On a class of solvable difference equations generalizing an iteration process for calculating reciprocals

The well-known first-order nonlinear difference equation $$ y_{n+1}=2y_{n}-xy_{n}^{2}, \quad n\in {\mathbb {N}}_{0}, $$ y n + 1 = 2 y n − x y n 2 , n ∈ N 0 , naturally appeared in the problem of computing the reciprocal value of a given nonzero real number x. One of the interesting features of the difference equation is that it is solvable in closed form. We show that there is a class of theoretically solvable higher-order nonlinear difference equations that include the equation. We also show that some of these equations are also practically solvable.

Note on constructing a family of solvable sine-type difference equations

We obtain a family of first order sine-type difference equations solvable in closed form in a constructive way, and we present a general solution to each of the equations.

On reduced models for gravity waves generated by moving bodies

In 1983, Tulin published a report proposing a framework for reducing the equations for gravity waves generated by moving bodies into a single nonlinear differential equation solvable in closed form (Proceedings of the 14th Symposium on Naval Hydrodynamics, 1983, pp. 19–51). Several new and puzzling issues were highlighted by Tulin, notably the existence of weak and strong wave-making regimes, and the paradoxical fact that the theory seemed to be applicable to flows at low speeds, ‘but not too low speeds’. These important issues were left unanswered, and despite the novelty of the ideas, Tulin’s report fell into relative obscurity. Now, 30 years later, we will revive Tulin’s observations, and explain how an asymptotically consistent framework allows us to address these concerns. Most notably, we demonstrate, using the asymptotic method of steepest descents, how the production of free-surface waves can be related to the arrangement of integration contours connected to the shape of the moving body. This approach provides a new and powerful methodology for the study of geometrically nonlinear wave–body interactions.