p-adic evolution pseudo-differential equations and p-adic wavelets

In the theory of p-adic evolution pseudo-differential equations (with time variable t is an element of R and space variable x is an element of Q(p)(n)), we suggest a method of separation of variables (analogous to the classical Fourier method) which enables us to solve the Cauchy problems for a wide class of such equations. It reduces the solution of evolution pseudo-differential equations to that of ordinary differential equations with respect to the real variable t. Using this method, we solve the Cauchy problems for linear evolution pseudodifferential equations and systems of the first order in t, linear evolution pseudo-differential equations of the second and higher orders in t, and semilinear evolution pseudo-differential equations. We derive a stabilization condition for solutions of linear equations of the first and second orders as t -> infinity. Among the equations considered are analogues of the heat equation and linear or non-linear Schrodinger equations. The results obtained develop the theory of p-adic pseudo-differential equations and can be used in applications.