Bifurcation Analysis at a Singular Soliton State in Fiber Couplers

Abstract

Abstract The system describes pulses in nonlinear fiber couplers. It has the family (U1+λ, -U1+λ), -1 < λ < ∞, of soliton states (that is, homoclinic solutions to the origin), where For λ ≥ 1, the equilibrium (0, 0) is not hyperbolic and therefore the soliton state (U1+λ, −U1+λ) can be qualified as “singular”. In N. Akhmediev and A. Ankiewicz [1], it is observed numerically that a branch of homoclinic solutions bifurcates subcritically at λ = 1 from the family (U1+λ, −U1+λ). The aim of the present paper is to give a rigorous proof of the existence of this bifurcation, as desired in A. Ambrosetti and D. Arcoya [3]. A particular feature of the present problem is that the linearized system at (U2, −U2) has a non-constant bounded solution that does not vanish at infinity. Hence the bifurcating homoclinic solutions have a transient “spatial” region where they are well described with the help of this bounded function. Moreover the decay to 0 is governed by two different scales, the larger one originating from the singular aspect of (U2, −U2). The existence proof developed here relies on the “broken geodesic” technique to match the inside transient region with the outside region.