John Hunter

Professor John Hunter studies singular perturbation methods and asymptotics for hyperbolic partial differential equations (PDEs). Non-linear PDEs are typically difficult to analyze and cannot be solved exactly. By the use of perturbation methods, one can sometimes obtain approximate solutions of the PDE in limiting regimes which retain the non-linear effects one wants to study. For instance, one can use the corresponding linearized PDE as a starting point, or one can perturb certain exact solutions of the non-linear PDE, such as traveling waves. Professor Hunter has studied various phenemona, such as shock waves, solitons, and integrability with this approach.

In particular, Professor Hunter has developed the method of non-linear geometrical optics [1]. Geometrical optics was originally developed to solve linear problems in optics and quantum mechanics. In the geometrical optics approximation, also called the WKB approximation, a wave or wave packet propagates along a set of rays. This fact greatly simplifies the analysis of wave propagation problems. Using the non-linear version of geometrical optics, Professor Hunter found a new kind of nonlinear resonance in coupled hyperbolic systems of PDEs, such as the coupling between sound waves and thermal fluctuations in air [3]. This resonance causes different types of waves to reflect off of each other, leading to complicated dynamics. In another application, he derived a fifth-order modification of the Korteweg-de Vries equation modelling the effect of surface tension on solitons in the theory of shallow water waves. He constructed solutions in which the solitons generate a small amplitude capillary wave, thereby decaying by radiating away their energy [2].

More recently, Professor Hunter has investigated a paradox, discovered by Von Neumann in the 1940's, involving the Mach reflection of weak shock waves in a compressible fluid [4]. When a weak shock wave reflects from a sufficiently acute wedge, it appears to produce three shock waves that meet at a triple point. According to the mathematical model, such a point is impossible - it cannot conserve mass, momentum, and energy. The apparent triple point must have an unknown local structure of very small but non-zero size. Computational solutions and theoretical analysis of a simplified asymptotic equation have led to a conjecture for this local structure. A successful resolution of the Von Neumann paradox would represent one of very few tangible results about shock waves in more than one spatial dimension.

Professor Hunter has advised and collaborated with several successful graduate students. With a former student, Ram Vedantham, he studied the propagation of non-linear sound waves through incompressible fluid flows. Another student, Binh Trong, has used non-linear geometrical optics to investigate the propagation of gravity waves in Einstein's theory of General Relativity and has shown that the resonant interaction of sound waves can generate a gravitational wave.

Selected publications

1. Weakly nonlinear high frequency waves (with J. B. Keller), Comm. Pure Appl. Math. 36 (1983), 547-569.

2. Existence of perturbed solitary wave solutions to a model equation for water waves (with J. Schuerle), Phys. D 32 (1988), 253-268.

3. Resonantly interacting weakly nonlinear hyperbolic waves II: several space variables (with A. Majda and R. Rosales), Stud. Appl. Math. 75 (1986), 187-226.

4. Mach reflection for the two dimensional Burgers equation, (with M. Brio), Phys. D 60 (1992), 194-207.

5. On a nonlinear hyperbolic variational equation: I. Global existence of weak solutions (with Y. Zheng), Arch. Rat. Mech. Anal. 129 (1995), 305-353.

Last updated: 0000-00-00