Integrable Nonlinear Evolution Equations in the Description of Waves in the Shallow-Water Long-Wave Approximation
almost one dimensional waves
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.
KeywordsCauchy Problem Solitary Wave Euler Equation Soliton Solution Nonlinear Evolution Equation
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