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On the form of local conservation laws for some relativistic field theories in 1+1 dimensions

We investigate the possible form of local translation invariant conservation laws associated with the relativistic field equations $\partial\bar\partialϕ_i=-v_i(\bphi)$ for a multicomponent field $\bphi$. Under the assumptions that (i)~the $v_i$'s can be expressed as linear combinations of partial derivatives $\partial w_j/\partialϕ_k$ of a set of functions $w_j(\bphi)$, (ii)~the space of functions spanned by the $w_j$'s is closed under partial derivations, and (iii)~the fields $\bphi$ take values in a simply connected space, the local conservation laws can either be transformed to the form $\partial{\bar{\cal P}}=\bar\partial\sum_j w_j {\cal Q}_j$ (where $\bar{\cal P}$ and ${\cal Q}_j$ are homogeneous polynomials in the variables $\bar\partialϕ_i$, $\bar\partial^2ϕ_i$,\ldots), or to the parity transformed version of this expression $\partial\equiv(\partial_t+\partial_x)/ \sqrt{2}\rightleftharpoons\bar\partial \equiv (\partial_t-\partial_x)/\sqrt{2}$.