In a two-component Bose-Einstein condensate there exist two types of motions with in-phase and counter-phase oscillations of the components. Correspondingly, they can be called waves of density and waves of polarization. It is shown that in the case of small difference between values of the nonlinear interaction constants the polarization waves are described by the Gardner equation [1]. This leads to essentially new dynamics compared with the case of the density waves described by the Korteweg-de Vries equation. In particular, it is found [2] that the flow of two-component condensate past a polarized obstacle leads to generation of oblique breathers, on the contrary to the case of a non-polarized obstacle yielding generation of oblique solitons. In the limit of equal nonlinear interaction constants the condensate dynamics is described by the Manakov system whose periodic solution is found [3] which explains the results of recent experiments [4].

[1] A. M. Kamchatnov, Y. V. Kartashov, P.-E. Larre, N. Pavloff, Nonlinear waves of polarization in two-component Bose-Einstein condensates, Phys. Rev. A 89, 033618 (2014).
[2] A. M. Kamchatnov, Y. V. Kartashov, Oblique breathers generated by a flow of two-component Bose-Einstein condensate past a polarized obstacle, Phys. Rev. Lett., 111, 140402 (2013).
[3] A. M. Kamchatnov, Periodic waves in two-component Bose-Einstein condensates with repulsive interactions between atoms, Europhys. Lett., 103, 60003 (2013).
[4] C. Hamner, J. J. Chang, P. Engels, M. A. Hoefer, Generation of dark-bright soliton trains in superfluid-superfluid counterflow, Phys. Rev. Lett., 106, 065302 (2011).