  In this work nonlinear pseudo-differential equations with the infinite number
of derivatives are studied. These equations form a new class of equations which
initially appeared in p-adic string theory. These equations are of much
interest in mathematical physics and its applications in particular in string
theory and cosmology.
  In the present work a systematical mathematical investigation of the
properties of these equations is performed. The main theorem of uniqueness in
some algebra of tempored distributions is proved. Boundary problems for bounded
solutions are studied, the existence of a space-homogenous solution for odd p
is proved. For even p it is proved that there is no continuous solutions and it
is pointed to the possibility of existence of discontinuous solutions.
Multidimensional equation is also considered and its soliton and q-brane
solutions are discussed.
