Some Questions in Nonlinear Differential Equations

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DMS-9732529PI: Rabinowitz We plan to study some problems in (a) dynamical systems,(b) geometry,and(c) partial differential equations.For (a),building on some recent research,we are using variational methods to construct a variety of homoclinic and heteroclinic orbits for Hamiltonian systems on the 2-torus and also on the n-torus when an additional symmetry is present.Some analogous questions are being pursued for (b) when minimal geodesics heteroclinic to a pair of periodic geodesics on the n-torus are being sought.Another problem of interest for (a) is the existence of homoclinic solutions for Hamiltonian systems which possess singular potentials, especially when the dimension of the configuration space is greaterthan two.Lastly for (c),there has been some recent progress by Craig and Wayne and by Bourgain on the existence of small amplitude periodic solutions for nonlinear wave equations.We are interested in whether variational tools can be employed to obtain large amplitude solutions in such settings. Dynamical systems and partial differential equations are used to model continuous phenonena arising in science,engineering,economics and many other fields.Of considerable interest are solutions of such equations that exist for all values of the time variable.Aside from equilibria,i.e. time independent solutions,the simplest solutions of this type are periodic ones.Periodic solutions return to their original state after one time period.The next simplest class of global in time solutions are so-called heteroclinic solutions which in the simplest cases will approach a pair of different equilibria or time periodic solutions as time goes forward or backward to infinity.(When the limit states are the same,the solutions are called homoclinics).Our research involves developing methods to find heteroclinic and homoclinic solutions for dynamical systems and finding periodic solutions for some special classes of partial differential equations.