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Existence of multi-travelling waves in capillary fluids

Part of: Partial differential equations Representations of solutions Equations of mathematical physics and other areas of application

Published online by Cambridge University Press:  19 August 2019

Corentin Audiard
Corentin Audiard*
Affiliation:
Laboratoire Jacques-Louis Lions (LJLL), Sorbonne université, CNRS, Université de Paris, F75005, Paris, France (corentin.audiard@upmc.fr)
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Abstract

We prove the existence of multi-soliton and kink-multi-soliton solutions of the Euler–Korteweg system in dimension one. Such solutions behave asymptotically in time like several travelling waves far away from each other. A kink is a travelling wave with different limits at ±∞. The main assumption is the linear stability of the solitons, and we prove that this assumption is satisfied at least in the transonic limit. The proof relies on a classical approach based on energy estimates and a compactness argument.

Keywords

traveling wavesEuler–Kortewegstability

MSC classification

Primary: 35Q35: PDEs in connection with fluid mechanics
Secondary: 35C07: Traveling wave solutions 35Q53: KdV-like equations (Korteweg-de Vries) 35Q31: Euler equations 35B35: Stability
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 150 , Issue 6 , December 2020 , pp. 2905 - 2936
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Copyright
Copyright © 2019 The Royal Society of Edinburgh

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References

1Audiard, C.. Small energy traveling waves for the Euler–Korteweg system. Nonlinearity 30 (2017), 3362–3399.CrossRefGoogle Scholar
2Audiard, C. and Haspot, B.. Global well-posedness of the Euler–Korteweg system for small irrotational data. Comm. Math. Phys. 351 (2017), 201–247.10.1007/s00220-017-2843-8CrossRefGoogle Scholar
3Barashenkov, I. V. and Makhankov, V. G.. Soliton-like ‘bubbles’ in a system of interacting bosons. Phys. Lett. A 128 (1988), 52–56.CrossRefGoogle Scholar
4Benzoni–Gavage, S., Danchin, R., Descombes, S. and Jamet, D.. Structure of Korteweg models and stability of diffuse interfaces. Interfaces Free Bound. 7 (2005), 371–414.CrossRefGoogle Scholar
5Benzoni–Gavage, S., Danchin, R. and Descombes, S.. Well-posedness of one-dimensional Korteweg models. Electron. J. Differ. Equ. (2006), 59, 35 pp. (electronic).Google Scholar
6Benzoni–Gavage, S., Danchin, R. and Descombes, S.. On the well-posedness for the Euler–Korteweg model in several space dimensions. Indiana Univ. Math. J. 56 (2007), 1499–1579.10.1512/iumj.2007.56.2974CrossRefGoogle Scholar
7Béthuel, F., Gravejat, P. and Smets, D.. Stability in the energy space for chains of solitons of the one-dimensional Gross–Pitaevskii equation. Ann. Inst. Fourier (Grenoble) 64 (2014), 19–70.10.5802/aif.2838CrossRefGoogle Scholar
8Carles, R., Danchin, R. and Saut, J.-C.. Madelung, Gross–Pitaevskii and Korteweg. Nonlinearity 25 (2012), 2843–2873.10.1088/0951-7715/25/10/2843CrossRefGoogle Scholar
9Combet, V.. Multi-existence of multi-solitons for the supercritical nonlinear Schrödinger equation in one dimension. Discrete Contin. Dyn. Syst. 34 (2014), 1961–1993.10.3934/dcds.2014.34.1961CrossRefGoogle Scholar
10Côte, R., Martel, Y. and Merle, F.. Construction of multi-soliton solutions for the L 2-supercritical gKdV and NLS equations. Rev. Mat. Iberoam. 27 (2011), 273–302.CrossRefGoogle Scholar
11Giesselmann, J., Lattanzio, C. and Tzavaras, A. E.. Relative energy for the Korteweg theory and related Hamiltonian flows in gas dynamics. Arch. Ration. Mech. Anal. 223 (2017), 1427–1484.10.1007/s00205-016-1063-2CrossRefGoogle Scholar
12Grillakis, M., Shatah, J. and Strauss, W.. Stability theory of solitary waves in the presence of symmetry. I. J. Funct. Anal. 74 (1987), 160–197.Google Scholar
13Le Coz, S. and Tsai, T.-P.. Infinite soliton and kink-soliton trains for nonlinear Schrödinger equations. Nonlinearity 27 (2014), 2689–2709.CrossRefGoogle Scholar
14Lin, L. and Tsai, T.-P.. Mixed dimensional infinite soliton trains for nonlinear Schrödinger equations. Discrete Contin. Dyn. Syst. 37 (2017), 295–336.Google Scholar
15Martel, Y. and Merle, F.. Multi solitary waves for nonlinear Schrödinger equations. Ann. Inst. H. Poincaré Anal. Non Linéaire 23 (2006), 849–864.Google Scholar
16Ming, M., Rousset, F. and Tzvetkov, N.. Multi-solitons and related solutions for the water-waves system. SIAM J. Math. Anal. 47 (2015), 897–954.Google Scholar
17Pego, R. L. and Weinstein, M. I.. Eigenvalues, and instabilities of solitary waves. Philos. Trans. Roy. Soc. London Ser. A 340 (1992), 47–94.Google Scholar
18Zakharov, V. E. and Shabat, A. B.. Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media. Ž. Èksper. Teoret. Fiz. 61 (1971), 118–134.Google Scholar