Bifurcation Analysis at a Singular Soliton State in Fiber Couplers

The system {-u '' + u = u(3) + lambda v, u(x)= u(-x) is an element of R, x is an element of R, -v '' + v = v(3) + lambda u, v(x) = v(-x) is an element of R, x is an element of R, describes pulses in nonlinear fiber couplers. It has the family (U1+lambda, -U1+lambda), -1 < lambda < infinity, of soliton states (that is, homoclinic solutions to the origin), where U1+lambda(x) = root 2(1 + lambda/cosh(root 1 + lambda x). For lambda >= 1, the equilibrium (0, 0) is not hyperbolic and therefore the soliton state (U1+lambda, -U1+lambda) 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 lambda = 1 from the family (U1+lambda, -U1+lambda), 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 (U-2, -U-2) 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 (U-2, -U-2). The existence proof developed here relies on the "broken geodesic" technique to match the inside transient region with the outside region.