Localized solutions of one-dimensional non-linear shallow-water equations with velocity

In dimensionless variables, the one-dimensional non-linear system of shallow-water equations over a non-flat bottom D(x) = c(x) with elevation component η(x, t) and velocity v(x, t) is given by ηt + ∂[v(η + D)]/∂x = 0, vt + vvx + ηx = 0. We introduce a parameter 0 < μ ≪ 1 and say that a function f(y) is localized in the μ-neighbourhood of a point a > 0 if f(a) = 1 + O(μ) and f(y) = o(μ) for |y − a| > μ1−δ, δ > 0. In the case when c(x) = x, we consider the Cauchy problem η|t=0 = η(x, μ), v|t=0 = v(x, μ) for our system, assuming that the initial data η, v are localized in a neighbourhood of the point x = a. Its solution is used to describe long waves running onto a shore [1], [2]. The following remarkable property (discovered in another form in [1], see also [2]) of the system under consideration can be established by direct differentiation.