Standing Waves on an Infinitely Deep Perfect Fluid Under Gravity

The existence of two-dimensional standing waves on the surface of an infinitely deep perfect fluid under gravity is established. When formulated as a second-order equation for a real-valued function w on the 2-torus and a positive parameter μ, the problem is fully nonlinear (the highest order x-derivative appears in the nonlinear term but not in the linearization at 0) and completely resonant (there are infinitely many linearly independent eigenmodes of the linearization at 0 for all rational values of the parameter μ). Moreover, for any prescribed order of accuracy there exists an explicit approximate solution of the nonlinear problem in the form of a trigonometric polynomial. Using a Nash-Moser method to seek solutions of the nonlinear problem as perturbations of the approximate solutions, the existence question can be reduced to one of estimating the inverses of linearized operators at non-zero points. After changing coordinates these operators become first order and non-local in space and second order in time. After further changes of variables the main parts become diagonal with constant coefficients and the remainder is regularizing, or quasi-one-dimensional in the sense of [22]. The operator can then be inverted for two reasons. First, the explicit formula for the approximate solution means that, restricted to the infinite-dimensional kernel of the linearization at zero, the inverse exists and can be estimated. Second, the small-divisor problems that arise on the complement of this kernel can be overcome by considering only particular parameter values selected according to their Diophantine properties. A parameter-dependent version of the Nash-Moser implicit function theorem now yields the existence of a set of unimodal standing waves on flows of infinite depth, corresponding to a set of values of the parameter μ>1 which is dense at 1. Unimodal means that the term of smallest order in the amplitude is cos x cos t, which is one of many eigenfunctions of the completely resonant linearized problem.

[1]  Lord Rayleigh,et al.  Deep Water Waves, Progressive or Stationary, to the Third Order of Approximation , 1915 .

[2]  Massimiliano Berti,et al.  Periodic Solutions of Nonlinear Wave Equations with General Nonlinearities , 2002, math/0211310.

[3]  E. A. Kuznetsov,et al.  Analytical description of the free surface dynamics of an ideal fluid (canonical formalism and conformal mapping) , 1996 .

[4]  C. B. Morrey.  Multiple Integrals in the Calculus of Variations , 1966 .

[5]  Min Chen,et al.  Standing waves for a two-way model system for water waves , 2005 .

[6]  M. Boussinesq.  Essai sur la théorie des eaux courantes , 1873 .

[7]  C. K. Thornhill,et al.  Part II. finite periodic stationary gravity waves in a perfect liquid , 1952, Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences.

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

[9]  A. K. Whitney,et al.  A semi-analytic solution for nonlinear standing waves in deep water , 1981, Journal of Fluid Mechanics.

[10]  Tosio Kato.  Perturbation theory for linear operators , 1966 .

[11]  Joseph B. Keller,et al.  Standing surface waves of finite amplitude , 1960, Journal of Fluid Mechanics.

[12]  Pavel I. Plotnikov,et al.  Nash-Moser Theory for Standing Water Waves , 2001 .

[13]  Zhaosheng Feng,et al.  Traveling solitary wave solutions to the generalized Boussinesq equation , 2003 .

[14]  198 ON THE THEORY OF OSCILLATORY WAVES , 2022 .

[15]  M. Berti,et al.  Multiplicity of periodic solutions of nonlinear wave equations , 2004 .

[16]  On the Standing Wave Problem in Deep Water , 2002 .

[17]  William Thomson,et al.  Mathematical and physical papers , 1880 .

[18]  T. Brooke Benjamin,et al.  Hamiltonian structure, symmetries and conservation laws for water waves , 1982, Journal of Fluid Mechanics.

[19]  Makoto Okamura.  Standing gravity waves of large amplitude in deep water , 2003 .

[20]  Dario Bambusi,et al.  Families of Periodic Solutions of Resonant PDEs , 2001, J. Nonlinear Sci..

[21]  J. Moser.  A rapidly convergent iteration method and non-linear partial differential equations - I , 1966 .

[22]  Haim Brezis,et al.  Periodic solutions of nonlinear vibrating strings and duality principles , 1983 .

[23]  R. Kannan,et al.  Periodic solutions of nonlinear wave equations , 1983 .

[24]  J. Toland,et al.  The semi-analytic theory of standing waves , 1987, Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences.

[25]  C. Eugene Wayne,et al.  Periodic and quasi-periodic solutions of nonlinear wave equations via KAM theory , 1990 .

[26]  L. Hörmander,et al.  The boundary problems of physical geodesy , 1976 .

[27]  Paul H. Rabinowitz,et al.  Periodic solutions of hamiltonian systems , 1978 .

[28]  Walter Craig,et al.  Newton's method and periodic solutions of nonlinear wave equations , 1993 .

[29]  V. Arnold.  SMALL DENOMINATORS AND PROBLEMS OF STABILITY OF MOTION IN CLASSICAL AND CELESTIAL MECHANICS , 1963 .

[30]  A. Craik.  THE ORIGINS OF WATER WAVE THEORY , 2004 .

[31]  G. Iooss,et al.  Multimodal Standing Gravity Waves: a Completely Resonant System , 2005 .

[32]  Standing waves on infinite depth , 2004 .

[33]  Geoffrey Ingram Taylor,et al.  An experimental study of standing waves , 1953, Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences.