Quasi-periodic solutions of Hamiltonian perturbations of 2D linear Schrödinger equations
The author's main problem is determination and characterization of solutions of integrable, but primarily linear, perturbed Hamiltonian equations. Specifically the author is interested in extending the classical KAM theory of existence of invariant tori. In this presentation the finite dimensional tori ``live'' in infinite-dimensional space. This may lead to small divisor computational instability problems. In one spatial dimension the natural frequencies are well-separated from each other. But in dimensions greater than one phenomenon of clustering of natural frequencies occurs, generally making the techniques used in derivation of KAM inapplicable. In fact the natural frequencies may have unbounded multiplicities. The author comments that at the present time not much is known about solutions of 3- or higher-dimensional nonlinear Schrödinger models. The case of known periodic solutions that is of persistence of \(n\)-dimensional tori is much easier. For example consider the equation: \(iu_t+Au+ \varepsilon\partial H/\partial\overline u=0\), where \(A\) is a selfadjoint operator, \(H(u, \widetilde u)\) is either a polynomial or real-analytic, and \(A=-\Delta+M_\alpha\) is the Laplacian plus a Fourier multiplier vector: \(M_\sigma\). Eigenfunction basis are exponentials \(e^{inx}\), and eigenvalues are \(\lambda_n =n^2+ \sigma_n\) with \(\lim_{n\to \infty}(\sigma_n) =0\). The author compares the Sturm-Liouville operator with \(A=d^2/dx^2 +V(x)\), where one encounters localization property of the eigenfunctions. This argument fails in dimensions higher than one. However in the special case when \(V(x)=V_1 (x_1)+V_2(x_2)+ \dots V_d(x_d)\) the solutions of unperturbed linear equation correspond to a flow on a \(b\)-dimensional torus. The aim of this paper is to generalize persistent properties of these tori in phase space. The author reduces weight functions, which allow some simplifications of this problem, and studies resulting dependency of the new eigenvalues on Fourier coefficients of the unperturbed problem. A version of (one-dimensional) Kuksin's theorem [\textit{S. B. Kuksin}, Nearly integrable Hamiltonian systems. Lect. Notes Math. 1556, Springer Verlag, Berlin, New York (1993; Zbl 0784.58028)] and a similar two-dimensional persistency theorem is proved. For sake of simplicity it is assumed that the Hamiltonian is a polynomial in \(u\) and \(\overline u\). A modification of arguments given in the polynomial case extending these results to the more general analytic cases is given at the end of this paper. Proofs, containing digression into related results and techniques, take more than sixty pages. Crucial to the proof is the Lyapunov-Schmidt decomposition technique used by W. Craig and C. E. Wayne in [\textit{W. Craig} and \textit{C. E. Wayne}, Periodic solutions of nonlinear Schrödinger equation and the Nash-Moser method, in Hamiltonian Mechanics, NATO Adv. Sci. Inst., Ser B, Physics, 331, Plenum Publ. New York, 1994]. These results are new, and also contain much material that is of importance in unrelated problems, such as a new use of Malgrange's preparation theorem, analysis of separation properties of large matrices, and several interesting inequalities.
scientific article; zbMATH DE number 1267995