# John Hunter

**Professor****Applied mathematics**

Ph.D., 1981, Stanford University**Refereed publications:** Via Math Reviews

**Web Page: ** http://www.math.ucdavis.edu/~hunter/

Email: hunter@math.ucdavis.edu

Office: MSB 3230

Phone: 601-4444 x4016

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.

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