Smale horseshoes and symbolic dynamics in perturbed nonlinear Schrödinger equations (Q1304404)

This paper is devoted to horseshoes in infinite dimensions. Here the author constructs Smale horseshoes and studies the corresponding symbolic dynamics for the following perturbed \((1+1)\)-dimensional cubic nonlinear Schrödinger system: \[ iq_t= q_{\xi\xi}+ 2[|q|^2- \omega^2]q+ i\varepsilon [-\alpha q+\widehat{D}^2q+ \Gamma] \tag{1} \] under even periodic boundary conditions \[ q(\zeta+1)= q(\zeta), \quad q(-\zeta)= q(\zeta). \] Here \(\varepsilon> 0\) is the perturbation parameter \((\omega, \alpha, \Gamma)\) are real constants, \(\alpha> 0\), \(\omega\in (\pi, 2\pi)\). The operator \(\widehat{D}^2\) is a ``regularized'' Laplacian, specifically given by \[ \widehat{D}^2q \equiv- \sum_{j=0}^\infty \beta_j k_j^2 \widehat{q}_j \cos k_j \zeta_j \] where \(\widehat{q}_j\) is the Fourier transform of \(q_k\) and \(k_j= 2\pi j\). The regularizing coefficient \(\beta_j\) is defined by \[ \beta_j= \begin{cases} \beta &\text{for }j\leq N,\\ \alpha_* k_j^{-2} &\text{for }j> N \end{cases} \] in which \(\beta\) and \(\alpha_*\) are positive constants and \(\alpha_*> 16\pi^2\beta\) with \(N\) a sufficiently large fixed positive integer.