1993/09/08 by N. J. Balmforth, Neil J. Balmforth, G. R. Ierley +5
Physics and Astronomy · #Chaotic Dynamics (nlin.CD) #FOS: Physical sciences #Nonlinear Photonic Systems #Nonlinear Waves and Solitons #Quantum chaos and dynamical systems #chao-dyn #nlin.CD
paper · pdf · doi:10.48550/arxiv.chao-dyn/9309002
22 pages, plain TEX. Figures available on request
arxiv created 1993/09/08 · openalex publication_date 1993/09/08 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We study a third-order nonlinear ordinary differential equation whose solutions, under certain specific conditions, are individual pulses. These correspond to homoclinic orbits in the phase space of the equation and we study the possible pulse types in some detail. Sufficiently close to the conditions under which a homoclinic orbit exists, the solutions take the form of trains of well-separated pulses. A measure of closeness to homoclinic conditions provides a small parameter for the development of an asymptotic solution consisting of superposed, isolated pulses. The solvability condition in the resulting singular perturbation theory is a \its timing map relating successive pulse spacings. This map of the real line onto itself, together with the known form of the homoclinic orbit, provides a concise and accurate solution of the equation.