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We study quantum double dark-solitons by constructing corresponding quantumstates in the Lieb-Liniger model for the one-dimensional Bose gas. Here weexpect that the Gross-Pitaevskii (GP) equation should play a central role inthe long distance mean-field behavior of the 1D Bose gas. We first introducenovel quantum states of a single dark soliton with a nonzero winding number. Weshow them by exactly evaluating not only the density profile but also theprofiles of the square amplitude and phase of the matrix element of the fieldoperator between the $N$-particle and $(N-1)$-particle states. For ellipticdouble dark-solitons, the density and phase profiles of the correspondingstates almost perfectly agree with those of the classical solutions,respectively, in the weak coupling regime. We then show that the scheme of themean-field product state is quite effective for the quantum states of doubledark solitons. Assigning the ideal Gaussian weights to a sum of the excitedstates with two particle-hole excitations we obtain double dark-solitons ofdistinct narrow notches with different depths. We suggest that the mean-fieldproduct state should be well approximated by the ideal Gaussian weighted sum ofthe low excited states with a pair of particle-hole excitations. The results ofdouble dark-solitons should be fundamental and useful for constructing quantummultiple dark-solitons.

Quantum dark solitons in the 1D Bose gas: From single to double dark-solitons