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The problem of space charge continues to be interesting in connection with the development of the Kaon and Neutron Facilities and the Superconducting Super Collider projects. In the former, the high average intensity of the proton beam can be reached by a super-high pick current in each acceleration cycle, giving rise to the painting procedure. In colliders, as is known, super luminosity can be reached by minimization of emittance growth under acceleration. In other words, the task consists of the beam acceleration with the maximum attainable density of current and with minimum achievable emittance growth. This is why this task arises when new accelerator projects are developed. There are at least two ways to investigate the space-charge effects on the designing stage: analytical and numerical simulations. Each has its own advantages and disadvantages. The numerical simulation, with its use of macro particles approximation, is very well developed but it requires a lot of computer time and has limited flexibility in the variation of the initial conditions. As a result, the numerical methods are restricted in optimization possibility. Analytical investigation gives a clearer idea about physical phenomena, but it must simplify the physical model and sometimes gives an incorrect interpretation of results. Analytical research can be divided into two types: the envelope equation method for self-consistent distribution, and the nonlinear equation for higher distribution moments. In our work we use numerical simulation as the base to achieve the correct results, but for interpretation of results the simplified analytical model is used as well

Passing through half-integer resonance due to space charge under different initial distributions