Boussinesq Equations and Other Systems for Small-Amplitude Long Waves in Nonlinear Dispersive Media. I: Derivation and Linear Theory

Summary. Considered herein are a number of variants of the classical Boussinesq system and their higher-order generalizations. Such equations were first derived by Boussinesq to describe the two-way propagation of small-amplitude, long wavelength, gravity waves on the surface of water in a canal. These systems arise also when modeling the propagation of long-crested waves on large lakes or the ocean and in other contexts. Depending on the modeling of dispersion, the resulting system may or may not have a linearization about the rest state which is well posed. Even when well posed, the linearized system may exhibit a lack of conservation of energy that is at odds with its status as an approximation to the Euler equations. In the present script, we derive a four-parameter family of Boussinesq systems from the two-dimensional Euler equations for free-surface flow and formulate criteria to help decide which of these equations one might choose in a given modeling situation. The analysis of the systems according to these criteria is initiated.

[1]  Jerry L. Bona,et al.  A Boussinesq system for two-way propagation of nonlinear dispersive waves , 1998 .

[2]  Jerry L. Bona,et al.  Comparison of model equations for small-amplitude long waves , 1999 .

[3]  P. Olver,et al.  HAMLTONIAN PERTURBATION THEORY AND WATER WAVES , 1984 .

[4]  Jerry L. Bona,et al.  Comparisons between model equations for long waves , 1991 .

[5]  M. A. Lavrent'ev,et al.  On the theory of long waves ; A contribution to the theory of long waves , 1954 .

[6]  L. Hörmander,et al.  The Analysis of Linear Partial Differential Operators I: Distribution Theory and Fourier Analysis , 1983 .

[7]  G. H. Keulegan,et al.  Mathematical theory of irrotational translation waves , 1940 .

[8]  F. Ursell,et al.  The long-wave paradox in the theory of gravity waves , 1953, Mathematical Proceedings of the Cambridge Philosophical Society.

[9]  Jerry L. Bona,et al.  A model for the two-way propagation of water waves in a channel , 1976, Mathematical Proceedings of the Cambridge Philosophical Society.

[10]  L. R. Scott,et al.  An evaluation of a model equation for water waves , 1981, Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences.

[11]  N. J. Zabusky,et al.  Shallow-water waves, the Korteweg-deVries equation and solitons , 1971, Journal of Fluid Mechanics.

[12]  Juan M. Restrepo,et al.  Stable and Unstable Solitary-Wave Solutions of the Generalized Regularized Long-Wave Equation , 2000, J. Nonlinear Sci..

[13]  T. Maxworthy,et al.  Experiments on collisions between solitary waves , 1976, Journal of Fluid Mechanics.

[14]  R. Sachs,et al.  On the integrable variant of the Boussinesq system: Painleve´ property, rational solutions, a related many-body system, and equivalence with the AKNS hierarchy , 1988 .

[15]  M. Lavrent'ev.  On the theory of long waves , 1975 .

[16]  Guido Schneider,et al.  The long‐wave limit for the water wave problem I. The case of zero surface tension , 2000 .

[17]  Min Chen,et al.  Exact solutions of various Boussinesq systems , 1998 .

[18]  L. Hörmander.  Distribution theory and Fourier analysis , 1990 .

[19]  T. Nishida,et al.  A mathematical justification for Korteweg-de Vries equation and Boussinesq equation of water surface waves , 1986 .

[20]  E. V. Krishnan,et al.  AN EXACT SOLUTION OF THE CLASSICAL BOUSSINESQ EQUATION , 1982 .

[21]  Min Chen,et al.  Equations for bi-directional waves over an uneven bottom , 2003, Math. Comput. Simul..

[22]  L. Schwartz.  Théorie des distributions , 1966 .

[23]  J. F. Toland,et al.  Solitary wave solutions for a model of the two-way propagation of water waves in a channel , 1981, Mathematical Proceedings of the Cambridge Philosophical Society.

[24]  Andrew Pickering,et al.  Factorization of the 'classical Boussinesq' system , 1994 .

[25]  J. F. Toland,et al.  Uniqueness and a priori bounds for certain homoclinic orbits of a Boussinesq system modelling solitary water waves , 1984 .

[26]  Thierry Colin,et al.  Rigorous derivation of korteweg-de vries-type systems from a general class of nonlinear hyperbolic systems , 2000 .

[27]  H. Segur,et al.  The Korteweg-de Vries equation and water waves. Part 2. Comparison with experiments , 1974, Journal of Fluid Mechanics.

[28]  Ragnar Winther,et al.  A Finite Element Method for a Version of the Boussinesq Equation , 1982 .

[29]  W. G. Pritchard,et al.  Experimental observations of edge waves , 1995, Journal of Fluid Mechanics.

[30]  J. Bona,et al.  Global existence of smooth solutions and stability of solitary waves for a generalized Boussinesq equation , 1988 .

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

[32]  dynamics. Boulder,et al.  Fluids and plasmas : geometry and dynamics , 1984 .

[33]  E. V. Krishnan,et al.  On shallow water waves , 1990 .

[34]  D. J. Kaup,et al.  A Higher-Order Water-Wave Equation and the Method for Solving It , 1975 .

[35]  J. Desanto.  Mathematical and numerical aspects of wave propagation , 1998 .

[36]  J. Bona,et al.  Solitary-wave solutions of nonlinear problems , 1990, Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences.

[37]  J. F. Toland,et al.  On solitary water-waves of finite amplitude , 1981 .

[38]  P. Brenner,et al.  The Cauchy problem for systems inLp andLp,α , 1973 .

[39]  W. G. Pritchard,et al.  Wave reflexion from beaches , 1980, Journal of Fluid Mechanics.

[40]  F. Smith,et al.  Conservative, high-order numerical schemes for the generalized Korteweg—de Vries equation , 1995, Philosophical Transactions of the Royal Society of London. Series A: Physical and Engineering Sciences.

[41]  J. Boussinesq,et al.  Théorie des ondes et des remous qui se propagent le long d'un canal rectangulaire horizontal, en communiquant au liquide contenu dans ce canal des vitesses sensiblement pareilles de la surface au fond. , 1872 .

[42]  J. Hammack.  A note on tsunamis: their generation and propagation in an ocean of uniform depth , 1973, Journal of Fluid Mechanics.

[43]  D. Peregrine.  Equations for Water Waves and the Approximation behind Them , 1972 .

[44]  L. R. Scott,et al.  Numerical schemes for a model for nonlinear dispersive waves , 1985 .

[45]  Jerry L. Bona,et al.  Dispersive Blowup of Solutions of Generalized Korteweg-de Vries Equations , 1993 .

[46]  L. R. Scott,et al.  A Comparison of Solutions of Two Model Equations for Long Waves. , 1983 .

[47]  J. Cooper.  SINGULAR INTEGRALS AND DIFFERENTIABILITY PROPERTIES OF FUNCTIONS , 1973 .

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

[49]  J. Bona,et al.  WAVE GROUPS IN A BARRED NEARSHORE , 1997 .

[50]  Jerry L. Bona,et al.  Interaction of internal waves with the seabed on continental shelves , 1991 .

[51]  Walter Craig,et al.  An existence theory for water waves and the boussinesq and korteweg-devries scaling limits , 1985 .

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

[53]  L. Hörmander.  The analysis of linear partial differential operators , 1990 .

[54]  L. Milne‐Thomson.  A Treatise on the Theory of Bessel Functions , 1945, Nature.

[55]  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.

[56]  Jerry L. Bona,et al.  Singularités dispersives de solutions d'équations de type Korteweg-de-Vries , 1986 .

[57]  Min Chen.  Solitary-wave and multi-pulsed traveling-wave solutions of boussinesq systems , 2000 .