  In this paper, we investigate the problem of electromagnetic wave propagation
in hyperbolic nonlinear media. To address this problem, we consider the scalar
hyperbolic nonlinear Schr\"odinger system and its coupled version, namely
hyperbolic Manakov type equations. These hyperbolic systems are shown to be
non-integrable. Then, we examine the propagation properties of both the scalar
and vector electromagnetic solitary waves by deriving their exact analytical
forms through the Hirota bilinear method. A detailed analysis shows that the
presence of hyperbolic transverse dispersion provides an additional degree of
freedom to prevent the formation of singularity in both the scalar and vector
solitary wave structures in this hyperbolic nonlinear media. Besides this, we
realize that the solitary waves in this media possess fascinating propagation
properties which cannot be observed in conventional nonlinear media. We believe
that the present study will be very useful in analyzing electromagnetic wave
propagation in hyperbolic nonlinear metamaterials.
