdRef">087220865</idno>
          <idno type="ISNI">0000000097740814</idno>
          <idno type="RNSR">199612400A</idno>
          <orgName>Laboratoire de Mathématiques Jean Leray</orgName>
          <orgName type="acronym">LMJL</orgName>
          <date type="end">2021-12-31</date>
          <desc>
            <address>
              <country key="FR"/>
            </address>
            <ref type="url">http://www.math.sciences.univ-nantes.fr/jeanleray/</ref>
          </desc>
          <listRelation>
            <relation active="#struct-97058" type="direct"/>
            <relation active="#struct-93263" type="indirect"/>
            <relation name="UMR6629" active="#struct-441569" type="direct"/>
          </listRelation>
        </org>
        <org type="regrouplaboratory" xml:id="struct-97058" status="OLD">
          <orgName>Université de Nantes - UFR des Sciences et des Techniques</orgName>
          <orgName type="acronym">UN UFR ST</orgName>
          <date type="end">2021-12-31</date>
          <desc>
            <address>
              <addrLine>2, rue de la Houssinière - BP 92208 - 44322 Nantes cedex 3</addrLine>
              <country key="FR"/>
            </address>
            <ref type="url">http://www.sciences-techniques.univ-nantes.fr/</ref>
          </desc>
          <listRelation>
            <relation active="#struct-93263" type="direct"/>
          </listRelation>
        </org>
        <org type="institution" xml:id="struct-93263" status="OLD">
          <idno type="IdRef">026403447</idno>
          <idno type="ROR">https://ror.org/03gnr7b55</idno>
          <orgName>Université de Nantes</orgName>
          <orgName type="acronym">UN</orgName>
          <date type="end">2021-12-31</date>
          <desc>
            <address>
              <addrLine>1, quai de Tourville - BP 13522 - 44035 Nantes cedex 1</addrLine>
              <country key="FR"/>
            </address>
            <ref type="url">http://www.univ-nantes.fr/</ref>
          </desc>
        </org>
      </listOrg>
      <listOrg type="projects">
        <org type="anrProject" xml:id="projanr-36344" status="VALID">
          <idno type="anr">ANR-13-BS01-0009</idno>
          <idno type="program">Blanc 2013</idno>
          <orgName>BoND</orgName>
          <desc>Frontières, numérique, dispersion.</desc>
          <date type="start">2013</date>
        </org>
        <org type="anrProject" xml:id="projanr-49748" status="VALID">
          <idno type="anr">ANR-10-LABX-0070</idno>
          <orgName>MILYON</orgName>
          <desc>Community of mathematics and fundamental computer science in Lyon</desc>
          <date type="start">2010</date>
        </org>
        <org type="anrProject" xml:id="projanr-38048" status="VALID">
          <idno type="anr">ANR-11-LABX-0020</idno>
          <idno type="program">Laboratoires d'excellence</idno>
          <orgName>LEBESGUE</orgName>
          <desc>Centre de Mathématiques Henri Lebesgue : fondements, interactions, applications et Formation</desc>
          <date type="start">2011</date>
        </org>
      </listOrg>
    </back>
  </text>
</TEI>