The reductive perturbation method and the Korteweg-de Vries hierarchy
The authors consider a two-dimensional inviscid incompressible fluid in a constant gravitational field. The space coordinates are denoted by \((x,z)\), and the corresponding components of the velocity \(\vec v\) by \((u,w)\). The gravitational acceleration \(g\) is in the negative \(z\) direction. If the fluid is irrotational, the equations describing it are \(\nabla^2 \varphi=0\) and \((p- p_0) /\rho=- {{\partial \varphi} \over {\partial t}}- {1\over 2} (\nabla \varphi)^2- gz\), where \(\rho\) is the fluid density, \(p\) is the pressure, \(\varphi\) is the velocity potential, and \(p_0\) is an integration constant. By using a reductive perturbation method with the introduction of an infinite sequence of slow time variables \(\tau_1, \tau_2, \tau_3, \dots\), the propagation of long surface waves in a shallow inviscid fluid is studied. The Korteweg-de Vries equation appears as the lowest order amplitude equation in slow variables. The paper is well written and organized.
scientific article; zbMATH DE number 790809