American Institute of Mathematical Sciences

February  2013, 33(2): 599-628. doi: 10.3934/dcds.2013.33.599

Propagation of long-crested water waves

 1 Department of Mathematics, Statistics and Computer Science, The University of Illinois at Chicago, 851 S. Morgan St. MC 249, Chicago, IL 60607, United States 2 Université de Bordeaux, Institut de Mathématiques de Bordeaux, UMR CNRS 5251 and INRIA Bordeaux Sud-Ouest, 351 cours de la Libération，33405 Talence, France 3 Université Paris-Est, Laboratoire d'Analyse et de Mathématiques Appliquées (UMR 8050), UPEC, UPEMLV, CNRS, 94010 Créteil, France

Received  July 2011 Revised  May 2012 Published  September 2012

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.
Citation: Jerry L. Bona, Thierry Colin, Colette Guillopé. Propagation of long-crested water waves. Discrete & Continuous Dynamical Systems - A, 2013, 33 (2) : 599-628. doi: 10.3934/dcds.2013.33.599
References:
 [1] M. Abramowitz and I. A. Stegun (ed.), "Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables,'', Dover, (1965).   Google Scholar [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.   Google Scholar [3] B. Alvarez-Samaniego and D. Lannes, Large time existence for 3D water-wavesand asymptotics,, Invent. Math, 171 (2008), 485.  doi: 10.1007/s00222-007-0088-4.  Google Scholar [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.  doi: 10.1098/rsta.1972.0032.  Google Scholar [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.  doi: 10.1017/S0305004100076945.  Google Scholar [6] J. L. Bona and M. Chen, A Boussinesq system for two-way propagation of nonlinear dispersive waves,, Phys. D, 116 (1998), 191.  doi: 10.1016/S0167-2789(97)00249-2.  Google Scholar [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.  doi: 10.1007/s00332-002-0466-4.  Google Scholar [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.  doi: 10.1088/0951-7715/17/3/010.  Google Scholar [9] J. L. Bona, T. Colin and D. Lannes, Long wave approximation for water waves,, Arch. Ration. Mech. Anal., 178 (2005), 373.  doi: 10.1007/s00205-005-0378-1.  Google Scholar [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.  doi: 10.1016/0022-247X(80)90098-0.  Google Scholar [11] J. L. Bona and H. Kalisch, Models for internal waves in deep water,, Discrete Contin. Dynam. Systems, 6 (2000), 1.   Google Scholar [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.   Google Scholar [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.   Google Scholar [14] J. L. Bona and N. Tzvetkov, Sharp well-posedness results for the BBM equation,, Discrete Contin. Dyn. Systems, 23 (2009), 1241.   Google Scholar [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.  doi: 10.1051/m2an:2007043.  Google Scholar [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, 23 (2009), 1191.   Google Scholar [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, (2007).   Google Scholar [18] B. B. Kadomtsev and V. I. Petviashvili, Stability of solitary waves in weakly dispersing media,, Doklady Akademii Nauk SSSR, 192 (1970), 753.   Google Scholar [19] T. Kato, On Nonlinear Schr\"odinger Equations. II. $H^s$ - Solutions and unconditional well-posedness,, J. Anal. Math., 67 (1995), 281.  doi: 10.1007/BF02787794.  Google Scholar [20] D. Lannes and J.-C. Saut, Weakly transverse Boussinesq systems and the KP approximation,, Nonlinearity, 19 (2006), 2853.  doi: 10.1088/0951-7715/19/12/007.  Google Scholar [21] M. Ming, J.-C. Saut and P. Zhang, Long time existence of solutions to Boussinesq systems,, Submitted., ().   Google Scholar [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.  doi: 0.1137/060654256.  Google Scholar [23] D. H. Peregrine, Calculation of the development of an undular bore,, J. Fluid Mechanics, 25 (1966), 321.  doi: 10.1017/S0022112066001678.  Google Scholar [24] S. V. Rajopadhye, Propagation of bores. II. Three-dimensional theory,, Nonlinear Anal., 27 (1996), 963.  doi: 10.1016/0362-546X(94)00358-O.  Google Scholar [25] S. V. Rajopadhye, Some models for the propagation of bores,, J. Differential Equations, 217 (2005), 179.   Google Scholar [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.   Google Scholar

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References:
 [1] M. Abramowitz and I. A. Stegun (ed.), "Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables,'', Dover, (1965).   Google Scholar [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.   Google Scholar [3] B. Alvarez-Samaniego and D. Lannes, Large time existence for 3D water-wavesand asymptotics,, Invent. Math, 171 (2008), 485.  doi: 10.1007/s00222-007-0088-4.  Google Scholar [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.  doi: 10.1098/rsta.1972.0032.  Google Scholar [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.  doi: 10.1017/S0305004100076945.  Google Scholar [6] J. L. Bona and M. Chen, A Boussinesq system for two-way propagation of nonlinear dispersive waves,, Phys. D, 116 (1998), 191.  doi: 10.1016/S0167-2789(97)00249-2.  Google Scholar [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.  doi: 10.1007/s00332-002-0466-4.  Google Scholar [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.  doi: 10.1088/0951-7715/17/3/010.  Google Scholar [9] J. L. Bona, T. Colin and D. Lannes, Long wave approximation for water waves,, Arch. Ration. Mech. Anal., 178 (2005), 373.  doi: 10.1007/s00205-005-0378-1.  Google Scholar [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.  doi: 10.1016/0022-247X(80)90098-0.  Google Scholar [11] J. L. Bona and H. Kalisch, Models for internal waves in deep water,, Discrete Contin. Dynam. Systems, 6 (2000), 1.   Google Scholar [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.   Google Scholar [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.   Google Scholar [14] J. L. Bona and N. Tzvetkov, Sharp well-posedness results for the BBM equation,, Discrete Contin. Dyn. Systems, 23 (2009), 1241.   Google Scholar [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.  doi: 10.1051/m2an:2007043.  Google Scholar [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, 23 (2009), 1191.   Google Scholar [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, (2007).   Google Scholar [18] B. B. Kadomtsev and V. I. Petviashvili, Stability of solitary waves in weakly dispersing media,, Doklady Akademii Nauk SSSR, 192 (1970), 753.   Google Scholar [19] T. Kato, On Nonlinear Schr\"odinger Equations. II. $H^s$ - Solutions and unconditional well-posedness,, J. Anal. Math., 67 (1995), 281.  doi: 10.1007/BF02787794.  Google Scholar [20] D. Lannes and J.-C. Saut, Weakly transverse Boussinesq systems and the KP approximation,, Nonlinearity, 19 (2006), 2853.  doi: 10.1088/0951-7715/19/12/007.  Google Scholar [21] M. Ming, J.-C. Saut and P. Zhang, Long time existence of solutions to Boussinesq systems,, Submitted., ().   Google Scholar [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.  doi: 0.1137/060654256.  Google Scholar [23] D. H. Peregrine, Calculation of the development of an undular bore,, J. Fluid Mechanics, 25 (1966), 321.  doi: 10.1017/S0022112066001678.  Google Scholar [24] S. V. Rajopadhye, Propagation of bores. II. Three-dimensional theory,, Nonlinear Anal., 27 (1996), 963.  doi: 10.1016/0362-546X(94)00358-O.  Google Scholar [25] S. V. Rajopadhye, Some models for the propagation of bores,, J. Differential Equations, 217 (2005), 179.   Google Scholar [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.   Google Scholar
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