[Paper Review] Interaction for Solitary Waves with a Phase Difference in a Nonlinear Dirac Model

Sihong Shao, Huazhong Tang|ArXiv.org|Apr 17, 2006

Nonlinear Photonic Systems13 references3 citations

TL;DR

This study investigates the interaction dynamics of solitary waves in a nonlinear Dirac model with a phase difference using a fourth-order Runge-Kutta discontinuous Galerkin (RKDG) method. Key findings include full repulsion in binary and ternary collisions depending on initial separation, repulsion followed by attraction and collapse in two-humped wave collisions, and distinct one- and two-overlap interaction patterns in ternary collisions of initially resting waves.

ABSTRACT

This paper presents a further numerical study of the interaction dynamics for solitary waves in a nonlinear Dirac field with scalar self-interaction by using a fourth order accurate Runge-Kutta discontinuous Galerkin method. Our experiments are conducted on the Dirac solitary waves with a phase difference. Some interesting phenomena are observed: (a) full repulsion in binary and ternary collisions and its dependence on the distance between initial waves; (b) repulsing first, attracting afterwards, and then collapse in binary and ternary collisions of initially resting two-humped waves; (c) one-overlap interaction and two-overlap interaction in ternary collisions of initially resting waves.

Motivation & Objective

• To investigate the interaction dynamics of Dirac solitary waves with an initial phase difference, particularly focusing on out-of-phase configurations.
• To extend prior numerical studies limited to in-phase solitary waves by introducing phase shifts in initial conditions.
• To analyze how phase differences influence collision outcomes such as repulsion, attraction, and collapse in binary and ternary interactions.
• To examine the role of wave structure (one-humped vs. two-humped) and initial parameters (Λ, velocity, separation) on interaction behavior.
• To validate the robustness and accuracy of the fourth-order RKDG method in simulating long-time dynamics without numerical oscillations.

Proposed method

• Employed a fourth-order accurate Runge-Kutta discontinuous Galerkin (RKDG) method for spatial and temporal discretization of the nonlinear Dirac equation with scalar self-interaction.
• Used discontinuous piecewise polynomial approximations for the spinor components ψ₁ and ψ₂, enabling high-order accuracy and stability.
• Applied explicit high-order Runge-Kutta time integration to ensure long-term numerical stability and conservation of energy and charge.
• Defined initial conditions as superpositions of solitary wave solutions with phase shifts: ψ(x,0) = e^{iθₗ}ψ̃ₗ^{ss}(x−xₗ,0) + e^{iθₘ}ψ̃ₘ^{ss}(x−xₘ,0) + e^{iθᵣ}ψ̃ᵣ^{ss}(x−xᵣ,0).
• Tracked energy and charge densities ρ_E and ρ_Q over time to analyze interaction features such as repulsion, attraction, and collapse.
• Simulated both binary and ternary collisions with varying initial phase differences (e.g., θₘ = π for out-of-phase middle wave), velocities, and Λ values (0.1 to 0.9).

Experimental results

Research questions

• RQ1 How does an initial phase difference of π affect the collision dynamics of Dirac solitary waves in binary and ternary interactions?
• RQ2 What is the dependence of repulsive forces on the initial separation distance between solitary waves with a phase shift?
• RQ3 Under what conditions does repulsion give way to attraction followed by collapse in two-humped solitary wave collisions?
• RQ4 How do one-overlap and two-overlap interaction patterns emerge in ternary collisions of initially resting waves with phase differences?
• RQ5 What role does the initial peak amplitude (controlled by Λ) play in determining the number of overlaps and the likelihood of collapse?

Key findings

• Full repulsion was observed in binary and ternary collisions of solitary waves with a phase difference of π, and the strength of repulsion increased as the initial separation decreased.
• In collisions of initially resting two-humped waves with a phase shift of π, waves first repel, then attract, and ultimately collapse into a singularity, with symmetry preserved throughout.
• For ternary collisions of initially resting waves, one-overlap and two-overlap interaction patterns emerged depending on the value of Λ: one overlap occurred at Λ = 0.5, while two overlaps were observed at Λ = 0.6 and 0.9.
• Collapse was observed in collisions of in-phase, equal-amplitude two-humped waves, but not in collisions of one-humped waves, indicating a strong dependence on wave structure.
• The macroscopic behavior of interaction dynamics was largely independent of initial phase difference when there was a significant disparity in peak amplitudes of the initial solitary waves.
• The fourth-order RKDG method demonstrated long-term numerical stability, preserved energy and charge conservation, and avoided spurious oscillations over extended simulation times.