We consider soliton solutions to equations describing a pair of tunnel-coupled nonlinear optical fibers with a phase-velocity mismatch between them. The analysis is based on the variational approximation, which is checked (with quite favorable results) against direct numerical solutions at selected values of the parameters. The results are presented in the form of curves demonstrating evolution of the energy in one of the soliton's components with increase of the mismatch, with the total energy of the soliton being fixed. Two bifurcations are found. One of them, which to our knowledge has not been observed in any form earlier, involves a termination, occurring at a finite value of the mismatch parameter, of the branch whose solitons have components of the opposite sign; this branch corresponds to the antisymmetric soliton in the model with no mismatch. The other, more important, bifurcation is perceived as the occurrence of a hysteresis-type behavior of another branch, whose solitons have the same signs of their components; this branch corresponds to both the symmetric and the asymmetric solitons in the model with no mismatch. The implications of this second bifurcation for the propagation of a subpicosecond soliton in a real-world dual-core fiber are also discussed.
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