Comptes Rendus
High-order evolution equation for nonlinear wave-packet propagation with surface tension accounting
Comptes Rendus. Mécanique, Volume 331 (2003) no. 3, pp. 197-201.

The nonlinear problem for propagation of wave-packets along the interface of two semi-infinite fluids is solved on the basis of multiple scale asymptotic expansions. Unlike all previous investigations dealing only with third-order approximations, here fourth-order approximation is developed. The corresponding solvability condition is obtained and the evolution equation in the case away from the cut-off wave number is derived. As a result, the nonlinear higher-order Schrödinger equation is obtained which contains the nonlinear part in a compact form. This equation is valid for a wide range of wave numbers. The stability diagram shows regions of stability and instability of capillary-gravity wave-packets.

Le problème non linéaire de la propagation de groupes d'ondes à l'interface de deux liquides semi-infinis est résolu en utilisant une méthode multi-échelles. Contrairement aux études antérieures développées qu'au troisième ordre, cet article consière une approximation au quatrième ordre. La condition de solvabilité correspondante est obtenue et l'équation d'évolution est formulée loin du nombre d'onde de coupure. Comme résultat on obtient une équation non linéaire de Schrödinger d'ordre élevé, dont la partie non linéaire est mise sous une forme compacte. Cette équation est utilisable pour un large intervalle de nombres d'onde. Le diagramme de stabilité met en évidence des domaines stables et instables de paquets d'ondes de gravité-capillarité.

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Accepted:
Published online:
DOI: 10.1016/S1631-0721(03)00043-3
Keywords: Waves, Wave-packets, Multiple scales, Fourth-order problem, Evolution equation
Mot clés : Ondes, Groupe d'ondes, Échelles multiples, Problème au quatrième ordre, Équation d'évolution

Igor Selezov 1; Olga Avramenko 2; Christian Kharif 3; Karsten Trulsen 4

1 Department of Wave Processes, Institute of Hydromechanics, NAS of Ukraine, Kiev, Ukraine
2 Mathematical Department, Kirovograd State Pedagogical University, Kirovograd, Ukraine
3 Institut de recherche sur les phénomènes hors équilibre, 49, rue F. Joliot-Curie, BP 146, 13384 Marseille cedex 13, France
4 University of Oslo, Department Mathematics, Oslo, Norway
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Igor Selezov; Olga Avramenko; Christian Kharif; Karsten Trulsen. High-order evolution equation for nonlinear wave-packet propagation with surface tension accounting. Comptes Rendus. Mécanique, Volume 331 (2003) no. 3, pp. 197-201. doi : 10.1016/S1631-0721(03)00043-3. https://comptes-rendus.academie-sciences.fr/mecanique/articles/10.1016/S1631-0721(03)00043-3/

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