[Paper Review] Black and gray spatial optical solitons with Kerr type nonlocal nonlinearity

Shigen Ouyang, Qi Guo|arXiv (Cornell University)|Nov 11, 2008

Nonlinear Photonic Systems27 references31 citations

TL;DR

This paper develops a generalized spectral renormalization method to numerically compute black and gray spatial optical solitons in Kerr-type nonlocal nonlinear media, focusing on exponentially decaying and Gaussian nonlocal response functions. The key contribution is the analytical derivation of nonlocal gray soliton tail behavior and a critical dependence of maximal transverse velocity on the characteristic nonlocal length, showing a constant velocity below a critical length and a decreasing velocity above it.

ABSTRACT

We develop one numerical method to compute black and gray solitons with Kerr-type nonlocal nonlinearity. As two examples of nonlocal cases, the gray soliton with exponentially decaying nonlocal response or with Gaussian nonlocal response are discussed. For such two nonlocal cases, the analytical form of the tails of nonlocal gray soliton is presented and the analytical relationship for the maximal transverse velocity of nonlocal gray soliton to the characteristic nonlocal length is obtained.

Motivation & Objective

• To develop a robust numerical method for computing black and gray solitons in nonlocal nonlinear optical media.
• To analyze the impact of nonlocal response functions—exponential and Gaussian—on soliton structure and dynamics.
• To derive analytical expressions for the tails of nonlocal gray solitons.
• To establish the relationship between the maximal transverse velocity of gray solitons and the characteristic nonlocal length.
• To resolve discrepancies in prior work regarding the monotonicity of maximal soliton velocity with nonlocality.

Proposed method

• A generalized spectral renormalization method is formulated to solve the nonlocal nonlinear Schrödinger equation (NNLSE) for gray and black soliton solutions.
• The method transforms the NNLSE into a system of equations for amplitude ψ(x) and phase φ(x), using the ansatz u(x,z) = ψ(x)ei(βz + φ(x)) and enforcing phase continuity via ψ²φ′ = const.
• Discrete Fourier transforms (DFT) are applied to handle spatial derivatives and nonlocal integrals in the governing equations.
• A fixed-point iteration scheme is implemented using projected equations for the amplitude perturbation χ(x) = η − ψ(x), with λ and θ as iterative variables.
• For Gaussian nonlocal response, the method derives analytical tail forms via asymptotic analysis of the linearized equation, yielding exponential or oscillatory decay depending on parameters.
• The method is validated by computing soliton profiles and velocity limits for both exponential and Gaussian nonlocality, with convergence monitored via residual norms.

Experimental results

Research questions

• RQ1 How does the characteristic nonlocal length influence the maximal transverse velocity of nonlocal gray solitons?
• RQ2 What analytical forms do the tails of nonlocal gray solitons take for exponential and Gaussian nonlocal response functions?
• RQ3 Does the maximal transverse velocity of nonlocal gray solitons monotonically decrease with increasing nonlocal length, as previously claimed?
• RQ4 Under what conditions do nonlocal gray solitons exhibit exponentially decaying versus exponentially decaying oscillatory tails?
• RQ5 How does the nonlocality profile affect the existence and stability of black soliton solutions?

Key findings

• Nonlocal gray solitons can exhibit either exponentially decaying tails or exponentially decaying oscillatory tails, depending on the nonlocal length and soliton velocity.
• For Gaussian nonlocal response, when the characteristic nonlocal length w < 1/η, the maximal transverse velocity is constant and equal to η, matching the local soliton velocity.
• When w > 1/η, the maximal transverse velocity decreases with increasing w and is given by √(1 + ln(η²w²))/w.
• For 1/√e η < w < 1/η, the soliton can have either tail type depending on the velocity µ, with a transition at µ = √(1 + ln(η²w²))/w.
• The analytical tail form for Gaussian nonlocality is derived via asymptotic solution of the linearized equation, showing dependence on w, η, and µ.
• The numerical method successfully computes both black and gray soliton solutions with high accuracy, confirming the analytical predictions for velocity and tail behavior.