Abstract
The paper studies linear differential operators in derivatives with respect to one variable. Such operators include, in particular, operators defined on infinite prolongations of evolutionary systems of differential equations with one spatial variable. In this case, differential operators in total derivatives with respect to the spatial variable are considered. In parallel, linear differential operators with one independent variable are investigated. The known algorithms for reducing the matrix to a stepwise or diagonal form are generalized to the operator matrices of both types. These generalizations are useful at points, where the functions, into which the matrix components are divided when applying the algorithm, are nonzero.In addition, the integral operator is defined as a multi-valued operator that is the right inverse of the total derivative. Linear operators that involve both the total derivatives and the integral operator are called integro-differential. An invertible operator in the integro-differential sense is an operator for which there exists a two-sided inverse integro-differential operator. A description of scalar differential operators that are invertible in this sense is obtained. An algorithm for checking the invertibility in the integro-differential sense of a differential operator and for constructing the inverse integro-differential operator is formulated.The results of the work can be used to solve linear equations for matrix differential operators arising in the theory of evolutionary systems with one spatial variable. Such operator equations arise when describing systems that are integrable by the inverse scattering method, when calculating recursion operators, higher symmetries, conservation laws and symplectic operators, and also when solving some other problems. The proposed method for solving operator equations is based on reducing the matrices defining the operator equation to a stepwise or diagonal form and solving the resulting scalar operator equations.
Highlights
Ëèíåéíûå äèôôåðåíöèàëüíûå îïåðàòîðûÀíàëîãè÷íîå ïðåäñòàâëåíèå ñóùåñòâóåò äëÿ ñå÷åíèé ðàññëîåíèÿ ζ, êîîðäèíàòû â ñëîÿõ êîòîðîãî áóäåì îáîçíà÷àòü ÷åðåç q1,
Äëÿ ðåøåíèÿ íåêîòîðûõ çàäà÷ òåîðèè äèôôåðåíöèàëüíûõ óðàâíåíèé íåîáõîäèìî ðåøàòü óðàâíåíèÿ íà ìàòðè÷íûå äèôôåðåíöèàëüíûå îïåðàòîðû
 îáùåì ñëó÷àå, îïåðàòîðû ðåêóðñèè ïðåäñòàâëÿþò ñîáîé ìàòðèöû, ïîëèíîìèàëüíûå îòíîñèòåëüíî ïîëíûõ ïðîèçâîäíûõ ïî ïðîñòðàíñòâåííûì ïåðåìåííûì è ïðàâî îáðàòíûì ê íèì îïåðàòîðîâ èíòåãðèðîâàíèÿ
Summary
Àíàëîãè÷íîå ïðåäñòàâëåíèå ñóùåñòâóåò äëÿ ñå÷åíèé ðàññëîåíèÿ ζ, êîîðäèíàòû â ñëîÿõ êîòîðîãî áóäåì îáîçíà÷àòü ÷åðåç q1, . Ord ∆ = r, åñëè ∆ | îïåðàòîð ïîðÿäêà íå âûøå r, íî íå ÿâëÿåòñÿ îïåðàòîðîì ïîðÿäêà íå âûøå r − 1. Ëèíåéíûé äèôôåðåíöèàëüíûé îïåðàòîð èç P = A â Q = A Äèôôåðåíöèàëüíûé îïåðàòîð ∆: P → Q íàçûâàþò (äâóñòîðîííå) îáðàòèìûì, åñëè ñóùåñòâóåò òàêîé äèôôåðåíöèàëüíûé îïåðàòîð ∆−1: Q → P, ÷òî êîìïîçèöèÿ ∆−1 ◦ ∆ åñòü òîæäåñòâåííîå îòîáðàæåíèå ìîäóëÿ P, à êîìïîçèöèÿ ∆ ◦ ∆−1 | òîæäåñòâåííîå îòîáðàæåíèå ìîäóëÿ Q.  ýòîì ñëó÷àå îïåðàòîð ∆−1 íàçûâàþò îáðàòíûì ê ∆.  êîîðäèíàòàõ ýëåìåíòû P, Q óäîáíî ïðåäñòàâëÿòü â âèäå ñòîëáöîâ ôóíêöèé, à ëèíåéíûé îïåðàòîð ∆: P → Q | â âèäå ìàòðèöû ñêàëÿðíûõ îïåðàòîðîâ. Íåòðóäíî äîêàçàòü, ÷òî îáðàòèìûé ëèíåéíûé äèôôåðåíöèàëüíûé îïåðàòîð çàäàåòñÿ êâàäðàòíîé ìàòðèöåé.
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
