REMARKS ON THE FULL DISPERSION KADOMTSEV-PETVIASHVLI EQUATION

We consider in this paper the Full Dispersion Kadomtsev-Petviashvili Equation (FDKP) introduced in [19] in order to overcome some shortcomings of the classical KP equation. We investigate its mathematical properties, emphasizing the differences with the Kadomtsev-Petviashvili equation and their relevance to the approximation of water waves. We also present some numerical simulations.

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

[2]  Ljudmila A. Bordag,et al.  Two-dimensional solitons of the Kadomtsev-Petviashvili equation and their interaction , 1977 .

[3]  D. Lannes,et al.  Long-wave short-wave resonance for nonlinear geometric optics , 2001 .

[4]  G. G. Stokes.  "J." , 1890, The New Yale Book of Quotations.

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

[6]  Zhifei Zhang,et al.  Long-Wave Approximation to the 3-D Capillary-Gravity Waves , 2010, SIAM J. Math. Anal..

[7]  Mark D. Groves,et al.  On the existence and stability of solitary-wave solutions to a class of evolution equations of Whitham type , 2012, 1203.3228.

[8]  Christof Sparber,et al.  Numerical Study of Oscillatory Regimes in the Kadomtsev–Petviashvili Equation , 2006, J. Nonlinear Sci..

[9]  Jean-Claude Saut,et al.  Dispersive Perturbations of Burgers and Hyperbolic Equations I: Local Theory , 2013, SIAM J. Math. Anal..

[10]  L. Chambers.  Linear and Nonlinear Waves , 2000, The Mathematical Gazette.

[11]  M. Weinstein.  Existence and dynamic stability of solitary wave solutions of equations arising in long wave propagation , 1987 .

[12]  Jean Bourgain,et al.  On the Cauchy problem for the Kadomstev-Petviashvili equation , 1993 .

[13]  C. Klein,et al.  A numerical approach to blow-up issues for dispersive perturbations of Burgers’ equation , 2014, 1401.1390.

[14]  Jean-Claude Saut,et al.  Numerical Study of Blow up and Stability of Solutions of Generalized Kadomtsev–Petviashvili Equations , 2010, Journal of Nonlinear Science.

[15]  Luc Molinet,et al.  On the Asymptotic Behavior of Solutions to the (Generalized) Kadomtsev–Petviashvili–Burgers Equations☆ , 1999 .

[16]  Jean-Claude Saut,et al.  Weakly transverse Boussinesq systems and the Kadomtsev–Petviashvili approximation , 2006 .

[17]  Rupert L. Frank,et al.  Uniqueness and Nondegeneracy of Ground States for (−Δ)^sQ+Q−Q^(α+1)=0 in R , 2015 .

[18]  J. Saut,et al.  Solitary waves of generalized Kadomtsev-Petviashvili equations , 1997 .

[19]  Francisco Gancedo,et al.  Singularity formations for a surface wave model , 2010, 1004.3975.

[20]  B. Kadomtsev,et al.  On the Stability of Solitary Waves in Weakly Dispersing Media , 1970 .

[21]  G. B. Whitham,et al.  Variational methods and applications to water waves , 1967, Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences.

[22]  David Lannes,et al.  The Water Waves Problem: Mathematical Analysis and Asymptotics , 2013 .

[23]  H. Takaoka,et al.  On the local regularity of the Kadomtsev-Petviashvili-II equation , 2001 .

[24]  J. Bona,et al.  Model equations for waves in stratified fluids , 1997, Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences.

[25]  J. Escher,et al.  Wave breaking for nonlinear nonlocal shallow water equations , 1998 .

[26]  M. Weinstein.  Nonlinear Schrödinger equations and sharp interpolation estimates , 1983 .

[27]  Jean-Claude Saut,et al.  Remarks on the Mass Constraint for KP-Type Equations , 2006, SIAM J. Math. Anal..

[28]  Jeffrey Rauch,et al.  The Linear Diffractive Pulse Equation , 2000 .

[29]  Pavel I. Naumkin,et al.  Nonlinear Nonlocal Equations in the Theory of Waves , 1994 .

[30]  David Lannes,et al.  Large time existence for 3D water-waves and asymptotics , 2007, math/0702015.

[31]  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.

[32]  Lizhong Peng,et al.  Decay estimates for a class of wave equations , 2008, 0802.3167.

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

[34]  H. Kalisch,et al.  Traveling waves for the Whitham equation , 2009, Differential and Integral Equations.

[35]  The water waves problem and its asymptotic regimes , 2013 .