[Paper Review] Stationary and moving breathers in (2+1)-dimensional O(3) nonlinear $\sigma$-model

Ф. Ш. Шокиров|arXiv (Cornell University)|May 3, 2016

Nonlinear Photonic Systems9 references3 citations

TL;DR

This paper presents analytical and numerical solutions for stationary and moving breathers in the (2+1)-dimensional O(3) nonlinear σ-model by leveraging a trial function for the 2D sine-Gordon equation and extending it via isotopic spin rotations. The key result is the demonstration of long-term stability (over 45,000 cycles) of both stationary and moving breather solutions, despite weak radiation losses, with energy loss below 5.15% and stable dynamics confirmed through numerical simulations with absorbing boundary conditions.

ABSTRACT

The formation and evolution of stationary and moving breather solutions in (2+1)-dimensional O(3) nonlinear $\sigma$-model are investigated. The analytical form of oscillating solutions for (2+1)-dimensional sine-Gordon equation, which evolve to periodic (breather) radially symmetric solutions is determined. On the basis of the found solutions by adding the rotations to the A3-field vector in isotopic space of S^2, the solutions for the O(3) nonlinear $\sigma$-model are obtained. By numerical study of the solutions dynamics their stability in a stationary and a moving state for quite a long time (45000 cycles), although in the presence of weak radiation is shown.

Motivation & Objective

• To investigate the formation and evolution of stationary and moving breather solutions in the (2+1)-dimensional O(3) nonlinear σ-model.
• To extend analytical solutions of the 2D sine-Gordon equation to the O(3) nonlinear σ-model via isotopic space rotations.
• To numerically validate the long-term stability of breather solutions under dynamic evolution.
• To analyze energy loss mechanisms due to radiation and assess the role of internal degrees of freedom in breather dynamics.

Proposed method

• Derivation of averaged Lagrangian density for the 2D sine-Gordon equation using fast-phase averaging, yielding effective dynamics for phase variables λ(t) and ν(t).
• Use of a trial function with hyperbolic secant envelopes and time-dependent amplitude and phase to model breather solutions.
• Extension of sine-Gordon solutions to the O(3) nonlinear σ-model by introducing a rotating component φ = φ₀ + ωt in the isotopic spin vector on S².
• Application of stereographic projection to map the S² field onto the complex plane for numerical implementation.
• Implementation of a three-layer explicit finite difference scheme with second-order accuracy in space and time.
• Incorporation of absorbing boundary conditions to suppress spurious reflections and model energy radiation.

Experimental results

Research questions

• RQ1 Can stable stationary and moving breather solutions be analytically and numerically constructed in the (2+1)-dimensional O(3) nonlinear σ-model?
• RQ2 How do isotopic spin rotations (φ = φ₀ + ωt) affect the dynamics and stability of breather solutions?
• RQ3 What is the long-term energy loss behavior of breathers under numerical evolution, and how does it depend on initial speed and rotation frequency?
• RQ4 How does the internal dynamics (e.g., oscillation frequency, rotation) influence the effective speed and shape of moving breathers?
• RQ5 To what extent do absorbing boundary conditions accurately simulate radiation losses in finite simulation domains?

Key findings

• Stable stationary breathers were numerically generated and maintained over 45,000 time cycles with energy loss of approximately 5.109%.
• Moving breathers with initial speed vₜ₀ ≈ 0.7071 remained stable over 57 time units, showing energy loss of 3.3731% for ω = 0.0 and 3.5401% for ω = 0.5.
• The average speed of the moving breather with ω = 0.0 remained approximately constant at 0.2666 over t = 30 to 45, indicating stable propagation.
• For ω = 0.5, the breather exhibited non-uniform speed evolution, with average speed increasing from 0.1666 to 0.1754 between t = 30 and t = 57, indicating dynamic influence from internal rotation.
• Maximum energy density oscillations in the planar section (x, y₀) showed reduced frequency and amplitude for ω = 0.5, correlating with speed fluctuations and increased energy loss.
• A good agreement was observed between analytical predictions and numerical simulations, with energy conservation maintained within 5.15% loss over 45,000 cycles.