to be done in this area. Face recognition is a problem that scarcely clamoured for attention before the computer age but, having surfaced, has involved a wide range of techniques and has attracted the attention of some fine minds (David Mumford was a Fields Medallist in 1974). This singular application of mathematical modelling to a messy applied problem of obvious utility and importance but with no unique solution is a pretty one to share with students: perhaps, returning to the source of our opening quotation, we may invert Duncan's earlier observation, 'There is an art to find the mind's construction in the face!'.

Linear and nonlinear waves, by G. B. Whitham. Pp.636. £50. 1999. ISBN 0 471 35942 4 (Wiley).

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Methods Of Modern Mathematical Physics

The theory of pseudo differential operators, discussed in § 1, is well suited for investigating various problems connected with elliptic differential equations. However, this theory fails to be adequate for studying equations of hyperbolic type, and one is then forced to examine a wider class of operators, the so-called Fourier integral operators (Egorov [1975], Hormander [1968, 1971, 1983, 1985], Kumano-go [1982], Shubin [1978], Taylor [1981], Treves [1980]).

Fourier Integral Operators

A Modern Introduction to the Mathematical Theory of Water Waves. By R. S. Johnson. Cambridge University Press, 1997. xiv+445 pp. Hardback ISBN 0 521 59172 4 £55.00; paperback 0 521 59832 X £19.95.

We consider the problem of global in time existence and uniqueness of solutions of the 2-D infinite depth full water wave equation. It is known that this equation has a solution for a time period [0,T/e] for initial data of the form e
 Ψ, where T depends only on Ψ. In this paper, we show that for such data there exists a unique solution for a time period [0,e
 T/e
 ]. This is achieved by better understandings of the nature of the nonlinearity of the full water wave equation.

Almost global wellposedness of the 2-D full water wave problem

We show that periodic traveling waves with sufficiently small amplitudes of the Whitham equation, which incorporates the dispersion relation of surface water waves and the nonlinearity of the shallow water equations, are spectrally unstable to long-wavelengths perturbations if the wave number is greater than a critical value, bearing out the Benjamin–Feir instability of Stokes waves; they are spectrally stable to square integrable perturbations otherwise. The proof involves a spectral perturbation of the associated linearized operator with respect to the Floquet exponent and the small-amplitude parameter. We extend the result to related, nonlinear dispersive equations.

Modulational Instability in the Whitham Equation for Water Waves

The solitary water wave problem is to find steady free surface waves which approach a constant level of depth in the far field. The main result is the existence of a family of exact solitary waves of small amplitude for an arbitrary vorticity. Each solution has a supercritical parameter value and decays exponentially at infinity. The proof is based on a generalized implicit function theorem of the Nash–Moser type. The first approximation to the surface profile is given by the “KdV” equation. With a supercritical value of the surface tension coefficient, a family of small amplitude solitary waves of depression with subcritical parameter values is constructed for an arbitrary vorticity.

Exact Solitary Water Waves with Vorticity

Wave breaking in the Whitham equation

Strichartz-type estimates for one-dimensional surface water-waves under surface tension are studied, based on the formulation of the problem as a nonlinear dispersive equation. We establish a family of dispersion estimates on time scales depending on the size of the frequencies. We infer that a solution u of the dispersive equation we introduce satisfies local-in-time Strichartz estimates with loss in derivative: where C depends on T and on the norms of the H s -norm of the initial data. The proof uses the frequency analysis and semiclassical Strichartz estimates for the linealized water-wave operator.

Strichartz Estimates for the Water-Wave Problem with Surface Tension

The classical deep-water wave problem is to find a periodic traveling wave with a free surface of infinite depth. The main result is the construction of a global connected set of rotational solutions for a general class of vorticities. Each nontrivial solution on the continuum has a wave profile symmetric around the crests and monotone between crest and trough.The problem is formulated as a nonlinear elliptic boundary value problem in an unbounded domain with a parameter. The analysis is based on generalized degree theory and the global theory of bifurcation. The unboundedness of the domain renders consideration of approximate problems with stronger compactness properties.

Vera Mikyoung Hur

Papers

Modulational Instability in the Whitham Equation for Water Waves

Exact Solitary Water Waves with Vorticity

Wave breaking in the Whitham equation

Strichartz Estimates for the Water-Wave Problem with Surface Tension

Global Bifurcation Theory of Deep-Water Waves with Vorticity