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We give, for the first time, self-consistent large-$N$ analytical solutionsof inhomogeneous condensates in the quantum ${\mathbb C}P^{N-1}$ model in thelarge-$N$ limit. We find a map from a set of gap equations of the ${\mathbbC}P^{N-1}$ model to those of the Gross-Neveu (GN) model (or the gap equationand the Bogoliubov-de Gennes equation), which enables us to find theself-consistent solutions. We find that the Higgs field of the ${\mathbbC}P^{N-1}$ model is given as a zero mode of solutions of the GN model, andconsequently only topologically nontrivial solutions of the GN model yieldnontrivial solutions of the ${\mathbb C}P^{N-1}$ model. A stable single solitonis constructed from an anti-kink of the GN model and has a broken (Higgs) phaseinside its core,in which ${\mathbb C}P^{N-1}$ modes are localized,with asymmetric (confining) phase outside. We further find a stable periodic solitonlattice constructed from a real kink crystal in the GN model,while theAblowitz-Kaup-Newell-Segur hierarchy yields multiple solitons at arbitraryseparations.

Self-Consistent Large-$N$ Analytical Solutions of Inhomogneous Condensates in Quantum ${\mathbb C}P^{N-1}$ Model