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We present the numerical results of the structure and dynamics of self-bound ground state arising solely because of presence of beyond mean field quantum fluctuation in spin-orbit coupled binary Bose-Einstein condensates in one dimension. Depending upon spin-obit and Rabi couplings, we obtain that the ground state exhibits either bright (plane) or stripe soliton nature. We find an analytical solution of soliton for non-zero SO coupling that matches quite well with the numerical results. Further, we investigate the dynamical stability of these solitons by adopting three protocols, such as adding initial velocity to each component, quenching of the SO and Rabi coupling parameters at initial and finite time, and allowing collision of the two spin-components from an equal and opposite direction velocity. Many interesting dynamical features of the solitons, like, mutifragamented, repelling, and breathing in time as well in both space and time, are observed. For given Rabi coupling frequency, the breathing frequency of the soliton increases upon increase in SO coupling and attains a maximum at the critical SO coupling parameters where phase transition from the bright to stripe soliton occurs. We find that the critical breathing frequency exhibits power law dependence on the Rabi coupling frequency with an exponent ∼ 0.16. The quenching of SO and Rabi coupling frequency leads to the appearance of dynamical phase transition from bright to stripe soliton, from breathing stripe to bright soliton, etc. In absence of the SO and Rabi couplings, depending upon the velocity of the up-spin and down-spin component the collision between them is either elastic or inelastic in nature which is consistent with the earlier numerical and experimental observation.

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