Kuzmak’s method - tour de force or tour de farce?

Abstract: In 1959 Kuzmak introduced a new method for constructing asymptotic solutions to ordinary differential equations that describe nonlinear oscillators. The approach is effectively a nonlinear variant of the WKB method. Kuzmak's method was later refined by Luke (1966) and can be viewed as a precursor to Whitham Modulation Theory for nonlinear waves. The method appears to be not so well known. In this talk we will show how it can be applied using the example of the simple, damped pendulum. As is well known, with no damping the problem can be solved exactly using elliptic functions, and the oscillation period depends on the amplitude (i.e. the oscillations are non-isochronous). With damping active, the basic asymptotic approach is one of multiple scales, with the fast time scale describing the oscillations and the slow time scale describing the gradual diminution in amplitude as energy slowly leaks away. In a key step, the fast time scale is chosen to effectively fix the oscillation period, and this allows a bounded solution to be constructed. We show how the key elements of the method work, providing sufficient details to fully construct the leading order solution.

Mathematicsfluid dynamics

Audience: researchers in the topic

Fluids and Structures Seminar @ UEA