Journal of Dynamics and Differential Equations, VoL 6, No. 1, 1994 Chaos and Integrabifity in a Nonfinear Wave Equation C. Grotta Ragazzo 1 Received March 19, 1993 We consider the parametrized family of equations O,u-axxu-au + Ilull ~ u = 0, xG(O, nL), with Dirichlet boundary conditions. This equation has finite- dimensional invariant manifolds of solutions. Studying the reduced equation to a four-dimensional manifold, we prove the existence of transversal homoclinic orbits to periodic solutions and of invariant sets with "chaotic" dynamics, provided that ,, ffi 2, 3, 4, .... For ~,ffi I we prove the existence of infinitely many first integrals pairwise in involution. KEY WORDS: Conservative wave equations; Hamiltonian systems; trans- versal homoclinic orbits; integrability. AMS SUBJECT CLASSIFICATIONS: 34C25-28, 34C37, 35L70, 58F05, 58F07, 58F13-14, 58F39. 1. INTRODUCTION Our goal in this paper is to analyze the integrability question for the following family of equations: O,,u- O~u- au + Ilull~" u -- 0 u~C2((O,nL),R)raC~ nL],R), a>O, ~>0 (1) 2 ,,,L u(O,t)=u(nL, t)=O, Ilull~,-- ~--Zj0 u2(x) dx Here, integrability generically means the existence of a sufficiently large number of independent first integrals pairwise in involution in some sense. ~Instituto de Matem~tica e Estatistica, Universidade de Sao Paulo, CP 20570, 01498, Sio Paulo, SP, Brasii, and Courant Institute of Mathematical Sciences, New York University, 251 Mercer Street, New York, New York 10012. 227 1040.7294/94/0100-0227507.00/0 9 1994PlenumPublishing Corporation