The focusing singularity of the Davey-Stewartson equations for gravity-capillary surface waves

Abstract We analyze the self-focusing of gravity-capillary surface waves modeled by the Davey-Stewartson equations. This system governs the coupling of the wave amplitude to the induced mean flow and is an anisotropic two-dimensional Schrodinger equation with a non local cubic nonlinearity. With accurate numerical simulations based on dynamic rescaling and asymptotic analysis we show that the dynamics of singularity formation is critical, as in the usual two-dimensional cubic Schrodinger equation, which is the deep water limit of the Davey-Stewartson equations. We also derive a sharp upper bound for the initial amplitude of the wave that prevents singularity formation.

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

[2]  Alan C. Newell,et al.  Optical turbulence: weak turbulence, condensates and collapsing filaments in the nonlinear Schro¨dinger equation , 1992 .

[3]  W. Ferguson,et al.  Fermi–Pasta–Ulam recurrence in the two‐space dimensional nonlinear Schrödinger equation , 1978 .

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

[5]  George Papanicolaou,et al.  Stability of isotropic singularities for the nonlinear schro¨dinger equation , 1991 .

[6]  R. Cipolatti.  On the existence of standing waves for a davey–stewartson system , 1992 .

[7]  Papanicolaou,et al.  Rate of blowup for solutions of the nonlinear Schrödinger equation at critical dimension. , 1988, Physical review. A, General physics.

[8]  M. Ablowitz,et al.  On the Boundary Conditions of the Davey-Stewartson Equation , 1990 .

[9]  A. Bers,et al.  Nonlinear development of lower hybrid cones , 1978 .

[10]  Elliott H. Lieb,et al.  On the lowest eigenvalue of the Laplacian for the intersection of two domains , 1983 .

[11]  Jean-Michel Ghidaglia,et al.  On the initial value problem for the Davey-Stewartson systems , 1990 .

[12]  Catherine Sulem,et al.  Local structure of the self-focusing singularity of the nonlinear Schro¨dinger equation , 1988 .

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

[14]  L. Redekopp,et al.  On two-dimensional packets of capillary-gravity waves , 1977, Journal of Fluid Mechanics.

[15]  Mark J. Ablowitz,et al.  On the evolution of packets of water waves , 1979, Journal of Fluid Mechanics.

[16]  Hiroaki Ono,et al.  Nonlinear Modulation of Gravity Waves , 1972 .

[17]  C. Liu,et al.  Self‐modulation of ion Bernstein waves , 1980 .

[18]  Walter A. Strauss,et al.  Existence of solitary waves in higher dimensions , 1977 .

[19]  M. Ablowitz,et al.  Solitons, Nonlinear Evolution Equations and Inverse Scattering , 1992 .