Article Contents
Article Contents

# Propagation of long-crested water waves

• The present essay is concerned with a model for the propagation of three-dimensional, surface water waves. Of especial interest will be long-crested waves such as those sometimes observed in canals and in near-shore zones of large bodies of water. Such waves propagate primarily in one direction, taken to be the $x-$direction in a Cartesian framework, and variations in the horizontal direction orthogonal to the primary direction, the $y-$direction, say, are often ignored. However, there are situations where weak variations in the secondary horizontal direction need to be taken into account.
Our results are developed in the context of Boussinesq models, so they are applicable to waves that have small amplitude and long wavelength when compared with the undisturbed depth. Included in the theory are well-posedness results on the long, Boussinesq time scale. As mentioned, particular interest is paid to 1000 the lateral dynamics, which turn out to satisfy a reduced Boussinesq system. Waves corresponding to disturbances which are localized in the $x-$direction as well as bore-like disturbances that have infinite energy are taken up in the discussion.
Mathematics Subject Classification: Primary: 35A01, 35A02, 35A09, 35Q35, 35Q51; Secondary: 35Q86, 45G15, 76B03, 76B15, 76B25.

 Citation:

•  [1] M. Abramowitz and I. A. Stegun (ed.), "Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables,'' Dover, New York, 1965. [2] A. A. Alazman, J. P. Albert, J. L. Bona, M. Chen and J. Wu, Comparisons between the BBM-equation and a Boussinesq system, Adv. Differential Equations, 11 (2006), 121-166. [3] B. Alvarez-Samaniego and D. Lannes, Large time existence for 3D water-wavesand asymptotics, Invent. Math, 171 (2008), 485-541.doi: 10.1007/s00222-007-0088-4. [4] T. B. Benjamin, J. L. Bona and J. J. Mahony, Model equations for long waves in nonlinear, dispersive media, Philos. Trans. Roy. Soc. London Ser. A, 272 (1972), 47-78.doi: 10.1098/rsta.1972.0032. [5] J. L. Bona and P. J. Bryant, A mathematical model for long waves generated by wavemakers innon-linear dispersive systems, Proc. Cambridge Philos. Soc., 73 (1973), 391-405.doi: 10.1017/S0305004100076945. [6] J. L. Bona and M. Chen, A Boussinesq system for two-way propagation of nonlinear dispersive waves, Phys. D, 116 (1998), 191-224.doi: 10.1016/S0167-2789(97)00249-2. [7] J. L. Bona, M. Chen and J.-C. Saut, Boussinesq equations and other systems for small-amplitude long waves in nonlinear dispersive media. I: Derivation and linear theory, J. Nonlinear Sci., 12 (2002), 283-318.doi: 10.1007/s00332-002-0466-4. [8] J. L. Bona, M. Chen and J.-C. Saut, Boussinesq equations and other systems for small-amplitude long waves in nonlinear dispersive media. II: The nonlinear theory, Nonlinearity, 17 (2004), 925-952.doi: 10.1088/0951-7715/17/3/010. [9] J. L. Bona, T. Colin and D. Lannes, Long wave approximation for water waves, Arch. Ration. Mech. Anal., 178 (2005), 373-410.doi: 10.1007/s00205-005-0378-1. [10] J. L. Bona and V. A. Dougalis, An initial and boundary value problem for amodel equation for the propagation of long waves, J. Math. Anal. Appl., 75 (1980), 503-522.doi: 10.1016/0022-247X(80)90098-0. [11] J. L. Bona and H. Kalisch, Models for internal waves in deep water, Discrete Contin. Dynam. Systems, Series B, 6 (2000), 1-20. [12] J. L. Bona, S. V. Rajopadhye and M. E. Schonbek, Models for propagation of bores. I. Two dimensional theory, Differential Integral Equations, 7 (1994), 699-734. [13] J. L. Bona, S. M. Sun and B. Y. Zhang, Conditional and unconditional well-posedness for nonlinear evolution equations, Adv. Differential Equations, 9 (2004), 241-265. [14] J. L. Bona and N. Tzvetkov, Sharp well-posedness results for the BBM equation, Discrete Contin. Dyn. Systems, Series A, 23 (2009), 1241-1252. [15] V. A. Dougalis, D. E. Mitsotakis and J.-C. Saut, On some Boussinesq systems in two space dimensions: theory and numerical analysis, M2AN Math. Model. Numer. Anal., 41 (2007), 825-854.doi: 10.1051/m2an:2007043. [16] V. A. Dougalis, D. E. Mitsotakis and J.-C. Saut, On initial-boundary value problems for a Boussinesq system of BBM-BBM type in a plane domain, Discrete Contin. Dynam. Systems, Series A, 23 (2009), 1191-1204. [17] I. S. Gradshteyn and I. M. Ryzhik, "Table of Integrals, Series, and Products,'' $7^{th}$ edition (prepared by Yu. V. Geronimus and M. Yu. Tseytlin, transl. ed. A. Jeffrey and D. Zwillinger), Elsevier/Academic Press, Amsterdam, 2007. [18] B. B. Kadomtsev and V. I. Petviashvili, Stability of solitary waves in weakly dispersing media, Doklady Akademii Nauk SSSR, 192 (1970), 753-756. (Russian) (transl. Soviet Phys. Dokl., 15 (1970), 539-541). [19] T. Kato, On Nonlinear Schr\"odinger Equations. II. $H^s$ - Solutions and unconditional well-posedness, J. Anal. Math., 67 (1995), 281-306.doi: 10.1007/BF02787794. [20] D. Lannes and J.-C. Saut, Weakly transverse Boussinesq systems and the KP approximation, Nonlinearity, 19 (2006), 2853-2875.doi: 10.1088/0951-7715/19/12/007. [21] M. Ming, J.-C. Saut and P. Zhang, Long time existence of solutions to Boussinesq systems, Submitted. [22] L. Molinet, J.-C. Saut and N. Tzvetkov, Remarks on the mass constraint for KP-type equations, SIAM J. Math. Anal., 39 (2007), 627-641.doi: 0.1137/060654256. [23] D. H. Peregrine, Calculation of the development of an undular bore, J. Fluid Mechanics, 25 (1966), 321-330.doi: 10.1017/S0022112066001678. [24] S. V. Rajopadhye, Propagation of bores. II. Three-dimensional theory, Nonlinear Anal., 27 (1996), 963-986.doi: 10.1016/0362-546X(94)00358-O. [25] S. V. Rajopadhye, Some models for the propagation of bores, J. Differential Equations, 217 (2005), 179-203. [26] J.-C. Saut and L. Xu, The Cauchy problem on large time for surface waves Boussinesq systems, J. Math. Pures Appl., 97 (2012), 635-662.