Cole-Hopf-Like-Transformation-for-Schr¨odinger-EquationsContaining-Complex-Nonlinearities.pdfVIP

Cole-Hopf Like Transformation for Schr¨odinger Equations Containing Complex Nonlinearities G. Kaniadakis, and A.M. Scarfone∗ Dipartimento di Fisica - Politecnico di Torino - Corso Duca degli Abruzzi 24, 10129 Torino, Italy; Istituto Nazionale di Fisica della Materia - Unit´a del Politecnico di Torino, - Italy (July 12, 2006) Abstract We consider systems, which conserve the particle number and are described by Schr¨odinger equations containing complex nonlinearities. In the case of canonical systems, we study their main symmetries and conservation laws. We introduce a Cole-Hopf like transformation both for canonical and non- canonical systems, which changes the evolution equation into another one containing purely real nonlinearities, and reduces the continuity equation to the standard form of the linear theory. This approach allows us to treat, in a unifying scheme, a wide variety of canonical and noncanonical nonlinear systems, some of them already known in the literature. PACS number(s): 02.30.Jr, 03.50.-z, 03.65.-w, 05.45.-a, 11.30.Na, 11.40.Dw Typeset using REVT X E ∗e-mail: kaniadakis@polito.it, scarfone@polito.it 1 I. INTRODUCTION Over the last few decades many nonlinear Schr¨odinger equations (NLSEs) have been proposed in order to test the fundamental postulates of quantum mechanics, like for in- stance, the Bialynicki-Birula and Mycielski equation [1], the Kostin