  A pair of nonlinear diffusion equations in Fourier space} is used to study
the dynamics of strong Alfv\'en-wave turbulence, from MHD to electron scales.
Special attention is paid to the regime of imbalance between the energies of
counter-propagating waves commonly observed in the solar wind (SW), especially
in regions relatively close to the Sun. In the collisionless regime where
dispersive effects arise at scales comparable to or larger than those where
dissipation becomes effective, the imbalance produced by a given injection rate
of generalized cross-helicity (GCH), which is an invariant, is much larger than
in the corresponding collisional regime described by the usual (or reduced)
magnetohydrodynamics. The combined effect of high imbalance and ion Landau
damping induces a steep energy spectrum for the transverse magnetic field at
sub-ion scales. This spectrum is consistent with observations in highly
Alfvenic regions of the SW, such as trailing edges, but does not take the form
of a transition range continued at smaller scales by a shallower spectrum. This
suggests that the observed spectra displaying such a transition result from the
superposition of contributions originating from various streams with different
degrees of imbalance. Furthermore, when imbalanced energy injection is
supplemented at small scales in an already fully developed turbulence, for
example under the effect of magnetic reconnection, a significant enhancement of
the imbalance at all scales is observed.
