Integrable Nonlinear Evolution Equations in the Description of Waves in the Shallow-Water Long-Wave Approximation

  • Decio Levi
Conference paper
Part of the The IMA Volumes in Mathematics and Its Applications book series (IMA, volume 25)


In this paper we discuss the nonlinear evolution equations which describe the evolution of the surface of an infinite shallow basin filled with a fluid subject just to the force of gravity. In the approximation of long waves at least three possible situations can arise:
  1. i)

    almost one dimensional waves

  2. ii)

    cylindrical waves

  3. iii)

    waves with equal characteristic length in any direction of the plane.


In case (i) we get that the surface amplitude evolves according to the Kadomtsev-Petviashvili equation, in case (ii) to a cylindrical Korteweg-de Vries equation and in case (iii) to a Korteweg-de Vries equation which depends parametrically on an extra variable. In all cases we discuss two different Cauchy problems, both of physical interest: in the first, and most common, the initial datum is prescribed at a fixed instant of time in all space. A second type of Cauchy problem is obtained when the initial datum is prescribed at one position for all time. Finally, for the case (iii) a large set of one and two soliton solutions are presented.


Cauchy Problem Solitary Wave Euler Equation Soliton Solution Nonlinear Evolution Equation 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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Copyright information

© Springer-Verlag New York Inc. 1990

Authors and Affiliations

  • Decio Levi
    • 1
    • 2
  1. 1.Dipartimento di FisicaUniversita’ di Roma “La Sapienza”RomaItaly
  2. 2.Istituto Nazionale di Fisica NucleareSezione di RomaItaly

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