1999/05/28 by I. V. Barashenkov, E. V. Zemlyanaya · 1 citation
Computer Science · Mathematics · Physics and Astronomy · #Advanced Fiber Laser Technologies #Classical mechanics #Dissipation #Geometry #Mathematical analysis #Mathematics #Nonlinear Dynamics and Pattern Formation #Nonlinear Photonic Systems #Nonlinear Schrödinger equation #Nonlinear system #Order (exchange) #Perturbation (astronomy) #Perturbation theory (quantum mechanics) #Physics #Point (geometry) #Quantum electrodynamics #Quantum mechanics #Singular perturbation #Soliton #nlin.PS #patt-sol
paper · pdf · doi:10.1016/s0167-2789(99)00055-x
8 pages in RevTeX; 5 figures in ps format included in the text. To be published in Physica D
arxiv created 1999/05/28 · openalex publication_date 1999/08/01 · arxiv updated 2015/06/26 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
It has been known for some time that solitons of the externally driven, damped nonlinear Schrödinger equation can only exist if the driver's strength, h, exceeds approximately (2/ π) γ, where γ is the dissipation coefficient. Although this perturbative result was expected to be correct only to the leading order in γ, recent studies have demonstrated that the formula hthr= (2 /π) γ gives a remarkably accurate description of the soliton's existence threshold prompting suggestions that it is, in fact, exact. In this note we evaluate the next order in the expansion of hthr(γ) showing that the actual reason for this phenomenon is simply that the next-order coefficient is anomalously small: hthr=(2/ π) γ+ 0.002 γ3. Our approach is based on a singular perturbation expansion of the soliton near the turning point; it allows to evaluate hthr(γ) to all orders in γ and can be easily reformulated for other perturbed soliton equations.