Integrable Shallow‐Water Equations and Undular Bores

On the basis of the integrable Kaup-Boussinesq version of the shallow-water equations, an analytical theory of undular bores is constructed. A complete classification for the problem of the decay of an initial discontinuity is made.

[1]  Gennady El,et al.  Decay of an initial discontinuity in the defocusing NLS hydrodynamics , 1995 .

[2]  M. Gregory Forest,et al.  Geometry and Modulation Theory for the Periodic Nonlinear Schrodinger Equation , 1986 .

[3]  B. Dubrovin,et al.  Hydrodynamics of weakly deformed soliton lattices. Differential geometry and Hamiltonian theory , 1989 .

[4]  D. Parsons,et al.  Wave-number conservation and succession of symmetries during a Whitham averaging , 1991 .

[5]  David W. McLaughlin,et al.  Multiphase averaging and the inverse spectral solution of the Korteweg—de Vries equation , 1980 .

[6]  D. Haar,et al.  Evolution of a Riemann wave in dispersive hydrodynamics , 1992 .

[7]  A. O. Smirnov.  Real finite-gap regular solutions of the Kaup-Boussinesq equation , 1986 .

[8]  B. Dubrovin,et al.  Hydrodynamics of soliton lattices , 1993 .

[9]  V. Maslov,et al.  Finite-zone, almost-periodic solutions in WKB approximations , 1981 .

[10]  On the characterization of a new soliton sector in the classical Boussinesq system , 1994 .

[11]  A. Gurevich,et al.  Nonstationary structure of a collisionless shock wave , 1973 .

[12]  Maxim V. Pavlov,et al.  Nonlinear Schrödinger equation and the bogolyubov-whitham method of averaging , 1987 .

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

[14]  D. Kinderlehrer,et al.  Oscillation theory, computation, and methods of compensated compactness , 1986 .

[15]  G. Whitham,et al.  Linear and Nonlinear Waves , 1976 .

[16]  Igor Krichever,et al.  Method of averaging for two-dimensional "integrable" equations , 1988 .

[17]  R. Stephenson.  A and V , 1962, The British journal of ophthalmology.

[18]  J. C. Luke.  A perturbation method for nonlinear dispersive wave problems , 1966, Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences.

[19]  L. Alonso,et al.  Soliton interaction with change of form in the classical Boussinesq system , 1992 .

[20]  G. El,et al.  Breaking of a Riemann wave in dispersive hydrodynamics , 1991 .

[21]  Roger H.J. Grimshaw,et al.  Slowly varying solitary waves. II. Nonlinear Schrödinger equation , 1979, Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences.

[22]  Kevin Barraclough,et al.  I and i , 2001, BMJ : British Medical Journal.

[23]  R. Grimshaw,et al.  Slowly varying solitary waves. I. Korteweg-de Vries equation , 1979, Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences.