Mathematical Sciences: Research in Nonlinear Wave Motion

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8822444 Segur This project includes three separate lines of research in nonlinear wave motion in evolutionary systems. The three ideas are largely independent. Part A concerns the development of an effective method to determine whether or not a particular solution of an evolution equation is stable to small changes in initial data. Stability theory advanced significantly after the discovery by Arnol'd (1965) and others of a systematic method to construct Lyapunov functionals. Even so, the methods currently available are unable to determine whether a flow as simple as stably statified, paralled shear flow is a stable solution of Euler's equations in two dimensions. One goal is to devise a generalization of current methods that might settle the question. The focus in part B is on the Kadomstev-Peviashvili (KP) equation with periodic boundary conditions. The equation is known to be completely integrable, so there is reason to believe that the initial value problem for the KP equation with periodic boundary conditions is solvable. The KP equation also provides accurate models of a class of water waves, and it is intimately connected with the theory of Riemann surfaces in algebraic geometry. In fact, the problem is so rich that progress in almost any direction would valuable. The objective in part C is to construct a new kinetic theory for a large collection of wavepackets that interact primarily through resonant triads. If successful, the theory would result in a Boltzmann-type equation, whose equilibrium solutions would provide an equilibrium spectrum of wave energies, analogous to the Maxwell-Boltzmann distribution for molecules. An earlier version of such a theory was given by Segur (1984); the work in this part of the project would correct a deficiency in that model.