Computer Algebra Applied to a Solitary Waves Study

We apply Computer algebra techniques, such as algebraic computations of resultants and discriminants, certified drawing (with a guaranteed topology) of plane curves, to a problem in Fluid dynamics: We investigate ``capillary-gravity'' solitary waves in shallow water, relying on the framework of the Serre-Green-Naghdi equations. So, we deal with 2 dimensional surface waves, propagating in a shallow water of constant depth. By a differential elimination process, the study reduces to describing the solutions of an ordinary non linear first order differential equation, depending on two parameters. The paper is illustrated with examples and pictures computed with the computer algebra system Maple.

[1]  Evelyne Hubert,et al.  Essential Components of an Algebraic Differential Equation , 1999, J. Symb. Comput..

[2]  P. Hartman.  Ordinary Differential Equations , 1965 .

[3]  Hisashi Okamoto,et al.  The mathematical theory of permanent progressive water-waves , 2001 .

[4]  Malcolm A. Grant.  Standing Stokes waves of maximum height , 1973 .

[5]  N. G. Parke,et al.  Ordinary Differential Equations. , 1958 .

[6]  Shijun Liao.  Do peaked solitary water waves indeed exist? , 2014, Commun. Nonlinear Sci. Numer. Simul..

[7]  Allen Parker,et al.  On the Camassa–Holm equation and a direct method of solution. III. N-soliton solutions , 2005, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[8]  Alfred R. Osborne,et al.  Chapter 3 - The Infinite-Line Inverse Scattering Transform , 2010 .

[9]  George Gabriel Stokes,et al.  Mathematical and Physical Papers vol.1: Supplement to a paper on the Theory of Oscillatory Waves , 2009 .

[10]  F. Serre,et al.  CONTRIBUTION À L'ÉTUDE DES ÉCOULEMENTS PERMANENTS ET VARIABLES DANS LES CANAUX , 1953 .

[11]  Evelyne Hubet,et al.  Detecting degenerate behaviours in first order algebraic differential equations , 1997 .

[12]  P. Milewski,et al.  Finite volume and pseudo-spectral schemes for the fully nonlinear 1D Serre equations , 2011, European Journal of Applied Mathematics.

[13]  Laureano González-Vega,et al.  Efficient topology determination of implicitly defined algebraic plane curves , 2002, Comput. Aided Geom. Des..

[14]  Vladimir Igorevich Arnold,et al.  Geometrical Methods in the Theory of Ordinary Differential Equations , 1983 .

[15]  Denys Dutykh,et al.  Efficient computation of steady solitary gravity waves , 2013, 1302.1812.

[16]  Jian Ming Yu,et al.  How to define singular solutions , 1993 .

[17]  M. Hamburger.  Ueber die singulären Lösungen der algebraischen Differentialgleichungen erster Ordnung. , 1893 .

[18]  Richard M. Cohn.  THE GENERAL SOLUTION OF A FIRST ORDER DIFFERENTIAL POLYNOMIAL , 1976 .

[19]  Fabrice Rouillier,et al.  On the Topology of Real Algebraic Plane Curves , 2010, Math. Comput. Sci..

[20]  P. M. Naghdi,et al.  A derivation of equations for wave propagation in water of variable depth , 1976, Journal of Fluid Mechanics.

[21]  Alfred R. Osborne,et al.  Nonlinear Ocean Waves and the Inverse Scattering Transform , 2010 .

[22]  Evelyne Hubert,et al.  The general solution of an ordinary differential equation , 1996, ISSAC '96.

[23]  Darryl D. Holm,et al.  An integrable shallow water equation with peaked solitons. , 1993, Physical review letters.