Invariant Properties of Third-Order Non-hyperbolic Linear Partial Differential Operators
A test in terms of invariants for the existence of a factorization of a bivariate, non-hyperbolic third-order Linear Partial Differential Operator (LPDO) which has a given factorization of its principal symbol is found. The invariants that are used are with respect to known gauge transformations, which is together with constructive factorization itself are essentially involved in modern exact integration algorithms. The invariants, and even a generating system of those were found in previous paper using Moving Frames methods.
In order to find the expressions in terms of invariants that guarantee the existence of a factorization of a certain type, we show that the operation of taking the formal adjoint can be also defined in terms of invariants, that is for equivalence classes of LPDOs, and explicit formulae defining this operation in the space invariants are obtained. The operation of formal adjoint is highly interesting for factorization of LPDOs for if the initial operator has a factorization, its adjoint has also one, and they are related. (informally, the factorization types are symmetric).
KeywordsEquivalence Class Gauge Transformation Invariant Property Computer Algebra System Principal Symbol
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