Local and nonlocal conserved vectors of the system of two-dimensional generalized inviscid Burgers equations

•First integrals of the system of 2D generalized inviscid Burgers equations are studied.•Noether׳s theorem and non-linearly self-adjoint method are used to construct conserved vectors.•The system of 2D Burgers equations is neither self-adjoint nor quasi self-adjoint.•Non-linearly self-adjoint condition is satisfied only through differential substitutions.•Obtained conserved vectors are nontrivial, independent and infinitely many.

The concept of non-linear self-adjointness for the construction of conservation laws has attracted a lot of interest in recent years. The most noteworthy aspect of it is the likelihood of explicitly constructing the conservation laws for any arbitrary systems of differential equations, in particular for those for which Noether׳s theorem is not applicable. In this study, we shall use both Noether׳s theorem and the non-linear self-adjoint method to construct local and nonlocal conserved vectors of the system of two-dimensional Burgers equations under consideration. The first integrals obtained not only give more credence to obtained results due to their generality with respect to any arbitrary functions of the velocity components but are also independent, nontrivial and infinitely many.