Solutions to the nonlinear Schrodinger equation with an anti-Hermitian term, localized on curves, and quasi steady vortex states

Abstract: Speaking about semiclassically localized solutions to the Schrödinger equation, we mean the class of asymptotic solutions that are obtained for the linear Schrödinger equation by the Maslov complex germ method [1,2,3]. Such solutions are localized in a neighbourhood of the trajectory in the phase space (point for any fixed time) that is determined by solutions to the Hamilton system (classical equations). Such approach was also generalized for nonlinear equations [4]. In our report, we consider the Cauchy problem where the solutions to the Schrödinger equation with a nonlocal nonlinearity are localized in a neighborhood of the evolving curve. Also, we add the anti-Hermitian terms that allows us to consider the dissipative effects. Such problem is solved using the transition to the space of variables of higher dimension, where we can apply elements of the Maslov complex germ method. Asymptotic solutions to the original problem are the projection of the solutions in the extended space to the original space. The formalism proposed becomes applicable to the problem of the vortex lattice formation in condensed media with collective excitations. It is shown that such process includes the semiclassical stage that is treated as the quasi steady vortex state. The evolution of such states is mainly determined by the slow deformation of the semiclassical localization curve. The report is based on the paper [5].

This is joint work with A.V. Shapovalov.

[1] V.P. Maslov, The Complex WKB Method for Nonlinear Equations (I. Linear Theory. Birkhauser Verlag, Basel, 1994).

[2] V.V. Belov, S.Y. Dobrokhotov, Semiclassical Maslov asymptotics with complex phases. I. General approach. Theor. Math. Phys. 92(2), 843–868 (1992).

[3] V.G. Bagrov, V.V. Belov, A.Y. Trifonov, Semiclassical trajectory-coherent approximation in quantum mechanics I. High-order corrections to multidimensional time-dependent equations of Schrödinger type. Ann. Phys. 246(2), 231–290 (1996).

[4] V.V. Belov, A.Y. Trifonov, A.V. Shapovalov, The trajectory-coherent approximation and the system of moments for the Hartree type equation. Int. J. Math. Math. Sci. 32(6), 325–370 (2002).

[5] Kulagin, A., Shapovalov, A. Semiclassical states localized on a one-dimensional manifold and governed by the nonlocal NLSE with an anti-Hermitian term. Eur. Phys. J. Plus 141, 14 (2026). doi.org/10.1140/epjp/s13360-025-07236-6

Russianmathematical physicsanalysis of PDEsclassical analysis and ODEsdynamical systemsnumerical analysisexactly solvable and integrable systemsfluid dynamics

Audience: researchers in the topic

Mathematical models and integration methods