Quasi-periodic solutions for a nonlinear wave equation
The work treats the existence of quasi-periodic solutions of the nonlinear wave equation \(u_{tt}-u_{xx} +mu=-f(u)\) on the finite \(x\)-interval \([0,\pi]\) with Dirichlet boundary condition \(u(t,0)=0= u(t,\pi)\), \(-\infty<t <\infty\). Here \(m>0\), and \(f\) is a real analytic, odd function of the form \[ f(u)=au^3+ \sum_{k\geq 5} f_ku^k,\quad a\neq 0. \] Taking \({\mathcal P}= H^1_0 ((0,\pi)) \times L^2((0,\pi))\) as phase space, one can write the above equation as Hamiltonian system with Hamiltonian \(H={1\over 2} \langle v,v\rangle +{1\over 2} \langle Au,u \rangle+ \int^\pi_0 g(u)d_x\), where \(A=\partial^2_x+m\), \(g(u)=\int^u_0f(s)ds\), and \(\langle , \rangle\) denotes the usual scalar product in \(L^2\). For the linear case, when \(f=0\), every solution is a superposition of harmonic oscillations \[ u(t,x)=\sum_{j\geq 1}q_j(x) \varphi_j(x), \quad q_j(t)= I_j \cos(\lambda_j t+\varphi^0_j), \] where \(\varphi_j(x)= \sqrt{{2\over\pi}} \sin jx\), \(\lambda_j= \sqrt {j^2+m^2}\). The motion is periodic, quasi-periodic, or almost periodic, respectively, depending on whether one, finitely many, or infinitely many modes are excited. In particular, for every choice \(J=\{j_1<j_2< \cdots <j_n\} \subset\mathbb{N}^n\), there is an invariant \(2n\)-dimensional linear subspace \[ E_J= \bigl\{(u,v)=(q_1\varphi_{j_1} +\cdots +q_n\varphi_{j_n}, p_i\varphi_{j_i} + \cdots +p_n\varphi_{jn})\bigr\} =\bigcup_{I\in\mathbb{P}^n} F_J(I) \] where \(\mathbb{P}^n= \{I\in\mathbb{R}^n\); \(I_j>0\), \(j=1, \dots,n\}\) is the positive quadrant in \(\mathbb{R}^n\) and \(F_J(I)= \{(u,v):q^1_j+ \lambda_j^{-2} p^2_j=I_j\), \(j=1,\dots,n\}\). For the case when \(f\neq 0\) one can expect the existence of a Cantor set \(b\subset \mathbb{P}^n\), a family of \(n\)-tori \(F_J[b]= \cup_{I\in b} F_J(I) \subset E_J\) over \(b\), and a Lipschitz continuous embedding \(\varphi:F_J[b] \to{\mathcal E}_J \subset P\) such that the restriction of \(\varphi\) to each \(F_Y(I)\) in the family is an embedding of a rotational \(n\)-torus for the nonlinear equation. This is actually the main result of the work obtained for the case when the index set \(J=\{j_1< \cdots <j_n\}\) with \(n\geq 2\) satisfies \(\min_{1 \leq i<n} j_{i+1} -j_i \leq n-1\). More precisely, the conclusion of the result states that there exists a Cantor manifold \({\mathcal E}_J\) of real analytic, linearly stable, diophantine \(n\)-tori for the nonlinear wave equation given by a Lipschitz continuous embedding \(\varphi:F_J[b] \to{\mathcal E}_J\), which is a higher order perturbation of the inclusion map \(\varphi_0:E_J\to \mathbb{P}\) restricted to \(F_J[b]\). The Cantor set \(b\) has full density at the origin, and \({\mathcal E}_J\) has a tangent space at the origin equal to \(E_J\). Moreover, \({\mathcal E}_J\) is contained in the space of real analytic functions on \([0,\pi]\).
scientific article; zbMATH DE number 919899