Invariant Properties of Third-Order Non-hyperbolic Linear Partial Differential Operators
Invariant Properties of Third-Order Non-hyperbolic Linear Partial Differential Operators
复制标题
三阶非双曲线性偏微分算子的不变性质
DOI：
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发表时间：
2009
期刊：
影响因子：
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通讯作者：
E. Shemyakova
中科院分区：
文献类型：
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作者：
E. Shemyakova
本文给出了二元非双曲三阶线性偏微分算子（LPDO）的因子分解存在性的一个不变量检验，该算子的主符号为给定的因子分解，所用的不变量是关于已知的规范变换的，它与构造性因子分解本身在现代精确积分算法中是必不可少的.不变量，甚至生成系统的那些被发现在以前的文件中使用移动框架方法。
为了找到表达式中的不变量，保证存在某种类型的因式分解，我们表明，采取正式的伴随操作也可以定义在不变量，即等价类的LPDO，并明确公式定义此操作中的空间不变量。形式伴随运算对于LPDO的因式分解是非常有趣的，因为如果初始算子有因式分解，则其伴随也有因式分解，并且它们是相关的。（非正式地，因子分解类型是对称的）。
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).