Generalized traveling-wave method, variational approach, and modified conserved quantities for the perturbed...

Area of Science:

• Nonlinear Dynamics
• Mathematical Physics

Background:

• The nonlinear Schrödinger equation (NLSE) is a fundamental model in various scientific fields.
• Perturbations to the NLSE often require complex analytical techniques for studying collective dynamics.
• Existing methods like Inverse Scattering Transform (IST) can be computationally intensive.

Purpose of the Study:

• To develop a generalized traveling wave method (GTWM) for analyzing the NLSE with general perturbations.
• To derive equations of motion for an arbitrary number of collective coordinates.
• To offer a computationally simpler alternative to existing perturbation theories.

Main Methods:

• The generalized traveling wave method (GTWM) is formulated using a Hamiltonian approach.
• The method requires only the Hamiltonian of the unperturbed system.
• The approach is demonstrated using 4 collective coordinates: soliton position, velocity, amplitude, and phase.

Main Results:

• The GTWM is shown to be equivalent to the Lagrangian formalism but requires less information.
• It yields identical equations of motion compared to IST-based perturbation theory.
• The results match the time evolution of the norm, first moment of the norm, momentum, and energy for the perturbed NLSE.

Conclusions:

• The GTWM provides an efficient and versatile framework for analyzing perturbed NLSE.
• This Hamiltonian-based method simplifies the derivation of collective coordinate equations.
• The GTWM is a valuable tool for understanding the dynamics of solitons in nonlinear systems.