Transient Free-Surface Hydrodynamics Using Rational Approximation of the Green’s Function

Rational approximations in the frequency domain are developed for the source function of linear free-surface hydrodynamics using the recently uncovered fourth-order ordinary differential equation (ODE) satisfied by the time-domain source function. The radiation problem for a floating body in deep water is formulated using a source plus wave-free potential expansion for the fluid. The inherent rational dependence on frequency of the wave-free potentials as well as the source approximation are used to develop a system of constant-coefficient ODE's for the radiation impedance which can be used to develop the motion of the body in a simple manner. The technique is applied to the heaving motion of a floating sphere with good results. The application to more general body geometries is explored by formulating the frequency-domain problem using the variational principle of Chen and Mei and exploiting its polynomial dependence on frequency.

TAU-FUNCTIONS FOR ALGEBRAIC GEOMETRICAL SOLUTIONS TO THE GENERAL ZAKHAROV–SHABAT HIERARCHY

The totality of all Zakharov–Shabat equations (equations of zero curvature with rational dependence on a parameter) form a hierarchy — GZS. Starting from an element of the Grassmannian we obtain expressions for the tau-functions of the algebraic-geometrical solutions of GZS in terms of theta-functions.

General solutions depending algebraically on arbitrary constants

In his famous lectures [7] Painlevé investigates general solutions of algebraic differential equations which depend algebraically on some of arbitrary constants. Although his discussions are beyond our understanding, the rigorous and accurate interpretation to make his intuition true would be possible. Successful accomplishments have been done by some authors, for example, Kimura [1], Umemura [8, 9]. From differential algebraic viewpoint in [5] the author introduces the notion of rational dependence on arbitrary constants of general solutions of algebraic differential equations, and in [6] clarifies the relation between it and the notion of strong normality. Here we aim at generalizing to higher order case the result in [4] that in the first order case solutions of equations depend algebraically on those of equations free from moving singularities which are determined uniquely as the closest ones to the given. Part of our result can be seen in [7].

Differential algebraic function fields depending rationally on arbitrary constants

The general solution of an algebraic differential equation depends on the initial conditions, though it is in general too difficult to make explicit the shape of the relationship. Painlevé studied in [8] algebraic differentia] equations of second order with the general solutions depending rationally on the initial conditions and the solvability of such equations. Giving the precise definition of the notion “rational dependence on the initial conditions”, Umemura [10] revived and generalized rigorously the discussion of Painlevé in the language of modern algebraic geometry. The theorem of Umemura is as follows; Let K be a differential field extension of complex number field C generated by a finite number of meromorphic functions on some domain in C. Let y be the general solution of a given algebraic differential equation over K. Suppose that y depends rationally on the initial conditions. Then it is contained in the terminal Km of a finite chain of differential field extensions: K = K0 ⊂ K1 ⊂… ⊂Km such that each Ki is strongly normal over Ki−1.