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Fully localised solitary-wave solutions of the three-dimensional gravity – capillary water-wave problem. (English) Zbl 1133.76010
Summary: The model Kadomtsev-Petviashvili equation suggests that the hydrodynamic problem for three-dimensional water waves with strong surface-tension effects admits a fully localised solitary wave which decays to the undisturbed state of water in every horizontal spatial direction. This prediction is rigorously confirmed for the full water-wave problem in the present paper. The theory is variational in nature. A simple but mathematically unfavourable variational principle for fully localised solitary waves is reduced to a locally equivalent variational principle with significantly better mathematical properties. The reduced functional is related to the functional associated with Kadomtsev-Petviashvili equation, and a nontrivial critical point is detected using the direct methods of the calculus of variations.

##### MSC:
 76B25 Solitary waves for incompressible inviscid fluids 76B45 Capillarity (surface tension) for incompressible inviscid fluids 76M30 Variational methods applied to problems in fluid mechanics 35Q51 Soliton equations
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##### References:
 [1] Ablowitz M.J., Segur H. (1979) On the evolution of packets of water waves. J. Fluid Mech. 92: 691–715 · Zbl 0413.76009 [2] Adams R.A., Fournier J.J.F. (2003) Sobolev Spaces 2nd edn. Elsevier, Oxford (Pure and applied mathematics series 140) [3] Amick C.J., Kirchgässner K. (1989) A theory of solitary water waves in the presence of surface tension. Arch. Ration. Mech. Anal. 105: 1–49 · Zbl 0666.76046 [4] Bona J., Chen H.Q. (2002) Solitary waves in nonlinear dispersive systems. Discrete Contin. Dyn. Syst. Ser. B 2: 313–378 · Zbl 1162.35438 [5] Brezis H., Nirenberg L. (1991) Remarks on finding critical points. Commun. Pure Appl. Math. 44: 939–963 · Zbl 0751.58006 [6] Buffoni B. (2004a) Existence and conditional energetic stability of capillary-gravity solitary water waves by minimisation. Arch. Ration. Mech. Anal. 173: 25–68 · Zbl 1110.76308 [7] Buffoni B. (2004b) Existence by minimisation of solitary water waves on an ocean of infinite depth. Ann. Inst. Henri Poincaré Anal. Non Linéaire 21: 503–516 · Zbl 1109.76013 [8] Buffoni B., Groves M.D. (1999) A multiplicity result for solitary gravity–capillary waves in deep water via critical-point theory. Arch. Ration. Mech. Anal. 146: 183–220 · Zbl 0965.76015 [9] Buffoni B., Groves M.D., Toland J.F. (1996) A plethora of solitary gravity–capillary water waves with nearly critical Bond and Froude numbers. Phil. Trans. R. Soc. Lond. A 354: 575–607 · Zbl 0861.76012 [10] Buffoni B., Dancer E.N., Toland J.F. (2000a) The regularity and local bifurcation of steady periodic water waves. Arch. Ration. Mech. Anal. 152: 207–240 · Zbl 0959.76010 [11] Buffoni B., Dancer E.N., Toland J.F. (2000b) The sub-harmonic bifurcation of Stokes waves. Arch. Ration. Mech. Anal. 152: 241–271 · Zbl 0962.76012 [12] Buffoni B., Séré E., Toland J.F. (2003) Surface water waves as saddle points of the energy. Calc. Var. PDE 2: 199–220 · Zbl 1222.76019 [13] Craig W. (2002) Nonexistence of solitary water waves in three dimensions. Phil. Trans. R. Soc. Lond. A 360: 2127–2135 · Zbl 1013.76015 [14] Craig W., Nicholls D.P. (2000) Traveling two and three dimensional capillary gravity water waves. SIAM J. Math. Anal. 32: 323–359 · Zbl 0976.35049 [15] de Bouard A., Saut J.-C. (1997) Solitary waves of generalized Kadomtsev-Petviashvili equations. Ann. Inst. H. Poincaré Anal. Non Linéaire 14: 211–236 · Zbl 0883.35103 [16] Groves M.D. (1998) Solitary-wave solutions to a class of fifth-order model equations. Nonlinearity 11: 341–353 · Zbl 0994.34034 [17] Groves M.D. (2001) An existence theory for three-dimensional periodic travelling gravity–capillary water waves with bounded transverse profiles. Physica D 152–153: 395–415 · Zbl 1037.76005 [18] Groves M.D., Haragus M. (2003) A bifurcation theory for three-dimensional oblique travelling gravity–capillary water waves. J. Nonlin. Sci. 13: 397–447 · Zbl 1116.76326 [19] Groves M.D., Mielke A. (2001) A spatial dynamics approach to three-dimensional gravity–capillary steady water waves. Proc. R. Soc. Edin. A 131: 83–136 · Zbl 0976.76012 [20] Groves M.D., Haragus M., Sun S.-M. (2002) A dimension-breaking phenomenon in the theory of gravity–capillary water waves. Phil. Trans. R. Soc. Lond. A 360: 2189–2243 · Zbl 1068.76007 [21] Hardy G., Littlewood J.E., Pólya G. (1988) Inequalities, paperback edn. Cambridge University Press, Cambridge [22] Hutson V., Pym J.S. (1980) Applicatons of Functional Analysis and Operator Theory. Academic, London · Zbl 0426.46009 [23] Kadomtsev B.B., Petviashvili V.I. (1970) On the stability of solitary waves in weakly dispersing media. Sov. Phys. Dokl. 15: 539–541 · Zbl 0217.25004 [24] Kichenassamy S. (1997) Existence of solitary waves for water-wave models. Nonlinearity 10: 133–151 · Zbl 0906.76012 [25] Kirchgässner K. (1988) Nonlinearly resonant surface waves and homoclinic bifurcation. Adv. Appl. Mech. 26: 135–181 · Zbl 0671.76019 [26] Levandosky S.P. (2000) A stability analysis of fifth-order water wave models. Physica D 125: 222–240 · Zbl 0934.35159 [27] Lions J.L., Magenes E. (1961) Probleme ai limiti non omogenei (III). Ann. Scuola Norm. Sup. Pisa 15: 41–103 [28] Lions P.L. (1984a) The concentration-compactness principle in the calculus of variations–the locally compact case, part 1. Ann. Inst. Henri Poincaré Anal. Non Linéaire 1: 109–145 · Zbl 0541.49009 [29] Lions P.L. (1984b) The concentration-compactness principle in the calculus of variations–the locally compact case, part 2. Ann. Inst. Henri Poincaré Anal. Non Linéaire 1: 223–283 · Zbl 0704.49004 [30] Luke J.C. (1967) A variational principle for a fluid with a free surface. J. Fluid Mech. 27: 395–397 · Zbl 0146.23701 [31] Mazya V. (1985) Sobolev Spaces. Springer-Verlag, New York [32] Mielke A. (1991) Hamiltonian and Lagrangian Flows on Center Manifolds. Springer-Verlag, Berlin · Zbl 0747.58001 [33] Moser J. (1976) Periodic orbits near an equilibrium and a theorem by Alan Weinstein. Commun. Pure Appl. Math. 29: 727–747 · Zbl 0346.34024 [34] Pego R.L., Quintero J.R. (1999) Two-dimensional solitary waves for a Benny–Luke equation. Physica D 132: 476–496 · Zbl 0935.35139 [35] Plotnikov P.I. (1992) Nonuniqueness of solutions of the problem of solitary waves and bifurcation of critical points of smooth functionals. Math. USSR Izvestiya 38: 333–357 · Zbl 0795.76017 [36] Sachs R.L. (1991) On the existence of small amplitude solitary waves with strong surface-tension. J. Differ. Eqn. 90: 31–51 · Zbl 0746.35039 [37] Stein E.M. (1970) Singular Integrals and Differentiability Properties of Functions. Princeton University Press, Princeton · Zbl 0207.13501 [38] Stoker J.J. (1957) Water Waves: The Mathematical Theory with Applications. Interscience, New York · Zbl 0078.40805 [39] Tajiri M., Murakami Y. (1990) The periodic solution resonance: solutions of the Kadomtsev–Petviashvili equation with positive dispersion. Phys. Lett. A 143: 217–220 [40] Turner R.E.L. (1984) A variational approach to surface solitary waves. J. Differ. Eqn. 55: 401–438 · Zbl 0574.76015 [41] Wang X.P., Ablowitz M.J., Segur H. (1994) Wave collapse and instability of solitary waves of a generalized Kadomtsev–Petviashvili equation. Physica D 78: 241–265 · Zbl 0824.35116 [42] Weinstein M.I. (1987) Existence and dynamic stability of solitary wave solutions of equations arising in long wave propagation. Commun. Part. Differ. Eqn. 12: 1133–1173 · Zbl 0657.73040
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