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PresentationsSemiclassical solutions to the nonlinear Schrodinger equation with a non-Hermitian term that are localized on few trajectoriesTomsk Polytechnic University, Russia, 634050, Tomsk, 30 Lenina av., phone: (3822) 418913, E-mail: aek8@tpu.ru 1Tomsk State University, Russia, 634050, Tomsk, 1 Novosobornaya sq., phone: (3822) 529843, E-mail: shpv@phys.tsu.ru The method of semiclassically concentrated states, which is based on the Maslov complex germ method [1], is a powerful tool for constructing asymptotic solutions to the nonlocal nonlinear Schrodinger equation [2]. In the work [3], it was shown that it can be generalized for the case when the equation operator includes a non-Hermitian term. That allows one to apply this method to the description of the evolution of open quantum systems. Such solutions are associated with the dynamical system that determines the weighted trajectory of the “classical” particles. This trajectory describes the localization domain of solutions. For the description of long-range interactions in a system, the localization domain must include at least few points for every fixed moment of time, i.e. the solutions must be localized on few trajectories. In this case, the asymptotic solution to the nonlinear Schrodinger equation turns out to be related with the dynamical system for few classical particles. We introduce own semiclassical wave function for every classical particles, which is termed quasiparticle. These functions turn out to be nonlinearity related with each other through the dynamical systems of ordinary differential equations. Using such approach, we managed to construct the approximate nonlinear evolution operator for the original nonlocal nonlinear Schrodinger equation with a non-Hermitian term.
The study is supported by Russian Science Foundation, project no. 23-71-01047, https://rscf.ru/en/project/23-71-01047/.
References 1. Maslov V.P. The Complex WKB Method for Nonlinear Equations. – Moscow: Nauka, 1977. 384 p. 2. Belov V.V., Trifonov A.Y., Shapovalov A.V. The trajectory-coherent approximation and the system of moments for the Hartree type equation // International Journal of Mathematics and Mathematical Sciences, Vol. 32, No. 6, 2002. p. 325-370. 3. Kulagin A.E., Shapovalov A.V. A Semiclassical Approach to the Nonlocal Nonlinear Schrodinger Equation with a Non-Hermitian Term // Mathematics, Vol. 12, No. 4, 2024. 580.
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