Lanterna Rally

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Published: 1997

Total Pages: 6

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This is the final report of a three-year, Laboratory-Directed Research and Development (LDRD) project at the Los Alamos National Laboratory (LANL). We derived and analyzed new shallow water equations for one-dimensional flows near the critical Froude number as well as related integrable systems of evolutionary nonlinear partial differential equations in one spatial dimension, while developing new directions for the mathematics underlying the integrability of these systems. In particular, we applied the spectrum generating equation method to create and study new integrable systems of nonlinear partial differential equations related to our integrable shallow water equations. We also investigated the solutions of these systems of equations on a periodic spatial domain by using methods from the complex algebraic geometry of Riemann surfaces. We developed certain aspects of the required mathematical tools in the course of this investigation, such as inverse scattering with degenerate potentials, asymptotic reduction of the angle representations, geometric singular perturbation theory, modulation theory and singularity tracking for completely integrable equations. We also studied equations that admit weak solutions, i.e., solutions with discontinuous derivatives in the form of comers or cusps, even though they are solutions of integrable models, a property that is often incorrectly assumed to imply smooth solution behavior. In related work, we derived new shallow water equations in two dimensions for an incompressible fluid with a free surface that is moving under the force of gravity. These equations provide an estimate of the long-time asymptotic effects of slowly varying bottom topography and weak hydrostatic imbalance on the vertically averaged horizontal velocity, and they describe the flow regime in which the Froude number is small -- much smaller even than the small aspect ratio of the shallow domain.