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Gravitational two solitons in Levi-Civita spacetime

Applying the Pomeransky inverse scattering method to the four-dimensional vacuum Einstein equations and using the Levi-Civita solution as a seed, we construct a two-soliton solution with cylindrical symmetry. In our previous work, we constructed the one-soliton solution with a real pole and showed that the singularities that the Levi-Civita background has on an axis can be removed by the choice of certain special parameters, but it still has unavoidable null singularities, as usual one-solitons do. In this work, we show that for the two-soliton solutions, any singularities can be removed by suitable parameter-setting and such solutions describe the propagation of gravitational wave packets. Moreover, in terms of the two-soliton solutions, we mention a time shift phenomenon, the coalescence and the split of solitons as the nonlinear effect of gravitational waves.