A direct method for the computation of polynomial conservation laws of polynomial systems of nonlinear partial differential equations (PDEs) in multi-dimensions is presented. The method avoids advanced differential-geometric tools. Instead, it is solely based on calculus, variational calculus, and linear algebra.Densities are constructed as linear combinations of scaling homogeneous terms with undetermined coefficients. The variational derivative (Euler operator) is used to compute the undetermined coefficients. The homotopy operator is used to compute the fluxes.The method is illustrated with nonlinear PDEs describing wave phenomena in fluid dynamics, plasma physics, and quantum physics. For PDEs with parameters, the method determines the conditions on the parameters so that a sequence of conserved densities might exist. The existence of a large number of conservation laws is a predictor for complete integrability. The method is algorithmic, applicable to a variety of PDEs, and can be implemented in computer algebra systems such as Mathematica, Maple, and REDUCE.

Using standard calculus, explicit formulas for one-, two-and three-dimensional homotopy operators are presented. A derivation of the one-dimensional homotopy operator is given. A similar methodology can be used to derive the multi-dimensional versions. The calculus-based formulas for the homotopy operators are easy to implement in computer algebra systems such as Mathematica, Maple, and REDUCE. Several examples illustrate the use, scope, and limitations of the homotopy operators. The homotopy operator can be applied to the symbolic computation of conservation laws of nonlinear partial differential equations (PDEs). Conservation laws provide insight into the physical and mathematical properties of the PDE. For instance, the existence of infinitely many conservation laws establishes the complete integrability of a nonlinear PDE.

A method for symbolically computing conservation laws of nonlinear partial differential equations (PDEs) in multiple space dimensions is presented in the language of variational calculus and linear algebra. The steps of the method are illustrated using the Zakharov-Kuznetsov and Kadomtsev-Petviashvili equations as examples.The method is algorithmic and has been implemented in Mathematica. The software package, ConservationLawsMD.m, can be used to symbolically compute and test conservation laws for polynomial PDEs that can be written as nonlinear evolution equations.The code ConservationLawsMD.m has been applied to multi-dimensional versions of the Sawada-Kotera, Camassa-Holm, Gardner, and Khokhlov-Zabolotskaya equations. theory. As Newell (1983) narrates, the study of conservation laws led to the discovery of the Miura transformation (which connects solutions of the KdV and modified KdV (mKdV) equations) and the Lax pair (Lax, 1968), i.e., a system of linear equations which are only compatible if the original nonlinear PDE holds. In turn, the Lax pair is the starting point for the IST (Ablowitz and Clarkson, 1991;Ablowitz and Segur, 1981) which has been used to construct soliton solutions, i.e., stable solutions that interact elastically upon collision.Conversely, the existence of many (independent) conserved densities is a predictor for complete integrability. The knowledge of conservation laws also aids the study of qualitative properties of PDEs, in particular, bi-Hamiltonian structures and recursion operators (Baldwin and Hereman, 2010). Furthermore, if constitutive properties have been added to close "a model," one should verify that conserved quantities have remained intact. Another application involves numerical solvers for PDEs (Sanz-Serna, 1982), where one checks if the first few (discretized) conserved densities are preserved after each time step.There are several methods for computing conservation laws as discussed by e.g.,

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