The evolution equations for nonlinear Scholte waves (finite amplitude elastic waves propagating along liquid/solid interface), which account for the second order nonlinearity of a liquid, are derived for the first time. For mathematical simplicity the nonlinearity of the solid, which influence is expected to be weak in the case of weak localization of the Scholte wave, is not taken into consideration. The analysis of these equations demonstrates that the nonlinear processes contributing to the evolution of the Scholte wave can be divided into two groups. The first group includes nonlinear processes leading to wave spectrum broadening which are common to bulk pressure waves in liquids and gases. The second group includes the nonlinear processes which are active only in the frequency down-conversion (leading to wave spectrum conservation or narrowing), which are specific to the confined nature of the interface wave. It is demonstrated that the nonlinear parameters, which characterize the efficiency of various nonlinear processes in the interface wave, strongly depend on the relative properties of the contacting liquid and solid (or, in other words, on the deviation of the Scholte wave velocity from the velocities of sound in liquid and in solid). In particular, the sign of the nonlinear parameter responsible for the second harmonic generation can differ from the sign of the nonlinear acoustic parameter of the liquid. It is also verified that there are particular liquid/solid combinations where the nonlinear processes, which are inactive in the frequency up-conversion, dominate in the evolution of the Scholte wave. In this case distortionless propagation of the finite amplitude harmonic interface wave is possible. The proposed theory should find applications in nonlinear acoustics, geophysics, and nondestructive testing.