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Semiclassical approach to the nonlocal nonlinear Schrödinger equation with a non-Hermitian term

A. E. Kulagin, A. V. Shapovalov

Abstract

The nonlinear Schödinger equation (NLSE) with a non-Hermitian term is the model for various phenomena in nonlinear open quantum systems. We deal with the Cauchy problem for the nonlocal generalization of multidimensional NLSE with a non-Hermitian term. Using the ideas of the Maslov method, we propose the method of constructing asymptotic solutions to this equation within the framework of semiclassically concentrated states. The semiclassical nonlinear evolution operator and symmetry operators for the leading term of asymptotics are derived. Our approach is based on the solutions of the auxiliary dynamical system that effectively linearize the problem under certain algebraic conditions. The formalism proposed is illustrated with the specific example of the NLSE with a non-Hermitian term that is the model of an atom laser. The analytical asymptotic solution to the Cauchy problem is obtained explicitly for this example.

Semiclassical approach to the nonlocal nonlinear Schrödinger equation with a non-Hermitian term

Abstract

The nonlinear Schödinger equation (NLSE) with a non-Hermitian term is the model for various phenomena in nonlinear open quantum systems. We deal with the Cauchy problem for the nonlocal generalization of multidimensional NLSE with a non-Hermitian term. Using the ideas of the Maslov method, we propose the method of constructing asymptotic solutions to this equation within the framework of semiclassically concentrated states. The semiclassical nonlinear evolution operator and symmetry operators for the leading term of asymptotics are derived. Our approach is based on the solutions of the auxiliary dynamical system that effectively linearize the problem under certain algebraic conditions. The formalism proposed is illustrated with the specific example of the NLSE with a non-Hermitian term that is the model of an atom laser. The analytical asymptotic solution to the Cauchy problem is obtained explicitly for this example.
Paper Structure (11 sections, 1 theorem, 86 equations, 2 figures)

This paper contains 11 sections, 1 theorem, 86 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Nonlocal NLSE with a non-Hermitian term
  3. Expectation of an operator over functions from $\mathcal{S}$
  4. Class of semiclassically concentrated functions
  5. Moments of functions from the class ${\mathcal{P}}_\hbar^t$
  6. Associated linear Schrödinger equation with dissipation
  7. Semiclassical symmetry operators
  8. Example
  9. Conclusion
  10. Hamilton--Ehrenfest system with dissipation of the second order
  11. Solution to the system \ref{['prim13']}

Key Result

Proposition 6.1

Let $\Psi(\vec{x},t,\hbar)\in {\mathcal{P}}_\hbar^t\left(Z(t),S(t),\sigma(t)\right)$ be a solution to the Cauchy problem for hartree1 with the initial condition and $\Phi(\vec{x},t,\hbar,{\bf C}[\varphi])$ be a solution to the Cauchy problem for aseq6, ${\bf C}={\bf C}[\varphi]$, with the initial condition Then, we have

Figures (2)

  • Figure 1: Dependence of $|\Psi^{(0)}(x,t)|^2$ on $x$ for various $t$ and $\hbar=0.2$ (solid lines). Dashed lines are for the respective numerical solution
  • Figure 2: Dependence of $|\Psi^{(0)}(x,t)|^2$ on $x$ for various $t$ and $\hbar=0.05$ (solid lines). Dashed lines are for the respective numerical solution

Theorems & Definitions (3)

  • Definition 2.1
  • Definition 4.1
  • Proposition 6.1