§\[sec::linear-theory\].

LINEAR THEORY {#sec::linear-theory}
=============

In a first order perturbation analysis, the second order terms of the amplitude functions are ignored. The evolution of modes is then governed by the linear parts of (\[eq:ODE-for-z-one\]) as $${\rm i} \textbf{\textit{M}}\cdot { d \over  d t}
\textbf{\textit{z}}(t) =
\textbf{\textit{C}}\cdot \textbf{\textit{z}}(t).
\label{eq:ODE-for-z-linear}$$ A general solution of (\[eq:ODE-for-z-linear\]) has the form $\textbf{\textit{z}}(t)=e^{-{\rm i}\omega t}\textbf{\textit{z}}_0$, which leads to the following linear eigenvalue problem $$\textbf{\textit{C}}(m)\cdot \textbf{\textit{z}}_0=
\omega \textbf{\textit{M}}(m)\cdot \textbf{\textit{z}}_0,
\label{eq:eigenvalue-problem-one}$$ for a prescribed azimuthal wavenumber $m$. Operating $\textbf{\textit{M}}^{-1}$ on (\[eq:eigenvalue-problem-one\]) yields $$\textbf{\textit{A}}(m)\cdot \textbf{\textit{z}}_0=
\omega \textbf{\textit{z}}_0,
\label{eq:eigenvalue-problem-two}$$ where $\textbf{\textit{A}}$ is a general non-symmetric matrix. A reduction to Hessenberg form followed by the QR algorithm [@Press01] gives all real and complex eigenvalues. Real eigenvalues correspond to van Kampen modes and complex eigenvalues, which occur in conjugate pairs, give growing/damping modes. I utilize the method of singular value decomposition for finding the eigenvectors and perform the decomposition $\textbf{\textit{A}}-\omega \textbf{\textit{I}}=\textbf{\textit{U}}\cdot
\textbf{\textit{S}}\cdot \textbf{\textit{V}}^T$ where $\textbf{\textit{I}}$ is the identity matrix and the diagonal matrix $\textbf{\textit{S}}$ is composed of the singular values $S_j$ ($j=1,2,\cdots,n_{\rm max}$). The column of $\textbf{\textit{V}}$ that corresponds to the smallest $S_j$ is the eigenvector associated with $\omega$.

Calculation of $\textbf{\textit{C}}$ and $\textbf{\textit{M}}$ involves evaluation of some definite integrals in the 