Cylindrical gravitational impulse wave

The description of gravitational waves as explosion and implosion waves as predicted by Weber and Wheeler [{\it Rev. Mod. Phys. {\bf 29} 509 (1957)}] in Einstein and Rosen spacetime, has recently been confirmed following observations by the LIGO-VIRGO scientific team [{\it Phys. Rev. Lett. {\bf 116 } 061102 (2016)}] resulting from the collision of two massive black holes. In this dynamics, we explore a new possibility in the construction of gravitational waves like explosion and implosion waves, the special case of Jordan and Ehlers spacetime, by studying the exact solutions of the Einstein field equations. For this purpose, we use the inverse scattering method of Pomeransky in association with the method of Piran et al. [{\it Phys. Rev. D {\bf 32} 3101 (1985)}] by solving the Einstein field equations in combination with the specific metric derived from Jordan and Ehlers in order to obtain a two-soliton solution with complex conjugate poles that we assimilate to the gravitational wave. Consequently, under certain conditions, we obtain the Einstein and Rosen waves and the Chandrasekhar transcendental waves.