Fully localised three-dimensional gravity-capillary solitary waves on water of infinite depth. (English) Zbl 1501.76016
Summary: Fully localised three-dimensional solitary waves are steady water waves which are evanescent in every horizontal direction. Existence theories for fully localised three-dimensional solitary waves on water of finite depth have recently been published, and in this paper we establish their existence on deep water. The governing equations are reduced to a perturbation of the two-dimensional nonlinear Schrödinger equation, which admits a family of localised solutions. Two of these solutions are symmetric in both horizontal directions and an application of a suitable variant of the implicit-function theorem shows that they persist under perturbations.
MSC:
| 76B25 | Solitary waves for incompressible inviscid fluids |
| 76B45 | Capillarity (surface tension) for incompressible inviscid fluids |
| 76M45 | Asymptotic methods, singular perturbations applied to problems in fluid mechanics |
| 35Q35 | PDEs in connection with fluid mechanics |
| 35Q55 | NLS equations (nonlinear Schrödinger equations) |
Keywords:
existence; perturbation method; two-dimensional nonlinear Schrödinger equation; symmetric localised solution; implicit-function theoremReferences:
| [1] | Ablowitz, MJ; Segur, H., On the evolution of packets of water waves, J. Fluid Mech., 92, 691-715 (1979) · Zbl 0413.76009 · doi:10.1017/S0022112079000835 |
| [2] | Berestycki, H.; Lions, P-L, Nonlinear scalar field equations, II, Existence of infinitely many solutions, Arch. Ration. Mech. Anal., 82, 347-375 (1983) · Zbl 0556.35046 · doi:10.1007/BF00250556 |
| [3] | Buffoni, B.; Groves, MD; Sun, SM; Wahlén, E., Existence and conditional energetic stability of three-dimensional fully localised solitary gravity-capillary water waves, J. Differ. Equ., 254, 1006-1096 (2013) · Zbl 1336.76008 · doi:10.1016/j.jde.2012.10.007 |
| [4] | Buffoni, B.; Groves, MD; Wahlén, E., A variational reduction and the existence of a fully localised solitary wave for the three-dimensional water-wave problem with weak surface tension, Arch. Ration. Mech. Anal., 228, 773-820 (2018) · Zbl 1397.35199 · doi:10.1007/s00205-017-1205-1 |
| [5] | Chang, S-M; Gustafson, S.; Nakanishi, K.; Tsai, T-P, Spectra of linearized operators for NLS solitary waves, SIAM J. Math. Anal., 39, 1070-1111 (2007) · Zbl 1168.35041 · doi:10.1137/050648389 |
| [6] | Craig, W.; Sulem, C., Numerical simulation of gravity waves, J. Comp. Phys., 108, 73-83 (1993) · Zbl 0778.76072 · doi:10.1006/jcph.1993.1164 |
| [7] | Dias, F.; Kharif, C., Nonlinear gravity and capillary-gravity waves, Ann. Rev. Fluid Mech., 31, 301-346 (1999) · doi:10.1146/annurev.fluid.31.1.301 |
| [8] | Groves, MD, An existence theory for gravity-capillary solitary water waves, Water Waves, 3, 213-250 (2021) · Zbl 1486.76017 · doi:10.1007/s42286-020-00045-7 |
| [9] | Groves, MD; Sun, S-M, Fully localised solitary-wave solutions of the three-dimensional gravity-capillary water-wave problem, Arch. Ration. Mech. Anal., 188, 1-91 (2008) · Zbl 1133.76010 · doi:10.1007/s00205-007-0085-1 |
| [10] | Hörmander, L., Lectures on Nonlinear Hyperbolic Differential Equations (1997), Heidelberg: Springer, Heidelberg · Zbl 0881.35001 |
| [11] | Iaia, J.; Warchall, H., Nonradial solutions of a semilinear elliptic equation in two dimensions, J. Differ. Equ., 119, 533-558 (1995) · Zbl 0832.35040 · doi:10.1006/jdeq.1995.1101 |
| [12] | Jones, C.; Küpper, T., On the infinitely many solutions of a semilinear elliptic equation, SIAM J. Math. Anal., 17, 803-835 (1986) · Zbl 0606.35032 · doi:10.1137/0517059 |
| [13] | Kwong, MK, Uniqueness of positive solutions of \(\Delta u - u + u^p=0\) in \({\mathbb{R}}^n\), Arch. Ration. Mech. Anal., 105, 243-266 (1989) · Zbl 0676.35032 · doi:10.1007/BF00251502 |
| [14] | Liu, Y.; Wei, J., Nondegeneracy, Morse index and orbital stability of the KP-I lump solution, Arch. Ration. Mech. Anal., 234, 1335-1389 (2019) · Zbl 1428.35458 · doi:10.1007/s00205-019-01413-5 |
| [15] | McLeod, K.; Troy, WC; Weissler, WB, Radial solutions of \({\Delta } u + f(u)=0\) with prescribed numbers of zeros, J. Differ. Equ., 23, 368-378 (1990) · Zbl 0695.34020 · doi:10.1016/0022-0396(90)90063-U |
| [16] | Obrecht, C.; Saut, J-C, Remarks on the full dispersion Davey-Stewartson systems, Commun. Pure Appl. Anal., 14, 1547-1561 (2015) · Zbl 1318.35085 · doi:10.3934/cpaa.2015.14.1547 |
| [17] | Parau, EI; Vanden-Broeck, J-M; Cooker, MJ, Nonlinear three-dimensional gravity-capillary solitary waves, J. Fluid Mech., 536, 99-105 (2005) · Zbl 1073.76010 · doi:10.1017/S0022112005005136 |
| [18] | Pelinovsky, D.; Schneider, G., Justification of the coupled-mode approximation for a nonlinear elliptic problem with a periodic potential, Appl. Anal., 86, 1017-1036 (2007) · Zbl 1132.41344 · doi:10.1080/00036810701493850 |
| [19] | Stefanov, A.; Wright, JD, Small amplitude traveling waves in the full-dispersion Whitham equation, J. Dyn. Differ. Equ., 32, 85-99 (2020) · Zbl 1436.34018 · doi:10.1007/s10884-018-9713-8 |
| [20] | Strauss, WA, Existence of solitary waves in higher dimensions, Commun. Math. Phys., 55, 149-162 (1977) · Zbl 0356.35028 · doi:10.1007/BF01626517 |
| [21] | Sulem, C.; Sulem, PL, The Nonlinear Schrödinger Equation. Applied Mathematical Sciences (1999), New York: Springer, New York · Zbl 0928.35157 |
| [22] | Weinstein, MI, Modulational stability of ground states of nonlinear Schrödinger equations, SIAM J. Math. Anal., 16, 472-491 (1985) · Zbl 0583.35028 · doi:10.1137/0516034 |
| [23] | Wheeler, M., Integral and asymptotic properties of solitary waves in deep water, Commun. Pure Appl. Math., 71, 1941-1956 (2018) · Zbl 1404.35371 · doi:10.1002/cpa.21786 |
| [24] | Zakharov, V.E.: Stability of periodic waves of finite amplitude on the surface of a deep fluid. Zh. Prikl. Mekh. Tekh. Fiz. 9, 86-94 (English translation J. Appl. Mech. Tech. Phys. 9, 190-194) (1968) |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
