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Existence of multi-travelling waves in capillary fluids

  • Corentin Audiard (a1)


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.



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1Audiard, C.. Small energy traveling waves for the Euler–Korteweg system. Nonlinearity 30 (2017), 33623399.
2Audiard, C. and Haspot, B.. Global well-posedness of the Euler–Korteweg system for small irrotational data. Comm. Math. Phys. 351 (2017), 201247.
3Barashenkov, I. V. and Makhankov, V. G.. Soliton-like ‘bubbles’ in a system of interacting bosons. Phys. Lett. A 128 (1988), 5256.
4Benzoni–Gavage, S., Danchin, R., Descombes, S. and Jamet, D.. Structure of Korteweg models and stability of diffuse interfaces. Interfaces Free Bound. 7 (2005), 371414.
5Benzoni–Gavage, S., Danchin, R. and Descombes, S.. Well-posedness of one-dimensional Korteweg models. Electron. J. Differ. Equ. (2006), 59, 35 pp. (electronic).
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), 14991579.
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), 1970.
8Carles, R., Danchin, R. and Saut, J.-C.. Madelung, Gross–Pitaevskii and Korteweg. Nonlinearity 25 (2012), 28432873.
9Combet, V.. Multi-existence of multi-solitons for the supercritical nonlinear Schrödinger equation in one dimension. Discrete Contin. Dyn. Syst. 34 (2014), 19611993.
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), 273302.
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), 14271484.
12Grillakis, M., Shatah, J. and Strauss, W.. Stability theory of solitary waves in the presence of symmetry. I. J. Funct. Anal. 74 (1987), 160197.
13Le Coz, S. and Tsai, T.-P.. Infinite soliton and kink-soliton trains for nonlinear Schrödinger equations. Nonlinearity 27 (2014), 26892709.
14Lin, L. and Tsai, T.-P.. Mixed dimensional infinite soliton trains for nonlinear Schrödinger equations. Discrete Contin. Dyn. Syst. 37 (2017), 295336.
15Martel, Y. and Merle, F.. Multi solitary waves for nonlinear Schrödinger equations. Ann. Inst. H. Poincaré Anal. Non Linéaire 23 (2006), 849864.
16Ming, M., Rousset, F. and Tzvetkov, N.. Multi-solitons and related solutions for the water-waves system. SIAM J. Math. Anal. 47 (2015), 897954.
17Pego, R. L. and Weinstein, M. I.. Eigenvalues, and instabilities of solitary waves. Philos. Trans. Roy. Soc. London Ser. A 340 (1992), 4794.
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), 118134.


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