Abstract: Jn this paper, the galactic spiral structure is studied by the galactic shock wave of interstellar gas with self-gravitation. The perturbed gravitation of stars is not a necessary condition for the existence of such shock. It is proved first of all that there exists solution of local shock wave even if the perturbed gravitation is absent. The condition |ω_η0|>σ is required for such solution. The spiral structure can only be explained by the shock solution when the difference of density between the regions of arm and interarm is larger. The grand design of shock wave with self-gravitation is obtained by the iterative method. The features of shock wave can be analyzed qualitatively in the velocity plane for a special perturbed gravitation which is used to simulate the self-gravitation . of interstellar gas. As the mass distribution in proto-galactic disk is irregular initially, the grand design of the galactic shock wave was developed by the processes of winding,growth of instability and overlapping of waves. Hence, it gives a complete figure about the origin, evolution and persistance.A lot of observed phenomena and classificational features of the galactic spiral structure can be explained by adopting these i-deas.

Key words: nonlinear differential equations, transformation, algorithm, analytical solution