Mathematical Sciences: Nonlinear Wave Equations

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The underlying theme of this mathematical research is the analysis of solutions of nonlinear partial differential equations, particularly in conservative evolution equations that arise in mathematical physics and in fluid dynamics. Emphasis will be on the study of the existence question for classes of equations, in particular, the existence of periodic solutions, and the study of a priori regularity and other properties of the evolution. The work is limited to three objectives. First, in the area of infinite dimensional dynamical systems and KAM theory, continuing research will be done in extending techniques of Hamiltonian perturbation applied to the wave equation to other equations and to investigate methods related to studies in higher space dimensions. The second direction involves smoothing properties of dispersive equations. It has been known for some time that solutions of dispersive evolution equations can be smoother than their initial conditions. But their quadratic norms can grow unbounded without cut-off restrictions in spatial directions. Work will be done to isolate the parts of the equations which contribute to the growth and try to develop estimates on the norm size. The final part of the project studies waves in free surfaces and interfaces. The problem, known as Stokes conjecture, concerns the solitary wave of extremal form. Such waves are known to be analytic except at their crest. It is the object of this study to show (i) that a Lipschitz singularity of interior angle 120 degrees must occur at the crest and (ii) that the extremal wave profile is monotone decreasing and convex on either side. Partial differential equations form the backbone of mathematical modeling in the physical sciences. Phenomena which involve continuous change such as that seen in motion, materials and energy are known to obey certain general laws expressible in terms of the interactions and relationships between partial derivatives. The key role of mathematics is not to state the relationships, but rather, to extract qualitative and quantitative meaning from them.