Stationary waves on nonlinear quantum graphs: General framework and canonical perturbation theory

Area of Science:

• Mathematical Physics
• Nonlinear Dynamics
• Graph Theory

Background:

• The stationary nonlinear Schrödinger equation (NLSE) models various physical phenomena.
• Solving NLSE on complex network structures like metric graphs is challenging.
• Existing methods often struggle with the nonlinearity and network topology.

Purpose of the Study:

• To develop a general framework for solving the stationary NLSE on metric graphs.
• To provide a method for analyzing wave propagation in 1D wire networks.
• To simplify the mathematical treatment of nonlinear systems on graphs.

Main Methods:

• Modeling 1D wires as a metric graph with vertex matching conditions.
• Deriving a nonlinear transfer operation for the wave function and its derivative.
• Expressing the nonlinear transfer for cubic NLSE using Jacobi elliptic functions.
• Reducing differential equations to a system of nonlinear algebraic equations.

Main Results:

• A formal solution is established for the stationary NLSE on metric graphs.
• The nonlinear transfer operation is explicitly defined for the cubic NLSE.
• The problem is successfully reduced to solving a finite set of algebraic equations.
• Canonical perturbation theory is applied to extract leading nonlinear corrections for small amplitudes.

Conclusions:

• The proposed framework offers an efficient method for solving NLSE on network structures.
• The use of Jacobi elliptic functions provides an explicit solution for the cubic NLSE.
• The approach facilitates the analysis of nonlinear wave phenomena in complex systems.