Abstract
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:
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i)
almost one dimensional waves
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ii)
cylindrical waves
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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.
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Levi, D. (1990). Integrable Nonlinear Evolution Equations in the Description of Waves in the Shallow-Water Long-Wave Approximation. In: Olver, P.J., Sattinger, D.H. (eds) Solitons in Physics, Mathematics, and Nonlinear Optics. The IMA Volumes in Mathematics and Its Applications, vol 25. Springer, New York, NY. https://doi.org/10.1007/978-1-4613-9033-6_7
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DOI: https://doi.org/10.1007/978-1-4613-9033-6_7
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