We consider nonlinear wave structures described by the modified Korteweg-de Vries equation, taking into account a small Burgers viscosity for the case of steplike initial conditions. The Whitham modulation equations are derived, which include the small viscosity as a perturbation. It is shown that for a long enough time of evolution, this small perturbation leads to the stabilization of cnoidal bores, and their main characteristics are obtained. The applicability conditions of this approach are discussed. Analytical theory is compared with numerical solutions and good agreement is found.
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