Nonlinear Matrix Differential Equations Research Articles

Impedance of a linear system with anisotropic coefficients

The concept of dual impedance on a vector space is introduced. First, a necessary condition is established for an impedance to exist under the linear mapping. Then, it is shown that a given matrix is an impedance if and only if it is a unique solution of a non-linear matrix differential equation. Finally, necessary and sufficient conditions are established for a lossless system. The last result is closely related to that obtained by Redheffer for scattering processes. An example, the non-uniform transmission-line problem is discussed. The general results can be applied to a multi-port system with anisotropic coefficients.

On the Polar Equations for Linear Systems and Related Nonlinear Matrix Differential Equations

On the Polar Equations for Linear Systems and Related Nonlinear Matrix Differential Equations

On the polar equations for linear systems and related nonlinear matrix differential equations

0. Introduction. This paper is devoted to laying the groundwork for a theory of matrix differential equations and associated linear systems. The fundamental problem is that of characterizing the solutions of general linear systems from the coefficient matrices themselves. Classically this requires a study of growth and decay properties, oscillation properties, and expansion properties; for the general problem the corresponding theories are far from complete. To develop convenient tools we present an elementary treatment of the transformation properties of matrix differential expressions in a form which directly reveals the connection to classical matrix theory. Applying this we derive polar equations for the general problem which provide a basis for separate studies of oscillation properties and growth-decay properties. There is an intrinsic connection between oscillation properties and expansion properties which will be detailed in a subsequent publication.

Generalization of the phase method to multi-channel potential scattering

The evaluation of theS-matrix in a multi-channel potential problem is reduced to the solution of a first-order nonlinear matrix differential equation. The two-channel problem is discussed in detail; firstorder differential equations for the eigenphase-shifts and the mixing parameter are obtained. Approximate expressions and bounds for the phaseshifts are also given.