Quasi-Periodic Solutions for 1D Nonlinear Wave Equation with a General Nonlinearity

In this paper, one–dimensional (1D) wave equation with a general nonlinearity utt−uxx +mu+f(u)= 0, m > 0 under Dirichlet boundary conditions is considered; the nonlinearity f is a real analytic, odd function and f(u)= au+ ∑ k≥r̄+1 f2k+1u , a 6=0 and r̄∈N. It is proved that for almost all m > 0 in Lebesgue measure sense, the above equation admits smallamplitude quasi-periodic solutions corresponding to finite dimensional invariant tori of an associated infinite dimensional dynamical system. The proof is based on infinite dimensional KAM theorem, partial normal form and scaling skills. 1 Statement of the main result In this paper, we are going to study the nonlinear wave equation utt−uxx +mu+f(u)= 0, m > 0 (1.1) on the finite x−interval [0,π] with Dirichlet boundary conditions u(t,0)= 0= u(t,π). (1.2) Here, m> 0 is a real parameter, sometimes referred to as “mass”, and f is a real analytic, odd function of u of the form f(u)= au + ∑ k≥r̄+1 f2k+1u , a 6=0 and r̄∈N. (1.3) As [24], we study this equation (1.1) as an infinite dimensional hamiltonian system on P =H1 0 ([0,π])×L2([0,π]) with coordinates u and v =ut. Let φj = √ 2 π sinjx, λj = √ j2 +m, j≥ 1 ∗The work was supported by the Special Funds for Major State Basic Research Projects of China (973 projects).