In this article, we demonstrate that embedded vector solitons (EVSs) indeed exist. In other words, we prove that there exist systems of nonlinear partial differential equations (NLPDEs) which have soliton solutions ( u , v ) , where each of the solitons, u ( z , t ) and v ( z , t ) , is an embedded soliton . We present two systems of NLPDEs with these peculiar EVSs: a system of two cubic-quintic nonlinear Schrödinger (NLS) equations, and a system of two complex modified Korteweg-de Vries (cmKdV) equations. These two systems are the first systems of NLPDEs with EVSs known to date. Moreover, we demonstrate that these two systems also possess dark vector solitons and a highly unusual type of vector soliton: mixed vector solitons, which are composed of a bright soliton and a dark one.
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