Weakly transverse Boussinesq systems and the Kadomtsev–Petviashvili approximation

We study here the asymptotic behaviour of weakly transverse water-waves in the long waves regime. It is well-known that the Kadomtsev–Petviashvili (KP) approximation describes formally the dynamics of the exact solutions of the water-waves equations. We provide here a rigorous justification of this approximation, showing that if solutions of the water-waves equations exist over a relevant time scale, then they are well approximated by the KP approximation. A nonphysical zero mass assumption, inherent to the structure of the KP equation, is however needed to obtain this result; this is the reason why we introduce a class of weakly transverse Boussinesq systems. These new systems provide a much more precise approximation than the KP equation and do not require any zero mass assumption.

[1]  Matsuno.  Nonlinear evolutions of surface gravity waves on fluid of finite depth. , 1992, Physical review letters.

[2]  Walter Craig,et al.  Hamiltonian long-wave approximations to the water-wave problem , 1994 .

[3]  Thierry Colin,et al.  Long Wave Approximations for Water Waves , 2005 .

[4]  Walter Craig,et al.  Nonlinear modulation of gravity waves: a rigorous approach , 1992 .

[5]  G. B. Whitham,et al.  Linear and Nonlinear Waves: Whitham/Linear , 1999 .

[6]  D. Lannes.  Secular growth estimates for hyperbolic systems , 2003 .

[7]  Min Chen,et al.  Boussinesq Equations and Other Systems for Small-Amplitude Long Waves in Nonlinear Dispersive Media. I: Derivation and Linear Theory , 2002, J. Nonlinear Sci..

[8]  J. Bona,et al.  Model equations for long waves in nonlinear dispersive systems , 1972, Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences.

[9]  Yongqian Han,et al.  Remarks on the generalized Kadomtsev-Petviashvili equations and two-dimensional Benjamin-Ono equations , 1996, Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences.

[10]  O. Nwogu.  Alternative form of Boussinesq equations for nearshore wave propagation , 1993 .

[11]  David Lannes,et al.  Well-posedness of the water-waves equations , 2005 .

[12]  David Lannes,et al.  Consistency of the KP approximation , 2003 .

[13]  David Lannes,et al.  Dispersive effects for nonlinear geometrical optics with rectification , 1998 .

[14]  H. Schäffer,et al.  Further enhancements of Boussinesq-type equations , 1995 .

[15]  Jean-Claude Saut,et al.  Remarks on the Mass Constraint for KP-Type Equations , 2007, SIAM J. Math. Anal..

[16]  Thierry Gallay,et al.  KP description of unidirectional long waves. The model case , 2001, Proceedings of the Royal Society of Edinburgh: Section A Mathematics.

[17]  Vladimir E. Zakharov,et al.  Stability of periodic waves of finite amplitude on the surface of a deep fluid , 1968 .

[18]  A. Nachbin,et al.  Asymptotic nonlinear wave modeling through the Dirichlet-to-Neumann operator , 2004 .

[19]  Lionel Paumond,et al.  A rigorous link between KP and a Benney-Luke equation , 2003, Differential and Integral Equations.

[20]  Walter Craig,et al.  The modulational regime of three-dimensional water waves and the Davey-Stewartson system , 1997 .

[21]  J. Bona,et al.  Boussinesq equations and other systems for small-amplitude long waves in nonlinear dispersive media: II. The nonlinear theory , 2004 .

[22]  David Lannes,et al.  THE LONG WAVE LIMIT FOR A GENERAL CLASS OF 2D QUASILINEAR HYPERBOLIC PROBLEMS , 2002 .