Gopala Krishna Srinivasan

An integrable system of ODE reductions of the stratified Boussinesq equations

We provide here a complete analysis of the system of six ODEs arising as reductions of the stratified Boussinesq equations. We demonstrate the complete integrability of this system and reduce the problem to quadratures. We determine the stable and unstable manifolds passing through the rest points and deduce certain qualitative features of the solutions of these ODE reductions.

A Note on the Jump Conditions for Systems of Conservation Laws

In [1], the authors have demonstrated the effect on the Rankine-Hugoniot conditions for a system of conservation laws driven by a singular forcing function and have applied their results to a problem in water waves. We analyze here a similar problem in several space dimensions, in which the singularity in the forcing term involves a simple layer potential supported along the singularity locus. A classical theorem in electrostatics appears as a special case.

On weakly nonlinear waves in media exhibiting mixed nonlinearity

We obtain the transport equations governing small amplitude high frequency disturbances, that include both quadratic and cubic nonlinearities inherent in hyperbolic systems of conservation laws. The coefficients of the nonlinear terms in the transport equation are obtained in terms of the Glimm interaction coefficients. For symmetric and isotropic systems the mean curvature of the wave front, which appears as the coefficient of the linear term in the transport equation, is shown to be related to the derivative of the ray tube area along the bicharacteristics; the amplitude of the disturbance is shown to become unbounded in the neighborhood of the point where the ray tube collapses. We also obtain a formula, akin to the one obtained by R. Rosales (1991), for the energy dissipated across shocks.