PHYSICAL REVIEW 8 VOLUME 47, NUMBER 2 1 JANUARY 1993-II Bunching of Auxons in a long Josephson junction with surface losses Boris A. Malomed* Department of Physics, Uniuersity of Illinois at Urbana C-hampaign, 1110 West Green Street, Urbana, Illinois (Received 6 April 1992) It is known that the sine-Gordon model of a long Josephson junction with a surface-loss term predicts a spatially oscillating trailing tail of a strongly contracted fluxon. In the present work, it is demonstrated that this should give rise to bunched (bound) states of fluxons, although the onset of the bunching drasti- cally differs from the standard situation recently investigated in terms of the perturbed nonlinear Schrodinger model. Influence of the bunching on the I-V characteristic of an annular junction with a finite number of trapped fluxons is analyzed qualitatively. It is demonstrated that the bunching may set in with a small hysteresis, and it increases voltage at a given current. The latter effect has been observed in most recent experiments with the annular junction. Creation of annular long Josephson junctions' (LJJ's) opens ways to observe new dynamical phenomena with fluxons (magnetic flux quanta, or Josephson vortices), which were not possible in traditional linear LJJ's with edges. In particular, the most recent experiments ' point out a possibility of bunching (clustering) of strongly con- tracted Auxons moving with a velocity close to the Swihart velocity. As is mentioned in Ref. 3, the bunching should be possible in the presence of the surface losses. The aim of the present Report is to analyze this possibili- ty briefly. The well-known perturbed sine-Gordon (SG) equation for the magnetic flux t() trapped in the LJJ is P„— P „+ sing= — aP, +/3$, f, — where a and /3 are small coefficients of the shunt and sur- face losses, and f is the current bias density. As is well known, the perturbation theory gives quite a satisfactory description of the fluxon dynamics apart from the region where the fluxon's velocity V is close to one (i. e. , to the Swihart velocity), so that the Lorentz contraction renders the surface-loss term in Eq. (1) comparable with the basic terms. A thorough numerical investigation of this region was performed in Ref. 6. It has been demonstrated that, at 1 — V2 &/3 /, the trailing "tail" of the fluxon becomes oscillating. With the decrease of the ratio (1 — V )//3 the oscillations grow and finally lead to an instability of the Auxon. The scenario ends with establishing the McCumber mode (the spatially homogeneous phase rota- tion) in the system. In the region where the Auxons are still stable, the os- cillating tails may give rise to bunching, i.e. , formation of two-Auxon and multifluxon bound states. Recently, a similar phenomenon was analyzed for nonlinear Schrodinger (NS) solitons in the presence of small dissi- pative terms. Formation of a stable bound state of two NS-like solitons has been lately observed in experiments with subcritical traveling-wave convection in a binary Auid filling a narrow annular channel. It will be demon- strated below that in the SG system (1) the bunching is drastically different from the "standard" situation con- sidered in Ref. 7. (3) Linearizing Eq. (1) far from the center of the fluxon (an exact form of which is actually unknown ) and looking for solutions in the form P„—exp(icx), one arrives at the well-known equation for ~, Ptt +(1 — V )tc — ate — 1=0, (2) in which it is implied that /3«1, and V is close to one. The parameter a competes with /3 if a)/3'/ . In real LJJ's, typical values may be a-0. 01, and /3) 0. 001. Thus, one may neglect u, and the corresponding term will be omitted in Eq. (2). Setting a =0, one immediately sees that Eq. (2) has a pair of complex roots at (1 — V ) &(1 — V )o — 3(/3/2) At 1 — V =(1 — V )o, the roots of Eq. (2) are 0) —i0) — (2//3)1/3 (Oi ( g/3) — I /3 At I V2 &(1 — V )o, the pair (tc„tc2) gives rise to the complex roots, the presence of which implies that the trailing edge of the Auxon is spatially oscillating. The real root ~3 cor- responds to the nonoscillating leading tail. The next step, following the lines of Ref. 7, is to calculate an effective potentia1 of interaction of two separated fluxons, pro- duced by overlapping of each Auxon with its mate's tail. This is how the spatial oscillations give rise to a set of bound (bunched) states of the solitons in the perturbed NS system. . However, in the present case the situation is different because the full interaction potential contains two terms, produced, respectively, by overlapping the leading Auxon with the leading tail of the trailing Auxon, and by overlapping the trailing Auxon with the trailing tail of the leading Auxon. Only the latter term is oscillat- ing, while the former one corresponds to the usual repul- sion between unipolar Auxons. Next, one notices that, at the point (3) where the oscillations set in, tc& ' & ~n'i z'~. Re- call that the root ~z corresponds to the nonoscillating leading tail of the Auxon amenable for the mutual repul- sion. Therefore, this inequality implies that at a large distance between the Auxons, at which bunched states may appear, the nonoscillating repulsive term in the full interaction potential decays more slowly than the oscil- lating one; hence the bunched states are not possible at all. 1993 The American Physical Society