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We derive a new variational principle, leading to a new momentum map and anew multisymplectic formulation for a family of Euler--Poincar\'e equationsdefined on the Virasoro-Bott group, by using the inverse map (also called`back-to-labels' map). This family contains as special cases the well-knownKorteweg-de Vries, Camassa-Holm, and Hunter-Saxton soliton equations. In theconclusion section, we sketch opportunities for future work that would applythe new Clebsch momentum map with $2$-cocycles derived here to investigate anew type of interplay among nonlinearity, dispersion and noise.

New variational and multisymplectic formulations of the Euler-Poincar\'e equation on the Virasoro-Bott group using the inverse map