Localized solutions of one-dimensional non-linear shallow-water equations with velocity $ c=\sqrt x$
Russian Mathematical Surveys
In dimensionless variables, the one-dimensional non-linear system of shallow-water equations over a non-flat bottom D(x) = c 2 (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| > µ 1−δ , δ > 0. In the case when c 2 (x) = x, we consider the Cauchy problem η|t=0 = η 0
... oblem η|t=0 = η 0 (x, µ), v|t=0 = v 0 (x, µ) for our system, assuming that the initial data η 0 , v 0 are localized in a neighbourhood of the point x = a. Its solution is used to describe long waves running onto a shore , . The following remarkable property (discovered in another form in , see also ) of the system under consideration can be established by direct differentiation. a solution of the original non-linear system. To study solutions of the linear system of equations on the semi-axis y 0 with singular coefficient c 2 = y we require these solutions to be bounded at y = 0. (This guarantees that they belong to the domain of the operator ∂ ∂y y ∂ ∂y .) Papers concerning the systems under consideration focus mainly on oscillating solutions. Localized solutions are studied in papers by Mazova, Pelinovsky, and their co-authors (see  and the bibliography there). The purpose of this note is to construct simple exact solutions of linear (and hence also non-linear) shallow-water equations and to interpret some results of . Assertion 2. Let A and b be arbitrary complex numbers with Re b ̸ = 0 and let P(k) be a polynomial. Then the functions and (N, U ) = P ∂ ∂τ N 0 , P ∂ ∂τ U 0 are continuous for y −(Re b) 2 /4 and are exact solutions of the linear shallow-water equations for y 0. If the Jacobi matrix for the transition from (τ, y) to (t, x) is non-singular, then by Assertion 1 these functions determine a parameter-dependent family of exact solutions of the initial non-linear system. Although Assertion 2 is proved by direct differentiation, we give another argument to clarify the derivation and properties of these solutions. If we take b = µβ/ √ a + 2i √ a and A = µ 3/2 (1 + i)/(2 √ a ), then our functions for τ = 0 are localized in an O(µ)-neighbourhood of the point y = a. Let us consider a more general Cauchy problem N |τ=0 = V ((y − a)/µ), U |τ=0 = 0 for the linear system. Here V (z) is a function of z with compact support. Since there is a parameter µ, one can construct an asymptotic formula This paper was written with the support of the Russian Foundation for Basic Research (grant no. 09-01-12063-ofi-m) and the Gruppo Nazionale per la Fisica Matematica (Italy).