Speed of wave packets and the nonlinear Schrödinger equation

Area of Science:

• Nonlinear Physics
• Wave Propagation
• Mathematical Physics

Background:

• The nonlinear Schrödinger equation (NLSE) is a fundamental model for weakly nonlinear wave packets.
• Understanding soliton behavior in NLSE is crucial for various physical phenomena.
• Previous analyses often assumed distinct group and phase velocities.

Purpose of the Study:

• To revisit the universal theory of weakly nonlinear wave packets described by the NLSE.
• To investigate soliton dynamics in the specific limit where group and phase velocities are nearly identical.
• To uncover novel behaviors and interactions of solitons under these conditions.

Main Methods:

• A multiple-scale analysis was performed, extending beyond all orders.
• The analysis focused on the regime where group and phase velocities are very close.
• The derived dynamics were compared to a pendulum equation with a periodic potential.

Main Results:

• Solitons (bright or dark) can propagate at speeds different from the group velocity.
• In a specific parameter range, soliton envelopes lock to carrier wave oscillations.
• Soliton speed becomes non-constant, exhibiting a periodic component due to an effective potential.

Conclusions:

• The derived dynamics are analogous to a pendulum, influencing soliton speed and interactions.
• Interactions between distant solitons may be significantly altered, potentially leading to new bound states.
• These findings are believed to have universal validity across a wide class of wave models.