The formulation of classical field theory in terms of variational bicomplexes seems very interesting and promising. See the following references:
1. I. Anderson : The Variational Bicomplex. Very nice explanation about the basics of variational bicomplexes.
http://deferentialgeometry.org/papers/The%20Variational%20Bicomplex.pdf
2. G. Giachetta et al. : Advanced Classical Field Theory. Also includes classical BV formalism in terms of variational bicomplexes.
http://www.worldscientific.com/worldscibooks/10.1142/7189
3. L. Dickey: Soliton equations and hamiltonian systems. This is mainly about integrable hierarchies, but the last two chapters discuss the connections with variational bicomplexes.
http://www.worldscientific.com/worldscibooks/10.1142/1109
4. Also, Dubrovin - Zhang’s famous paper (book?) about higher genus Gromov - Witten invariants in terms of integrable hierarchies related to semisimple Frobenius manifolds also uses this notion.
https://arxiv.org/abs/math/0108160
Enjoy!
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