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Periodic solutions for the Schrödinger equation with nonlocal smoothing nonlinearities in higher dimension. (English) Zbl 1172.35071

The paper aims to prove the existence of small-amplitude time-periodic solutions to a \(D\)-dimensional equation similar to the Gross-Pitevskii equation, but with nonlinear terms subjected to the smoothing:
\[ iv_t - \Delta v + \mu v = |\Phi(v)|^2\Phi(v) + F(x,\Phi(v),\Phi^{*}(v)). \]
It is implied that the expansion of the additional nonlinear term, \(F\), in powers of \(\Phi\) and \(\Phi^{*}\), starts with terms of power higher than \(3\), the cubic term in the equation being written separately. The smoothing is defined in terms of the Fourier transform of a function \(v(\mathbf{r}\)), \((\Phi(u))_{\mathbf{k}} =|k|^{-2s}u_{\mathbf{k}}\), with some positive \(s\). The equation is formulated in a \(D\)-dimensional square, with Dirichlet conditions at the boundaries. In the case of \(D=2\), the nonlinearity may be of the most general type, while for \(D\geq 3\), a condition is imposed that the nonlinearity must be of a Hamiltonian type.
It should be stressed that the regularized equation, written in this form, has no direct physical applications, but makes it possible to produce a rigorous proof of the existence of time-periodic solutions. The analysis uses the Lyapunov-Schmidt decomposition, aiming to separate resonant and nonresonant parts of the general solution. To handle the resonant part, it is necessary to treat properly small divisors in a vicinity of the resonances. This is done by means of the so-called modified Lindstedt series, within the framework of the KAM-type theory. The resultant solutions, whose existence is eventually proved, can be realized as wave packets, in the sense that they are obtained as continuation of linear solutions which combine an arbitrary number of the resonant modes.

MSC:

35Q55 NLS equations (nonlinear Schrödinger equations)
37K55 Perturbations, KAM theory for infinite-dimensional Hamiltonian and Lagrangian systems
35B10 Periodic solutions to PDEs
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