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We prove the existence of periodic solutions of the nonlinear wave equationsatisfying either Dirichlet or periodic boundary conditions on the interval 10, rr]. The coefficients of the eigenfunction expansion of this equation satisfy a nonlinear functional equation. Using a version of Newton's method, we show that this equation has solutions provided the nonlinearity g(x, u ) satisfies certain generic conditions of nonresonance and genuine nonlinearity.

Invariant manifolds and the long-time asymptotics of the Navier-Stokes and vorticity equations on R Abstract We construct finite-dimensional invariant manifolds in the phase space of the Navier-Stokes equation on R 2 and show that these manifolds control the long-time behavior of the solutions. This gives geometric insight into the existing results on the asymptotics of such solutions and also allows one to extend those results in a number of ways.

The Korteweg‐de Vries equation, Boussinesq equation, and many other equations can be formally derived as approximate equations for the two‐dimensional water wave problem in the limit of long waves. Here we consider the classical problem concerning the validity of these equations for the water wave problem in an infinitely long canal without surface tension. We prove that the solutions of the water wave problem in the long‐wave limit split up into two wave packets, one moving to the right and one to the left, where each of these wave packets evolves independently as a solution of a Korteweg‐de Vries equation. Our result allows us to describe the nonlinear interaction of solitary waves. © 2000 John Wiley & Sons, Inc.

In this paper the nonlinear wave equationis studied. It is shown that for a large class of potentials, v(x\ one can use KAM methods to construct periodic and quasi-periodic solutions (in time) for this equation.