COMPUTATION AND STABILITY OF FLUXONS IN A SINGULARLY PERTURBED SINE-GORDON MODEL OF THE JOSEPHSON-JUNCTION

New numerical results are reported regarding the solutions of a singularly perturbed sine-Gordon equation, modeling magnetic flux quanta (fluxons) in long Josephson tunnel junctions with nonzero surface impedance. Previous authors, hampered by an inadequate numerical shooting method, conjectured that the fluxon branch of solitary wave solutions terminates at a critical value of the bias current. In the present paper, a new numerical boundary value solver is developed and then used to show that the critical value of the bias current corresponds not to a termination point but to a turning point in the bifurcation diagram. Plotting a suitable norm of the solution against the bias current, we find that the solution curve turns back at the critical value and then oscillates via a sequence of turning points to a limiting value of the bias current below the value at the first turning point. These conclusions are confirmed, a post priori, by adapting current numerical procedures described in the literature, e.g., by Beyn, Doedel, Friedman and others. Furthermore, a careful numerical stability analysis of these solutions shows that none of the solutions past the first turning point is stable and that the solution acquires one additional instability per turning point. Multiple fluxon solutions, i.e., solitary wave solutions that connect fixed points separated by integer multiples of 2pi are also studied. The solution curves for the multiple fluxons exhibit the same qualitative behavior as those for the single fluxon case. However, the first turning point occurs at a value of the bias current which is less than the critical value of the single fluxon. If we denote by gamma(j)* the value of the bias current at the first turning point of the j-fluxon family, these numerical results suggest: for gamma > gamma1*, no fluxons exist: for gamma2* < gamma less-than-or-equal-to gamma1*, only single fluxons exist; for gamma(k+1)* < gamma less-than-or-equal-to gamma(k)*, solution from the j-fluxon family exist only for j = 1, 2, ..., k. The most physically relevant conclusion is that no fluxons exist past gamma1*, which agrees with previous reports based on numerical simulations in the time-domain.