2007/03/12 by Yi Zhu, Jianke Yang · 4 citations
Physics and Astronomy · #Advanced Fiber Laser Technologies #Nonlinear Photonic Systems #Nonlinear Waves and Solitons #nlin.CD #nlin.PS
paper · pdf · doi:10.1103/physreve.75.036605
published as Phys. Rev. E 75, 036605 (2007) · 22pages, 15 figures
openalex publication_date 2007/03/12 · arxiv created 2007/03/14 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Weak interactions of solitary waves in the generalized nonlinear Schrödinger equations are studied. It is first shown that these interactions exhibit similar fractal dependence on initial conditions for different nonlinearities. Then by using the Karpman-Solov'ev method, a universal system of dynamical equations is derived for the velocities, amplitudes, positions, and phases of interacting solitary waves. These dynamical equations contain a single parameter, which accounts for the different forms of nonlinearity. When this parameter is zero, these dynamical equations are integrable, and the exact analytical solutions are derived. When this parameter is nonzero, the dynamical equations exhibit fractal structures which match those in the original wave equations both qualitatively and quantitatively. Thus the universal nature of fractal structures in the weak interaction of solitary waves is analytically established. The origin of these fractal structures is also explored. It is shown that these structures bifurcate from the initial conditions where the solutions of the integrable dynamical equations develop finite-time singularities. Based on this observation, an analytical criterion for the existence and locations of fractal structures is obtained. Lastly, these analytical results are applied to the generalized nonlinear Schrödinger equations with various nonlinearities such as the saturable nonlinearity, and predictions on their weak interactions of solitary waves are made.