We study both linear and nonlinear surface waves localized at the interface separating a left-handed medium (i.e. the medium with both negative dielectric permittivity and magnetic permeability) and a conventional (or right-handed) dielectric medium. We demonstrate that the interface can support both TE- and TM-polarized surface waves - surface polaritons, and we study their properties. We describe the intensity-dependent properties of nonlinear surface waves in three different cases, i.e. when both LH and RH media are nonlinear and when either of the media is nonlinear. In the case when both media are nonlinear, we find two types of nonlinear surface waves, one with the maximum amplitude at the interface, and the other one with two humps. In the case when one medium is nonlinear, only one type of surface wave exists, which has the maximum electric field at the interface, unlike waves in right-handed materials where the surface-wave maximum is usually shifted into a self-focussing nonlinear medium. We study the energy flow near the surface and demonstrate the specific vortex-like structure of temporal pulses propagating along the interface, as well as discuss the possibility of tuning the wave group velocity in both the linear and nonlinear cases. We show that the group-velocity dispersion, which leads to the pulse broadening, can be balanced by nonlinearity of the media resulting in the soliton propagation.
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